Module 4: Quantum Mechanics PDF

MODULE 4: QUANTUM MECHANICS This module, primarily based on the University Physics textbook, introduces quantum mechanics with a strong emphasis on the wave-like behavior of particles. It focuses on simple mathematical equations to facilitate understanding. It does not delve into the deriva...

MODULE 4: QUANTUM MECHANICS This module, primarily based on the University Physics textbook, introduces quantum mechanics with a strong emphasis on the wave-like behavior of particles. It focuses on simple mathematical equations to facilitate understanding. It does not delve into the derivation of the Schrödinger equation but instead explores its physical interpretations and how it relates to Newton's laws in classical mechanics. The discussion begins with a free particle and progresses to particles influenced by forces in bound states, such as electrons in atoms. By solving the Schrödinger equation, we uncover possible energy levels and the probabilities of locating particles in various regions, including the intriguing phenomenon of particles passing through barriers—an idea not permitted in classical mechanics. We then extend to the understanding of three-dimensional scenarios, starting with a particle confined to a cubical box. The module then examines the hydrogen atom, demonstrating how its wave functions illuminate atomic structure and chemical behavior without requiring the quantization seen in the Bohr model. The key concepts include quantum numbers for labeling states, intrinsic electron spin, and the exclusion principle, which asserts that no two electrons can occupy the same quantum state. The module concludes by discussing characteristic x-ray spectra and introduces quantum entanglement, highlighting its significance in the emerging field of quantum computing. 4.1 WAVE FUNCTIONS AND THE ONE-DIMENSIONAL SCHRÖDINGER EQUATION At atomic and subatomic levels, electrons cannot be described as classical point particles; wave characteristics must be considered. The Bohr model of the hydrogen atom attempted to combine classical particle behavior with wave principles using the de Broglie relationship for allowed orbits. However, the Heisenberg uncertainty principle indicates that this hybrid description is insufficient. Quantum mechanics provides a purely wave-based description of particles, replacing classical descriptions based on coordinates and velocity. This new framework resembles classical wave motion, where wave functions describe motion. For example, in transverse waves on a string, the wave function 𝑦(𝑥, 𝑡) represents displacement over time, allowing derivation of velocity and acceleration. Similar approaches apply to sound waves and electromagnetic waves. The wave function becomes central to quantum mechanics, denoted by the Greek letter psi (Ψ). § Uppercase Ψ represents a function of all spatial coordinates and time. § Lowercase 𝜓 represents a function of spatial coordinates only. The wave function Ψ(𝑥, 𝑦, 𝑧, 𝑡) contains all information about a quantum particle, analogous to the wave function for mechanical waves. This is a transition from classical models, like the Bohr model of the hydrogen atom, to a purely wave-based description of particles. The wave function, denoted by the Greek letter psi (ψ), serves as the fundamental descriptor in quantum mechanics, capturing all relevant information about a particle. We will first focus on one-dimensional motion, where the wave function's properties resemble those of classical waves on a string, governed by a wave equation. This sets the stage for extending quantum mechanics to three-dimensional problems, such as analyzing the hydrogen atom and understanding more complex atomic structures and properties, which will be discussed later in this module. One-Dimensional Waves on a String A wave function represents a particle's motion along the x-axis, dependent on position (𝑥) and time (𝑡). Any wave function 𝑦(𝑥, 𝑡) must satisfy the wave equation: [4.1] Where 𝑣 is the peed of the wave, which is constant regardless of wavelength.  Eq. [4.1] is a partial second order differential equation (PDE), which means that it is second- order in both time and space. This indicates that the behavior of the string's displacement depends not only on its current state but also on its rate of change and the rate of change of that rate (acceleration). This complexity captures the dynamic nature of wave motion. ! ! "($,&) § The left side of the equation, !$ ! , represents the acceleration of points on the string. ( ! ! "($,&) § The right side, ) ! !& ! , relates to the curvature of the string at a given point. The curvature of the string influences how disturbances propagate through the medium, and is related to the amplitude of the wave. The relationship, however, is more complex than a direct proportionality, which means it is not linear. Larger amplitudes correspond to greater displacements and sharper bends in the string, leading to higher curvature. This can affect how energy is distributed along the string and how the wave propagates. The curvature also plays a role in determining how the wave reflects, refracts, or interacts with other waves. Regions of high curvature (peaks and troughs) can create more complex interference patterns when multiple waves overlap. An example of a wave function is a sinusoidal wave on a string. [4.2] *+ Where 𝑘 = is the wave number , 𝜔 = 2𝜋𝑓 is the angular frequency, and A and B are constants , defining amplitude and phase of the wave. The first and second derivatives are calculated to check if they satisfy the wave equation, leading to the relationship: 𝑣 = 𝜆𝑓 [4.3] Quantum Mechanical Wave Equation for Particle Waves In quantum mechanics, particles can be classified into two main categories based on their energy and confinement: free particles and bound states. Each type exhibits distinct properties and behaviors. A free particle is one that is not subject to any external forces and is not confined to a specific region in space, which means it can move freely in its environment. Bound states, on the other hand, refer to particles that are confined to a specific region in space due to the presence of a potential well or other constraining forces. Table 4.1. Comparison of Free Particles and Bound States Characteristics Free Particles Bound States Energy Levels Continuous: It can take any value Quantized: The energy levels of a within a certain range. They are not particle in a bound state are discrete restricted to discreet levels and can rather than continuous, leading to vary smoothly. specific allowed states. Wave Function Plane waves (delocalized): This Localized (standing waves): The wave means that they extend infinitely in function for a bound state is typically space without a defined position, localized and can take various forms, which indicates that the particle such as the solutions to the associated with the plane wave has Schrödinger equation for a potential an equal likelihood of being found well. anywhere along the wave's path. Position Uncertainty High: Free particles have well- Low (localized): They have wave defined momentum but an functions that decay outside the region uncertain position, as described by of confinement, indicating a higher the Heisenberg uncertainty probability of finding the particle within principle. that region. Potential Energy Usually zero or constant It varies (potential wells) Examples Electrons in vacuum Electrons in atoms, nucleons.  Free particles illustrate the principles of wave-particle duality in an unconfined state, while bound states illustrate how confinement leads to quantization and localized behavior. This framework is essential for analyzing atomic, molecular, and nuclear systems, providing insights into the nature of matter at microscopic scales. Quantum mechanics requires a different wave equation than the classical one for strings, due to different relationships between wave speed and wave number. For a free particle, the energy is purely kinetic: [4.4] The De Broglie relationships relate energy and momentum to wave properties: [4.5] So that for a free particle: [4.6] Equation [4.6] highlights a key difference in the behavior of particle waves compared to classical waves on a string: § for particle waves, the angular frequency 𝜔 is proportional to the square of the wave number 𝑘 § for string waves, 𝑣 is directly proportional to 𝑘. This necessitates the formulation of a quantum-mechanical wave equation that aligns with this relationship. Hence, a sinusoidal wave function 𝜓(𝑥, 𝑡) is assumed, similar to that of classical waves. In this context, the wave function describes a free particle with mass 𝑚, where momentum 𝑝 is related to 𝑘 and energy 𝐸 is connected to 𝑣, both moving in the positive x-direction, which leads to a new wave function; [4.7] The proposed form of the Quantum Mechanical Wave Equation is: [4.8] Equation [4.8] is a one-dimensional Schrödinger Equation (SE) for a free particle. This is developed by Erwin Schrödinger in 1926, which incorporates both real and imaginary units, reflecting the fundamental aspects of quantum mechanics. The imaginary unit 𝑖 in the Schrödinger equation is crucial for allowing the wave function to exhibit a complex behavior. This imaginary unit captures the oscillatory nature of quantum states, reflecting the dual wave-particle nature of matter, where particles display wave-like characteristics. The complex wave function enables the principle of superposition, allowing multiple states to coexist and interfere. This interference can result in observable phenomena such as diffraction patterns and resonance. Wave Function in Exponential Form When solving the Schrödinger equation, the solutions typically involve exponential functions with complex arguments, which lead to sinusoidal functions (sine and cosine) that are fundamental to wave mechanics. Although the wave function itself is complex, all physical observables must be real quantities. For instance, the probability density, which indicates the likelihood of locating a particle in a specific state, is a real number. The interplay between real and imaginary components encapsulates the oscillatory behavior and wave-like properties of particles, which are crucial for understanding unique quantum phenomena, including superposition and entanglement. Using Euler's formula, the exponential form of the wave function is: [4.9] This describes a free particle's wave moving in the direction of the wave number 𝑘. § positive 𝑘 indicates motion in the positive 𝑥 − 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛, with momentum 𝑝 = ℏ𝑘 and energy ℏ!. ! 𝐸 = ℏ𝜔 = */ § negative 𝑘 indicates that the momentum and hence its motion is in the negative 𝑥 − 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛. *+ For a negative value of k, the wavelength is 𝜆 = |.|. § Figure 4.1. Spatial wave function of a free particle of definite momentum. Characteristics of the Wave Function for a Free Particle 1. The wave function 𝜓(𝑥, 𝑡) for a free particle is complex, making its physical interpretation challenging. Unlike earlier physical phenomena, imaginary numbers are now involved. 2. It can be expressed as a plane wave as in Equation [4.9]. 3. The wave function must be normalized. This ensures the total probability of finding the particle is 100%. This serves as a challenge for plane waves, requiring the use of wave packets. [4.10] Wave packets are superpositions of multiple sinusoidal waves with varying wave numbers and amplitudes to create a more localized wave function compared to a simple wave, which extends over a range. By combining many waves, one can form a wave packet that has a single maximum, resembling both a particle (localized) and a wave (periodic structure). A wave packet can be expressed mathematically, showing the relationship between the wave function and the amplitude of the different wave numbers. The width of the amplitude function 𝐴(𝑘)affects the localization of the wave packet due to the uncertainty principle: § a narrow range of wave numbers leads to a broader wave packet § a broader range results in a more localized packet. Wave packets differ in matter waves and in light waves.  § Light waves maintain their shape while traveling since all wavelengths move at the same speed in a vacuum. § Matter waves vary in speed based on their wavelength, causing the wave packet's shape to change over time. This behavior illustrates the different properties of matter waves compared to light waves. Figure 4.2: Effect of the varying function 𝐴(𝑘) in the wave packet expression. 4. Probability Density: § ∣ 𝜓(𝑥, 𝑡) ∣* 𝑑𝑥 gives the probability of finding the particle between 𝑥 and 𝑥 + 𝑑𝑥. § ∣ 𝜓 ∣* is often referred to as the probability density or probability distribution function. 5. Probability can be interpreted as follows: § ∣ 𝜓(𝑥, 𝑡) ∣* 𝑑𝑥 describes the probability distribution of finding a particle in space, similar to how electric field magnitudes relate to photon distribution in electromagnetic waves. § The intensity of radiation in patterns is proportional to the square of the electric field, paralleling how ∣ 𝜓 ∣* indicates probability density for particles. 6. Heisenberg Uncertainty Principle: § For a particle with definite momentum 𝑝 = ℏ𝑘, the uncertainty in position 𝛥𝑥 must be infinite:  ℏ 𝛥𝑥𝛥𝑝 ≥ [4.11] 2 § This means we cannot pinpoint the particle’s location along the x-axis. 7. Calculation of Probability Distribution: § The probability density function ∣ 𝜓(𝑥, 𝑡) ∣* is calculated using the complex conjugate: [4.12] § This indicates equal likelihood of finding the particle anywhere along the x-axis, leading to an infinite integral and non-normalizability. 8. Definite Energy and Time Uncertainty: § The wave function describes a particle with definite energy 𝐸 and zero uncertainty in energy (𝛥𝐸 = 0), resulting in infinite time uncertainty (𝛥𝑡). 9. Limitations of the Wave Function: § A wave function of the form 𝜓(𝑥, 𝑡) = 𝐴𝑒 1(.$2)&) isn't a realistic model because it suggests no localization of the particle. § A more localized wave function can be constructed by superposing multiple sinusoidal functions. Figure 4.3. Superposing a large number of sinusoidal waves with different wave numbers and appropriate #$ amplitude creates a wave pulse with amplitude 𝜆!" = % and is localized within a region of space ∆𝑥. !" One Dimensional Schrödinger Equation and Stationary States The one-dimensional Schrödinger equation primarily addresses free particles with zero potential energy. However, in real scenarios like electrons in atoms, potential energy significantly influences particle behavior. To account for this, we use a generalized form of the Schrödinger equation that incorporates a non-zero potential energy function 𝑈(𝑥). [4.13] § The generalized equation relates kinetic and potential energy to total energy, similar to classical mechanics (𝐾 + 𝑈 = 𝐸). When 𝑈(𝑥) = 0, the equation simplifies to the free-particle version. The generalized equation relates kinetic and potential energy to total energy, analogous to classical mechanics. Predictions from this equation align with experimental results, validating its use. The concept of stationary states arises from wave functions representing definite energy levels. While sinusoidal wave functions work for free particles, they don’t satisfy the generalized equation in potential fields. Instead, stationary state wave functions can be expressed as the product of a time- independent function and a time-dependent exponential factor. This leads to a time-independent Schrödinger equation that focuses solely on the spatial component of the wave function, which can be used to solve for various physical scenarios to determine allowed energy levels and corresponding wave functions. Note that the term "stationary state" refers to the probability distribution being constant over time, not to the particle being motionless. [4.14] Table 4.2. Comparison between TDSE and TISE Characteristic Time-Dependent Schrödinger Time-Independent Schrödinger s Equation (TDSE) Equation (TISE) Equation Variables Depends on both position 𝑟 and time 𝑡 Depends only on position 𝑟 Describes time evolution of quantum states, while providing a Determines stationary states and Purpose comprehensive understanding of how energy levels states change over time. Systems with time-dependent Systems with time-independent Use Cases potentials or interactions potentials Time-independent wave functions Solution Type Time-dependent wave functions 𝛹(𝑟, 𝑡). 𝜓(𝑟). Energy Time-evolving state represented as Energy eigenvalues 𝐸 associated with Representatio superposition of eigenstates 𝜓(𝑟). n Example Quantum dynamics, scattering Bound states in potential wells, Applications problems harmonic oscillators  4.2. PARTICLE IN A BOX Using the time-independent Schrödinger equation we can describe a "particle in a box." We begin with the case of a free particle, where the energy can take any value, and then transitions to a confined particle between two rigid walls (𝑎𝑡 𝑥 = 0 𝑎𝑛𝑑 𝑥 = 𝐿). [4.15] Figure 4.4. Newtonian view of a particle in a box In this model, the potential energy is infinite outside the box, and zero within, meaning the wave function must be zero at the boundaries and continuous throughout the region. This leads to specific boundary conditions for the wave function: § Wave function 𝜓(𝑥) must be zero at boundaries 𝑥 = 0 𝑎𝑛𝑑 𝑥 = 𝐿. § Continuity of the wave function and its first derivative is required. Figure 4.5. Potential-energy function of a particle in a box To find the stationary-state wave functions, we recognize that the general solution involves a superposition of waves moving in opposite directions, similar to standing waves. By applying Euler's formula, the wave function can be expressed in terms of sine and cosine functions.  Figure 4.6. Normal nodes of vibration for a string with length L The boundary conditions lead to the requirement that 𝑘𝐿 = 𝑛𝜋 (where 𝑛 is a positive integer), resulting in quantized values of wave number 𝑘 and wavelength 𝜆. Each allowed wave function has nodes at the walls, with the length of the box corresponding to an integral number of half-wavelengths. Thus, the particle exhibits quantized energy levels and wave functions, reflective of similar principles seen in classical systems like vibrating strings. Energy Levels of a Particle in a box 3!. ! The energy levels for a particle in a box are quantized and given by the expression 𝐸 = */ = 3" ! 4 *+ 4 */ , 𝑤ℎ𝑒𝑟𝑒 𝑝 = ℏ𝑘 = *+ N , O = ,. From this, we get [4.16] This indicates that inside the box, where the potential energy is zero, all energy is kinetic. [4.17] Each energy level corresponds to a quantum number 𝑛, with associated values for momentum. The wave functions for the particle in a box are expressed: [4.18] This illustrates that they resemble standing waves similar to those on a vibrating string.  Figure 4.7. (a)Energy level diagram for a particle in a box, (b) wave functions of a particle in a box The energy levels are depicted in Figure 4.7, showing that they increase according to 𝑛* , leading to greater spacing between higher levels. Importantly, a particle in a box cannot have zero energy; if 𝐸 = 0 were allowed, it would imply 𝑛 = 0, resulting in a non-physical zero wave function. This restriction stems from the Heisenberg uncertainty principle, which indicates that a zero-energy state would require infinite position uncertainty. Consequently, the allowed stationary states maintain a non-zero momentum uncertainty, consistent with the finite position uncertainty intrinsic to the wave-like nature of particles. Probability and Normalization of Wave functions In one dimension, the probability of finding a particle within a small interval 𝑑𝑥 around a position 𝑥 is |𝜓(𝑥)|* 𝑑𝑥, which is: [4.19] Unlike in classical mechanics, where a particle is equally likely to be found anywhere in the box, the quantum mechanical probability distribution varies, leading to points where the probability is zero. This aligns with the uncertainty principle, indicating that exact position measurement isn't possible, and the particle is localized between 𝑥 = 0 𝑎𝑛𝑑 𝑥 = 𝐿.  Figure 4.8. Graph of |𝜓(𝑥)| 𝑎𝑛𝑑 |𝜓(𝑥)|#. The normalization condition requires that the total probability of finding the particle within the box equals 1, formulated as: [4.20] A wave function is considered normalized if it incorporates a constant C that ensures this total probability condition is met. For normalized wave functions, |𝜓(𝑥)|* 𝑑𝑥 directly represents the probability of finding the particle in the interval 𝑥 𝑎𝑛𝑑 𝑥 + 𝑑𝑥. Example: Electron in a box For an electron in a box of length 5.0 × 102(5 𝑚: o First energy level: 𝐸( = 1.5 𝑒𝑉. o Second energy level: 𝐸* = 6.0 𝑒𝑉. o Energy difference: 𝐸* − 𝐸( = 4.5 𝑒𝑉. For a proton in a much smaller box (𝑤𝑖𝑑𝑡ℎ 1.1 × 102(6 𝑚): o First energy level: 𝐸( =1.7 eV. o Second energy level: 𝐸* = 6.8 𝑒𝑉. o Energy difference: 𝐸* − 𝐸( = 5.1 𝑒𝑉. Thus, § The wave functions for stationary states are sinusoidal standing waves with nodes at each end. § Each state is characterized by a quantum number 𝑛 and has energy proportional to 𝑛*. § There are infinitely many stationary states possible within the box. For the stationary-state wave functions for a particle in a box, its normalized state is:  [4.21] 4.3. POTENTIAL WELLS AND BOUND STATES A potential well is defined as a potential-energy function 𝑈(𝑥)that has a minimum, allowing particles to exhibit periodic motion within it. In Newtonian mechanics, a particle can vibrate back and forth in such wells. The Schrödinger equation is used to describe these systems quantum mechanically. Figure 4.9. A square potential well Figure 4.9 is a square potential well characterized by a potential energy function that is constant (usually set to zero) within a specific interval and takes a higher constant value outside that interval. It illustrates how particles behave when confined to specific regions, showcasing both quantization and the probabilistic nature of quantum systems. Its implications extend across multiple domains in physics, providing insight into the behavior of confined systems. Specifically it serve as models in various fields: § Understanding fundamental quantum behavior in Quantum Mechanics. § Modeling electrons in materials and quantum dots, which is crucial in Solid State Physics. § Describing nucleons in atomic nuclei that gives rise to Nuclear Physics. A Finite Square Well is a model featuring a potential well that is zero within a certain range and has a finite height outside this range. It represents scenarios such as an electron in a metallic sheet. Within the well, the particle can move freely, while it must overcome a potential barrier to escape. Unlike the infinite square well, where particles are strictly confined and energy levels are infinite, the finite square well allows for a more realistic description of particles, as it acknowledges the possibility of tunneling and the existence of bound and unbound states. It demonstrates the interplay between confinement, energy levels, and quantum effects like tunneling, making it a fundamental concept across various fields of physics. § Quantum Dots: The finite square well model is useful in describing electrons in quantum dots, which are nanoscale semiconductor particles that confine electrons in three dimensions. § Nuclear Physics: It can also model the behavior of nucleons within an atomic nucleus, where strong forces create a potential well for protons and neutrons. § Solid State Physics: Understanding electronic properties in materials, particularly in semiconductors, where potential wells influence electron behavior. In quantum mechanics, a bound state occurs when a particle's total energy is less than the potential energy outside the well. The solutions to the Schrödinger equation yield wave functions that are sinusoidal within the well and exponential outside it, with the wave functions needing to match at the boundaries to maintain continuity. For a finite square well, unlike an infinite well, only a limited number of bound states exist. As the potential depth decreases, the energy levels also decrease compared to those in an infinitely deep well, and the particle has a non-zero probability of being found outside the well in classically forbidden regions. This also describes a particle that is confined within a potential well of finite depth. The energy levels in a finite well depend on the depth of the well relative to the ground level energy of an infinite well. If the well is very deep, the energy levels closely resemble those of an infinite well. [4.22] The principles of finite potential wells are applicable in quantum dots, where electrons behave similarly to particles in a finite potential well. When illuminated, these electrons can absorb energy and transition between levels, demonstrating quantum behavior. Figure 4.10: (a) wave functions for the three bound states for a particle in a finite potential well of depth 𝑈& = 6𝐸'()*+. (b) Energy-level for the system. In Figure 4.10 the horizontal brown lines for each wave function in (a) corresponds to 𝜓 = 0; while the vertical lines indicates the energy of each bound state. Meanwhile all energies greater than U0 are possible in (b). Note that states with 𝐸 > 𝑈5 form a continuum. Table 4.3. Comparison between finite potential well and infinite potential well Characteristics Finite Potential Well Infinite Potential Well A well with finite depth and finite A well with infinitely high walls, Definition width creating a strict confinement Wavefunctions may decay outside Wavefunctions are zero outside the Wavefunctions the well; continuity at boundaries well; sinusoidal inside Wave Function Does not vanish at boundaries; Vanishes at boundaries; shorter Behavior longer wavelengths wavelengths Potential inside the well is negative Potential Energy Potential outside the well is infinite; (or lower); outside is positive (or Profile potential inside is zero higher) Particle can exist outside the well Particle cannot exist outside the Particle Behavior (with some probability) well (strictly confined)  Discrete energy levels; all states are Discrete energy levels; some states bound within the well; higher energy Energy Levels may be unbound; lower energy levels with no bound state lower levels than zero Ground state energy influenced by Ground state energy fixed for given Energy Relationship depth 𝑈5 width Finite number of bound states, Number of Bound dependent on 𝑈5 𝑎𝑛𝑑 𝐸(2789 = + ! ℏ! Infinite number of bound states States (ground energy level of infinitely */:! deep well) Probability in Some probability of finding particle No probability of finding particle Forbidden Regions outside the well outside the well Form a continuum for energies Free-Particle States Discrete energy levels only greater than 𝑈5 Depends on the well depth and Simplified condition based on well Quantization width; more complex due to ;^ℏ! +! Condition dimensions (e.g., 𝐿): 𝐸; = tunneling effects */:! Tunneling effects are significant; can Tunneling No tunneling since walls are infinite penetrate barriers Particle in a box, basic quantum Examples Quantum dots mechanics models 4.4. POTENTIAL BARRIERS AND TUNNELING A potential barrier is a region in a potential-energy function characterized by a maximum. In classical mechanics, a particle cannot surpass this barrier if its total energy is lower than the barrier height, as this would result in negative kinetic energy, which is impossible. Conversely, in quantum mechanics, a particle may "tunnel" through a barrier even if its energy is less than the barrier height, allowing it to appear on the other side without losing energy. Figure 4.11: Possible wave function for a particle tunneling through the potential energy barrier. For a rectangular potential barrier with height 𝑈5 and width 𝐿, where potential energy is zero outside the barrier, the solutions to the Schrödinger equation reveal different behaviors. Outside the barrier, the wave function is sinusoidal, while inside it is exponential. The wave function must be continuous at the boundaries of the barrier, leading to a non-zero probability of finding the particle on the other side, which depends on the barrier's width and the particle's energy. The tunneling probability 𝑇 can be approximated as:  [4.23] where 𝐺 and 𝑘 are defined in terms of the particle's energy and the barrier height. This probability decreases rapidly with increasing barrier width and depends critically on the energy difference between the particle's energy and the barrier height. Tunneling has several significant practical applications: 1. Electrical Connections: When copper wires are twisted together or a switch is closed, electrons can tunnel through a thin layer of insulating copper oxide, allowing current to flow. 2. Tunnel Diodes: These semiconductor devices exploit tunneling, enabling rapid switching (within picoseconds) by varying the potential barrier height. 3. Josephson Junctions: Comprising two superconductors separated by a thin oxide layer, these junctions allow electron pairs to tunnel, resulting in unique circuit properties. They are essential for precise voltage standards, measuring small magnetic fields, and advancing quantum computing. 4. Scanning Tunneling Microscopes (STM): STMs utilize electron tunneling to image surfaces at the atomic level. A sharp conducting needle is positioned close to a surface, allowing electrons to tunnel when the needle is at a positive potential. The tunneling current, sensitive to the gap width, is used to map the surface as the needle scans. 5. Nuclear Physics: Tunneling is crucial for nuclear fusion, as it allows nuclei to overcome their electrical repulsion and fuse, a process that powers stars like the sun. Alpha particles also escape unstable nuclei via tunneling, which influences the radioactivity of materials. 4.5 THE HARMONIC OSCILLATOR The harmonic oscillator is a fundamental system that describes oscillatory motion, crucial in both classical and quantum physics. It models phenomena ranging from sound vibrations to molecular movements. In classical mechanics, a harmonic oscillator involves a particle moving under a restoring force proportional to its displacement, represented mathematically as 𝐹𝑥 = −𝑘′𝑥, where 𝑘′ is the force ( constant. The potential energy is given by 𝑈 = 𝑘 = 𝑥 * , leading to sinusoidal motion with specific * frequencies. In quantum mechanics, the Schrödinger equation is applied to the harmonic oscillator. The energy levels of the quantum harmonic oscillator are quantized and can be expressed as: [4.24] where 𝑛 is the quantum number, ℏ is the reduced Planck's constant, and ω is the angular frequency. This reveals that the energy levels are spaced by a constant amount, consistent with the assumptions made by Planck regarding electromagnetic radiation. Boundary conditions dictate that the wave functions must approach zero as the position x approaches infinity, leading to specific forms for the wave functions. Viable solutions exhibit behaviors that ensure they decay to zero at large displacements, with only certain energy levels satisfying these conditions. [4.25] The lowest energy state, or ground state, occurs at 𝑛 = 0, which is a half-integer multiple of the energy unit, reflecting the non-zero minimum energy of quantum systems. The quantum mechanical analysis of the harmonic oscillator begins with the time-independent Schrödinger equation, which describes the system's wave functions. Unlike systems with boundary walls, the harmonic oscillator has no physical boundaries, leading to unique boundary conditions. The following are some key points to remember. 1. Schrödinger Equation: The equation for the harmonic oscillator incorporates a potential term ( 𝑘𝑥 * with solutions representing the wave functions for the system's possible states. * 2. Boundary Conditions: As x approaches infinity, wave functions must approach zero, which requires careful consideration of their behavior in the classically forbidden regions. 3. Possible Behaviors: Analyzing the wave function 𝜓(𝑥), four behaviors are examined based on the sign of its second derivative: Figure 4.12: Possible behaviors of harmonic oscillators Figure 4.12 shows possible behaviors of harmonic oscillator wave function in the region ( 𝑘′𝑥 * > 𝐸 * > ! $($) o Curve a: The slope of 𝜓(𝑥) is positive at point x. Since > 0, the function curves >$ ! upward increasingly steeply and goes to infinity. This increase to infinity violates the boundary condition that 𝜓(𝑥) → 0 𝑎𝑠 |𝑥| → ∞. Hence, this is not a viable wave function. > ! $($) o Curve b: The slope of 𝜓(𝑥) is negative at point x, and has a large positive value. >$ ! Hence the slope changes rapidly from negative to positive and keeps on increasing – again this wave approaches infinity. This rapid increase from negative to positive also violates the boundary conditions. > ! $($) o Curve c: The slope of 𝜓(𝑥) is negative at point x. But >$ ! has a small positive value, so the slope increases only gradually as 𝜓(𝑥) decreases to zero and crosses over the > ! $($) negative values, which makes negative. The curve concaves downward and heads >$ ! to negative infinity, which also violates the boundary conditions. > ! $($) o Curve d: If the slope of 𝜓(𝑥) at point x is negative , and positive value of >$ ! at this point is neither too large nor too small, the curve bends just enough to glide asymptotically to >?($) > ! $($) the x-axis. In this case 𝜓(𝑥), , 𝑎𝑛𝑑 all approaches zero asymptotically, which >$ >$ ! is acceptable and leads to valid solutions. This however, occurs only for certain very special values of energy. The boundary conditions dictate that valid energy levels and occur at specific values 𝐸; , defined as: [4.26] where 𝑛 is the quantum number. This confirms that energy levels are spaced evenly, consistent with Planck’s assumptions. Figure 4.13: Energy levels of the harmonic oscillator Thus the quantum harmonic oscillator is a system that has infinitely many energy levels, which correspond to the amplitudes of classical oscillators. The analysis emphasizes the importance of boundary conditions in determining allowed states in quantum mechanics. Table 4.4. Comparing Quantum and Newtonian Oscillators Aspect Quantum Oscillator Newtonian Oscillator Hermite functions (exponential × Simple harmonic functions Wave Functions polynomial) (sine/cosine) 1 Ground State Energy 𝐸5 = 𝑈 𝐸 = 0 (particle at rest) 2 Wave functions normalized to Normalization Not applicable total probability = 1 Energy Levels Discrete levels (quantized) Continuous energy levels Wave Function Penetrates forbidden regions Confined to allowed regions Penetration Varies with quantum state; shows Uniform distribution based on Probability Distributions peaks and zeros speed Heisenberg Uncertainty Minimum uncertainty in position No such constraint Principle and momentum Approximation of Other Typically requires parabolic Can approximate near equilibrium Potentials shape for harmonic Photon Absorption Wavelength depends on energy Varies based on classical levels principles 4.6. MEASUREMENT IN QUANTUM MECHANICS Consider a particle confined to a one-dimensional box. In the classical picture, the particle's momentum can be precisely measured, and it can have any value within a certain range. However, in the quantum mechanical description, the particle's momentum is uncertain. When we measure the momentum, we force the particle into a specific momentum state. The following are some key concepts that should be noted. 1. Wave Function Collapse: The act of measurement causes the wave function to collapse into an eigenstate corresponding to the measured value. 2. Uncertainty Principle: The Heisenberg uncertainty principle states that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa. 3. Quantum Superposition: Before measurement, a quantum particle can exist in a superposition of multiple states.  4. Quantum Entanglement: Entangled particles can exhibit correlated behavior, even when separated by large distances. Concept Description Used to calculate wave functions and energy levels of quantum Schrödinger Equation particles; represents a probability distribution. Particle in a Box Model with boundaries (0 𝑡𝑜 𝐿); allows discrete energy levels 𝐸;. - Example: Hockey puck. Classical Measurement - Momentum is defined and measurable without altering its state. - Minor changes occur from photon interactions. - Wave function reflects superposition of momentum states (both positive and negative). Quantum Measurement - Measurement collapses wave function to a definite momentum state. - Upon measuring momentum, wave function collapses to either 𝑝$ = +ℏ𝑘; 𝑜𝑟 𝑝$ = −ℏ𝑘; - This collapse means it has many-worlds interpretation. This means Measurement Outcomes that each measurement creates parallel timelines, outcomes exist in separate branches of the universe. This phenomenon illustrates the fundamental unpredictability in quantum mechanics. - 50% probability for each outcome in repeated experiments. - Measurement changes the wave function. Wave Function Collapse - Measurements of energy do not alter the wave function. Quantum measurements can fundamentally alter the state of the system, showcasing the unpredictability of quantum mechanics Discussion Questions: 1. If quantum mechanics replaces the language of Newtonian mechanics, why don’t we have to use wave functions to describe the motion of macroscopic objects such as tennis balls and cars? 2. Why must the wave function of a particle be normalized? 3. In your Waves and optics class a standing wave was represented ass a superposition of two waves traveling in opposite directions. Can the wave functions for a particle in a box also be thought of as a combination of two traveling waves? Why or why not? What physical interpretation does this representation have? Explain. 4. For a particle in a finite potential well, is it correct to say that each bound state of definite energy is also a state of definite wavelength? Is it a state of definite momentum? Explain. 5. How do the quantized energy levels in the infinite square well arise, and what is their physical significance? 6. Describe Heisenberg's uncertainty principle and its significance in quantum mechanics. How does it challenge classical concepts of measurement? 7. How Does Quantum Mechanics Change Our Understanding of Reality? 8. In classical (Newtonian) mechanics, the total energy E of a particle can never be less than the potential energy U because the kinetic energy K cannot be negative. Yet in barrier tunneling a particle passes through regions where E is less than U. Is this a contradiction? Explain. 9. Qualitatively, how would you expect the probability for a particle to tunnel through a potential barrier to depend on the height of the barrier? Explain. 10. What Is the Significance of the Schrödinger Equation? Sample Problems 1. Find the first two energy levels for an electron confined to a one-dimensional box 5.0 x 10-10 m across (about the diameter of an atom).  2. An electron is trapped in a square well 0.50 nm across (roughly five times a typical atomic diameter). (a) Find the ground-level energy𝐸(2789 if the well is infinitely deep. (b) Find the energy levels if the actual well depth U0 is six times the ground-level energy found in part (a). (c) Find the wavelength of the photon emitted when the electron makes a transition from the n = 2 level to the n = 1 level. In what region of the electromagnetic spectrum does the photon wavelength lie? (d) If the electron is in the n = 1 (ground) level and absorbs a photon, what is the minimum photon energy that will free the electron from the well? In what region of the spectrum does the wavelength of this photon lie? 3. A 2.0 eV electron encounters a barrier 5.0 eV high. What is the probability that it will tunnel through the barrier if the barrier width is (a) 1.00 nm and (b) 0.50 nm? 4. A sodium atom of mass 3.82 𝑥 102*@ kg vibrates within a crystal. The potential energy increases by 0.0075 eV when the atom is displaced 0.014 nm from its equilibrium position. Treat the atom as a harmonic oscillator. (a) Find the angular frequency of the oscillations according to Newtonian mechanics. (b) Find the spacing (in electron volts) of adjacent vibrational energy levels according to quantum mechanics. (c) What is the wavelength of a photon emitted as the result of a transition from one level to the next lower level? In what region of the electromagnetic spectrum does this lie? 5. An electron is moving as a free particle in the -x-direction with momentum that has magnitude 4.50 𝑥 102*6 𝑘𝑔 𝑚/𝑠. What is the one-dimensional time-dependent wave function of the electron? 4.7. THE SCHRÖDINGER EQUATION IN THREE DIMENSIONS: UNDERSTANDING THE ATOMIC SPECTRA The three-dimensional Schrödinger equation extends the one-dimensional version to describe how quantum particles behave in three-dimensional space. In this context, the wave function represents the state of the particle and depends on time and three spatial coordinates 𝜓(𝑥, 𝑦, 𝑧, 𝑡) while the potential energy 𝑈(𝑥, 𝑦, 𝑧) varies with these coordinates as well. The equation accounts for the kinetic energy of the particle as it moves in three dimensions and includes a potential energy that varies based on the particle's position. The norm-squared value of the wave function, ∣ 𝜓(𝑥, 𝑡) ∣* , gives the probability of finding the particle in a specific volume of space, and this must be normalized so that the total probability of finding the particle somewhere is equal to one. A particle that can move in three dimensions has a momentum of three dimensions as well 𝑝$ , 𝑝" , 𝑎𝑛𝑑 𝑝A. Hence its kinetic energy is: [4.27] For states with a definite energy, the wave function can be split into a time-independent part that describes the spatial behavior of the particle and a time-dependent part. This leads to a simpler form of the equation that is useful for solving problems like those related to the hydrogen atom. [4.28] 4.8. PARTICLE IN A THREE-DIMENSIONAL BOX  Consider a particle confined in a three-dimensional cubical box of side 𝐿, such as an electron in a metal cube. Figure 4.15: A particle confined in a cubical box. Figure 4.15 shows a particle confined in a cubical box with walls at 𝑥 = 0, 𝑥 = 𝐿, 𝑦 = 0, 𝑦 = 𝐿, 𝑧 = 0, 𝑎𝑛𝑑 𝑧 = 𝐿. The potential energy is set to zero inside the box and infinite outside, leading to a spatial wave function that must be zero outside the box. The stationary states are described by the time- independent Schrödinger equation, simplified using the method of separation of variables. This involves expressing the wave function as a product of functions dependent on each spatial coordinate. [4.29] Boundary conditions require the wave function to equal zero at the box walls. By applying separation of variables, the problem reduces to solving three ordinary differential equations, each corresponding to one spatial dimension. The solutions resemble those for a one-dimensional box but include three quantum numbers (𝑛$ , 𝑛" , 𝑛A ) representing motion along each axis. The wave functions for the stationary states are derived, showing the probability distribution has nodes where finding the particle is impossible. As quantum numbers increase, more nodes appear, resembling standing waves in a cavity, with specific regions (or "dead spots") where the probability of finding the particle is zero. The allowed energy levels for a particle of mass 𝑚 in a cubical box of side 𝐿 are given by the equation: [4.30] where ℎ is Planck’s constant and the quantum numbers 𝑛$ , 𝑛" , 𝑛A can each take values of 1, 2, 3, and so on. Most energy levels correspond to multiple sets of quantum numbers, indicating degeneracy—where multiple quantum states have the same energy. For instance, states like (2,1,1), (1,1,2), 𝑎𝑛𝑑 (1,2,1) can have identical energy levels due to the symmetry of the cubical box. This symmetry leads to degeneracy, which contrasts with the one-dimensional case where each energy level corresponds to a unique state. Degeneracy refers to the phenomenon where two or more distinct quantum states share the same energy level. This means that multiple sets of quantum numbers can correspond to the same energy eigenvalue. When a system exhibits degeneracy, it indicates that there are multiple ways to achieve the same energy state, leading to important implications for physical properties such as entropy and statistical mechanics.  Figure 4.16: Six lowest energy level diagram for a three-dimensional cubical box. Since degeneracy is a consequence of symmetry, we can remove this by making the box asymmetric by having different lengths 𝐿$ , 𝐿" , 𝑎𝑛𝑑 𝐿A. In such cases, the energy levels are expressed as: [4.31] resulting in non-degenerate states. The differences between the three-dimensional case and the one- dimensional case are notable: § The three-dimensional wave functions are products of three functions, § The one-dimensional wave functions are single functions. Additionally, three quantum numbers are required for three-dimensional states, whereas only one is needed for one-dimensional states. In three dimensions, many energy levels are degenerate, and probability distributions can exhibit surfaces where the probability of finding the particle is zero, unlike the one-dimensional case where these are simply points. Table 4.5. Comparison of one-dimensional and three-dimensional quantum systems Feature One-Dimensional System Three-Dimensional System Wave Function Single function of xxx Product of three functions: 𝑋(𝑥)𝑌(𝑦)𝑍(𝑧) Quantum Numbers One quantum number (𝑛) Three quantum numbers 𝑛B , 𝑛C , 𝑛D No degeneracy; each Often degenerate; multiple states can Energy Levels energy level corresponds share the same energy level to a unique state 𝜓(0, 𝑦, 𝑧) = 𝜓(𝐿, 𝑦, 𝑧) Boundary Conditions 𝜓(0) = 𝜓(𝐿) = 0 = 0 𝑎𝑛𝑑 𝑠𝑖𝑚𝑖𝑙𝑎𝑟 𝑓𝑜𝑟 𝑦 𝑎𝑛𝑑 𝑧 Probability density function Probability density can be zero on Probability Distribution is zero at specific points surfaces or planes Symmetry Typically less symmetric More symmetric due to equal dimensions  4.9 THE HYDROGEN ATOM The hydrogen atom's behavior is better understood through quantum mechanics, particularly via the Schrödinger equation, which addresses the limitations of the Bohr model. In the Bohr model, electrons orbit in circular paths with quantized angular momentum, successfully predicting energy levels based on spectral data. However, it faces several conceptual issues: a. it combines classical and quantum ideas. b. It lacks a clear explanation for photon emission and absorption. c. It cannot be applied to multi-electron atoms. d. It predicts incorrect magnetic properties. e. It inaccurately represents the electron as a point particle. It represents the electron as a localized point particle, inconsistent with modern quantum views. To analyze the hydrogen atom more accurately, the Schrödinger equation is applied, replacing the electron mass with the reduced mass of the electron-nucleus system. The potential energy function is spherically symmetric and depends solely on the radial distance 𝑟. The solution involves separating variables in spherical coordinates, yielding three ordinary differential equations for the radial, polar, and azimuthal components of the wave function. The boundary conditions dictate that the radial function approaches zero at large distances, indicating localized bound states, while the angular functions must remain finite and periodic. The solutions include exponential functions and polynomials for 𝑅(𝑟), and sinusoidal terms for the angular components, leading to quantized energy levels given by 𝐸;. These energy levels match those predicted by the Bohr model when the electron mass is replaced with the reduced mass. [4.32] The Schrödinger analysis confirms the energy-level scheme of hydrogen, demonstrating its mathematical and conceptual advantages over the Bohr model while also providing deeper insights into the atom's behavior. Quantization of Orbital Angular Momentum The quantization of orbital angular momentum in quantum mechanics refers to the fact that only certain discrete values of the magnitude and components of orbital angular momentum are allowed. This is a direct result of solving the Schrödinger equation, unlike in the Bohr model where it lacked fundamental justification. The magnitude 𝐿 of the orbital angular momentum is quantized according to the orbital quantum number 𝑙, and for a given energy level [4.33]  Figure 4.17: (a) When 𝑙 = 2, the magnitude of the angular momentum vector 𝐿 66⃗ is √6ℏ = 2.5ℏ, but 𝐿 66⃗ does not 66⃗ have a definite direction. In this semiclassical vector picture, 𝐿 makes an angle of 35.3 with the z-axis when the z- component has its maximum value of 2ℏ. (b) These cones show the possible directions of 6𝐿6⃗ for different values of 𝐿, In the Bohr model, each energy level corresponded to a specific angular momentum value, but in quantum mechanics, each energy level allows for multiple possible angular momenta. The component of angular momentum along the z-axis, 𝐿A is also quantized and determined by the magnetic quantum number 𝑚( , which can range from −𝑙 𝑡𝑜 𝑙. However, 𝐿A is always smaller than or equal to 𝐿, and the precise direction of the angular momentum vector cannot be known due to the uncertainty principle. The z-axis is chosen arbitrarily for measurement, but the results hold for any direction. [4.34] For the hydrogen atom, the energy of the state is determined by the principal quantum number 𝑛, while the magnitude of the orbital angular momentum is determined by 𝑙, and its component along the z-axis is determined by 𝑚E ,. These states are degenerate, meaning multiple states can share the same energy, and the angular momentum quantum numbers are linked to the shapes and symmetry of the electron wave functions. The quantum numbers 𝑛, 𝑙, 𝑎𝑛𝑑 𝑚E correspond to specific spectroscopic notations and "shells" (such as the K, L, M shells) used in the description of atomic structure.  Table 4.6. Quantum States of the Hydrogen Atom As shown in the table, states are labeled based on their quantum numbers using spectroscopic notation (e.g., 1s, 2p, 3d), while electron shells are named as K, L, M, N, etc., corresponding to 𝑛 = 1,2,3,4, respectively. Electron Probability Distribution: The Schrödinger equation describes the electron in an atom not as a point particle moving in a precise orbit, but as a probability distribution surrounding the nucleus. This distribution is three- dimensional, making it more complex to visualize compared to the simpler circular orbits of the Bohr model. The following are some key concepts to note: 1. Radial Probability Distribution Function: The radial probability distribution 𝑃(𝑟) represents the probability of finding the electron at various distances from the nucleus. 2. Wave Function and Normalization: The wave function 𝜓 is normalized so that the total probability of finding the electron anywhere in space is 100%. For wave functions that depend on angles 𝜃 and 𝜙, the probability is averaged over all angles. 3. Probability Distributions for Hydrogen Atom: Graphs of 𝑃(𝑟) show the probability of finding the electron at different distances, with peaks at specific radii. For states with the highest possible angular momentum 𝑙 for a given 𝑛 (such as 1s, 2p, 3d, etc.), the electron is most likely found at a distance matching the Bohr model prediction. [4.35] [4.36] For hydrogen, graphs of P(𝑟) display peaks, indicating the most probable distances where the electron might be found. In states with the highest possible angular momentum for a given energy level (like 1s, 2p, or 3d), the electron is most likely found at distances corresponding to the Bohr model’s predictions. Additionally, in some cases, there are regions of zero probability (nodes) at certain radii, reflecting surfaces where the electron is not found.  Figure 4.118: Radial probability distribution function Electron Cloud Representation In three-dimensional visualizations, such as electron cloud representations, darker regions indicate higher probability densities. For s-states, the probability distribution depends only on the radial distance and is spherically symmetric. However, for other states, the distribution also depends on angular coordinates, with regions of zero probability occurring at specific angles and distances. These visualizations provide a more complete picture of the electron’s likely positions around the nucleus. § For s-states, the probability distribution depends only on the radial distance 𝑟 and is spherically symmetric. § For other states, the probability distribution depends on both 𝑟 and 𝑢, with zero probability at certain angles 𝑢 and radii 𝑟. Figure 4.19: Three-dimensional probability distribution functions |𝜓|# for the spherically symmetric 1𝑠, 2𝑠, 𝑎𝑛𝑑 3𝑠 hydrogen-atom wave functions.  Figure 4.20: Cross sections of three-dimensional probability distributions for a few quantum states of the hydrogen atom. They are not to the same scale. Mentally rotate each drawing about the z-axis to obtain the three- dimensional representation of |𝜓|#. For example, the 2𝑝, 𝑚' = ±1 probability distribution looks like a fuzzy donut. 4.9 THE ZEEMAN EFFECT The Zeeman effect refers to the splitting of atomic energy levels and their corresponding spectral lines when atoms are placed in a magnetic field. This phenomenon, discovered by Dutch physicist Pieter Zeeman in 1896, provided experimental evidence for the quantization of angular momentum. It occurs because magnetic forces affect the motion and energy levels of charged particles within the atom, such as orbiting electrons. The magnetic moment of an orbiting electron interacts with the external magnetic field, causing changes in energy levels based on the orientation of the magnetic moment, which is related to the quantum number ml. This quantum number determines the number of distinct energy states that an electron can occupy in the presence of a magnetic field, splitting previously degenerate energy levels into multiple closely spaced lines. The analysis of the Zeeman effect can be done by reviewing the concept of magnetic dipole moment. The magnetic moment of an orbiting electron arises from its motion around the nucleus, which can be modeled as a current loop. The magnetic moment (𝜇⃗ = 𝐼𝐴⃗) is proportional to the angular momentum (L) of the electron, [4.37] [4.38] F G with the ratio ( : = */) known as the gyromagnetic ratio. This leads to the concept of the Bohr magneton𝜇H , a unit of magnetic moment. [4.39] The interaction energy for the Bohr magneton can be expressed as:  [4.40] In a magnetic field, the magnetic moment interacts with the field, altering the energy of the electron's orbit depending on the orbital magnetic quantum number (ml). This interaction causes energy level splitting, known as the Zeeman effect, which removes the degeneracy of electron energy states in the absence of a magnetic field. Spectral lines associated with electron transitions between energy levels are split into closely spaced lines, although the effect is small even for significant magnetic fields. Figure 4.21: An energy-level diagram for hydrogen showing how the levels are split when the electron’s orbit al magnetic moment interact with the external magnetic field. The Schrödinger model refines the Bohr model, particularly for states with zero angular momentum (e.g., the s-state), where the orbital magnetic moment is zero. However, the gyromagnetic ratio remains the same in both models. In the absence of a magnetic field, d states with quantum numbers 𝑙 = 2 and magnetic quantum numbers 𝑚( = −2, −1,0,1,2 are degenerate, meaning they have the same energy. When a magnetic field is applied, these states split, with adjacent levels separated by equal energy differences given by 𝛥𝐸 = 𝜇H 𝐵 where 𝜇H is the Bohr magneton and 𝐵 is the strength of the magnetic field. Without a magnetic field, a transition from a 3d to a 2p state would result in a single spectral line corresponding to the photon energy 𝐸1 − 𝐸I. However, with the levels split due to the magnetic field, only three possible photon energies are observed. This limitation arises from selection rules, which require that the angular momentum quantum number 𝑙 changes by 1 and the magnetic quantum number m1 changes by 0 or ±1. These selection rules restrict the number of allowed transitions, producing only three possible photon energies despite the five possible initial and final energy levels. This process is known as the normal Zeeman effect, which accounts for the orbital angular momentum of the electron but does not yet include the effect of electron spin. 4.10 THE ELECTRON SPIN  The Schrödinger equation, while successful in predicting hydrogen atom energy levels, fails to account for certain experimental observations regarding electron behavior. One key discrepancy is the discovery of additional spectral line splitting in a magnetic field, beyond the three equally spaced lines explained by the normal Zeeman effect. This phenomenon was initially called the anomalous Zeeman effect. Additionally, some energy levels exhibit splitting even without an external magnetic field. For example, high-resolution studies of the hydrogen spectrum reveal closely spaced lines, known as multiplets, and sodium's orange-yellow spectral line appears as a doublet. These observations suggest that the Schrödinger equation, in its original form, doesn't fully explain atomic behavior, hinting at the influence of electron spin. Figure 4.22. Illustration of the normal and the anomalous Zeeman effect. The Stern-Gerlach Experiment The Stern–Gerlach experiment in 1922, conducted by Otto Stern and Walter Gerlach, was a groundbreaking experiment that provided crucial evidence for the quantization of angular momentum. It involved sending a beam of silver atoms through an inhomogeneous magnetic field and observing their deflection. This revealed anomalies in atomic behavior when a beam of neutral atoms passed through a nonuniform magnetic field. Atoms were deflected based on their magnetic moment orientation, demonstrating the quantization of angular momentum. The surprising finding was that some atomic beams split into an even number of components, suggesting half-integer angular momentum values, something unexplained by the Bohr model. Classical Prediction: According to classical physics, the beam of silver atoms should have spread out continuously, forming a single band on the detector screen. This is because classical theory predicts a continuous range of orientations for the magnetic moment of the atoms. Quantum Mechanical Reality: The experiment revealed a surprising result: the beam split into two distinct bands. This indicated that the magnetic moment of the silver atoms could only have two discrete orientations, corresponding to two possible values of angular momentum.  Figure 4.23: The Stern-Gerlach Experiment Significance of the experiment 1. It provided strong evidence for the quantization of angular momentum, a fundamental concept in quantum mechanics. 2. It demonstrated that the properties of particles at the atomic scale cannot be explained by classical physics and require a quantum mechanical description. 3. It also introduced the concept of spin, an intrinsic form of angular momentum that is not associated with orbital motion. In 1925, Samuel Goudsmit and George Uhlenbeck proposed the concept of electron spin. They hypothesized that electrons, much like spinning spheres of charge, possess additional spin angular momentum and magnetic moment. This spin, separate from orbital angular momentum, could explain the observed energy-level anomalies, such as additional Zeeman shifts in the presence of a magnetic field. An analogy compares electron spin to the Earth's movements: the Earth orbits the sun (orbital angular momentum) and rotates on its axis (spin angular momentum). Similarly, an electron not only has orbital angular momentum from its motion around the nucleus but also intrinsic spin angular momentum from its own rotation. Although electron spin is fundamentally quantum-mechanical and not fully explainable by classical models like the Bohr model, the spinning-sphere analogy helps conceptualize this intrinsic property of electrons. Electron spin, which is intrinsic and independent of orbital motion, plays a crucial role in understanding atomic behavior, as demonstrated by various spectroscopic observations and experiments. Spin Quantum Numbers The spin quantum number describes the quantized spin angular momentum (𝑆⃗) of an electron. This was the result of the Stern-Gerlach experiment, which revealed that electrons possess an intrinsic ( ( property called spin. This spin is quantized, meaning it can only have two values: + * 𝑜𝑟 − *. This is represented by the spin magnetic quantum number, 𝑚J. ( The z-component of this spin, denoted as Sz, can have only two values: + ℏ 𝑜𝑟 − 1/2ℏ, 𝑤ℎ𝑒𝑟𝑒 ℏ is * the reduced Planck’s constant. These correspond to “spin up” and “spin down” orientations. The following are some important concepts in spin quantum numbers: 1. Spin Angular Momentum: Electrons have a spin angular momentum, S, which is quantized. Its ( magnitude is given by 𝑆 = q𝑠(𝑠 + 1) 𝑤ℎ𝑒𝑟𝑒 𝑠 =. The electron is referred to as "𝑠𝑝𝑖𝑛 − * ( * 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒".  [4.41] ( ( 2. Spin Magnetic Quantum Number: This quantum number, ms can only be +* 𝑜𝑟 − *. ( 3. Spin Orientation: Electrons can be in two spin states: "spin up” (𝑚J = + * ) or "spin down" ( (𝑚J = − *). 4. Spin Magnetic Moment: The spin of an electron gives rise to a spin magnetic moment, which interacts with external magnetic fields. This is proportional to spin angular momentum. The Z- G component is given by 𝑚A = −2.002322. N*/O. 𝑆A. [4.42] 5. Gyromagnetic Ratio: The ratio for electron spin is approximately twice that of orbital angular G momentum. Paul Dirac’s equation predicts this ratio as exactly 2.N*/O). 6. Quantum Electrodynamics (QED): This theory provides a highly accurate prediction of the electron's spin magnetic moment. QED predicts a gyromagnetic ratio value of 2.00231930436182(52)), confirming Dirac's result. To fully describe an electron's state in a hydrogen atom, four quantum numbers are needed 1. Principal quantum number (𝑛): Determines the energy level. 2. Orbital angular momentum quantum number (𝑙): Determines the shape of the orbital. 3. Magnetic quantum number (𝑚E ): Determines the orientation of the orbital. 4. Spin magnetic quantum number (𝑚J ) Determines the spin orientation of the electron. Spin-Orbit Coupling Spin-orbit coupling is a phenomenon in quantum mechanics where the interaction between an electron's spin (spin angular momentum) and its orbital motion (orbital angular momentum) causes a splitting of energy levels. This effect arises because, from the electron's perspective, the nucleus appears to orbit around it. This apparent motion of charge creates a magnetic field that interacts with the electron's spin magnetic moment. Cause of Spin-Orbit Coupling: 1. The orbital motion of an electron generates a magnetic field in the electron’s frame of reference. 2. This magnetic field interacts with the electron's spin magnetic moment. 3. The interaction causes energy level splitting, even in the absence of an external magnetic field, because of the two possible orientations of the electron’s spin (spin-up or spin-down). 4. The energy level splitting leads to closely spaced spectral lines. Spin-orbit coupling is a phenomenon that can be understood more rigorously using the Schrödinger equation. The interaction energy is expressed through the scalar product of the electron’s orbital angular momentum (𝐿) and spin angular momentum (𝑆), leading to spin-orbit coupling. Effect:  1. Spin-orbit coupling leads to slight differences in energy levels, causing what is known as fine structure in atomic spectra. 2. An example of this can be observed in sodium atoms, where spin-orbit coupling causes a small energy difference between two closely spaced excited levels. This results in a doublet in sodium’s emission spectrum, with lines at 589.0 nm and 589.6 nm (known as the sodium D-line doublet). Example: A common example of spin-orbit coupling is the sodium doublet. In the sodium atom, the two closely spaced, lowest excited energy levels are split due to spin-orbit coupling. This splitting results in two closely spaced spectral lines at 589.0 nm and 589.6 nm. Spin-orbit coupling is a subtle but important effect in atomic physics, influencing the energy levels and spectra of atoms. The total angular momentum Orbital and spin Angular momenta can be combined in different ways. One involves the addition of the electron's orbital angular momentum (𝐿v⃗) and spin angular momentum (𝑆⃗) to form the total angular momentum (𝐽⃗): [4.43] The magnitude of (𝐽): is expressed by the total angular momentum quantum number 𝑗, with possible values given by: [4.44] ( ( The quantum number 𝑗 can take 𝑣𝑎𝑙𝑢𝑒𝑠 𝑗 = 𝑙 + * 𝑗 𝑜𝑟 𝑗 = 𝑙 − * , where 𝑙 is the orbital quantum number ( and 𝑠 = *is the spin quantum number. For a given 𝑙, there are two possible sates for 𝑗: ( § Parallel configuration: 𝑗 = 𝑙 + = * ( § Antiparallel configuration: 𝑗 = 𝑙 − * ( K o For example, 𝑖𝑓 𝑙 = 1, 𝑗 𝑐𝑎𝑛 𝑏𝑒 𝑜𝑟. * * Spectroscopic notation uses symbols like ⬚*𝑃# 𝑎𝑛𝑑 ⬚*𝑃$ : ! ! § The superscript indicates the number of possible spin orientations. § The letter (𝑃) corresponds to the orbital angular momentum (𝑙 = 1). § The subscript is the total angular momentum quantum number (𝑗). The combination of spin-orbit coupling, magnetic effects, and relativistic corrections causes fine structure, which leads to small shifts in energy levels. Fine structure removes the degeneracy of energy levels, meaning states with the same principal quantum number 𝑛 but different 𝑗 values have slightly different energies. [4.45] The fine-structure constant 𝛼 is a fundamental dimensionless constant that characterizes the ( strength of these relativistic and quantum effects: 𝛼 ≈ (KM.56. Additional hyperfine structure occurs due to the interaction between the nucleus’s magnetic dipole moment and the electron's magnetic moments, leading to much smaller energy splittings. Example: a. In the hydrogen atom, when 𝑛 = 2 𝑎𝑛𝑑 𝑙 = 1: ( K The total angular momentum quantum number 𝑗 can be 𝑗 = 𝑜𝑟 𝑗 =. * * These two 𝑗-states lead to slightly different energy levels, creating a fine structure in the hydrogen spectrum. This splitting results in two spectral lines instead of one for transitions involving the 𝑛 = 2 energy level. b. Hyperfine Structure in Hydrogen: In the ground state of hydrogen, a hyperfine splitting occurs, separating two energy levels by only 5.9 × 10 − 6 𝑒𝑉. The photon emitted in the transition between these states has a wavelength of 21 cm, used by radio astronomers to map cold interstellar hydrogen gas. Figure 4.24: Images comparing galaxies as seen in visible light and in radio wavelengths In Figure 4.24, (a) shows a visible-light image where three distant galaxies appear unconnected. However, they are actually linked by immense streams of hydrogen gas, which are invisible in the optical spectrum. This hidden connection is revealed in (b), a false-color image captured using a radio telescope tuned to the 21 cm wavelength emitted by hydrogen atoms, highlighting the intergalactic gas that bridges these galaxies. 4.11 THE MANY-ELECTRON ATOMS AND THE EXCLUSION PRINCIPLE The analysis of atomic structure typically begins with hydrogen, the simplest atom with just one electron. However, in many-electron atoms (where an atom has Z electrons and Z protons), the complexity increases significantly. Each electron interacts not only with the nucleus but also with every other electron, making the Schrödinger equation for such atoms extremely complex, especially as Z increases. Exact solutions for these multi-electron systems are not feasible, even for simple atoms like helium (which has two electrons). However, approximation methods exist. The simplest approximation treats each electron as if it only interacts with the nucleus, ignoring electron-electron interactions. In this model, each electron has a wave function similar to that of a hydrogen atom, but the nuclear charge is adjusted to 𝑍G , where Z is the atomic number. The energy levels are modified accordingly, but this method provides only a rough approximation, as electron-electron interactions are critical in many- electron atoms. Thus, this model has limited quantitative accuracy. [4.46] The Central-Field Approximation  The Central-Field Approximation is a more refined and useful model for understanding the structure of many-electron atoms. In this approach, all the electrons together are considered to form a spherical charge cloud, which creates an averaged, spherically symmetric electric field around the nucleus. Each individual electron is then seen as moving within this averaged electric field, resulting in a corresponding spherically symmetric potential energy function, 𝑈(𝑟). This model allows us to use one-electron wave functions, similar to the hydrogen atom. However, ( instead of the typical O potential seen in hydrogen, a different function 𝑈(𝑟).is used to represent the averaged-out potential. The angular parts of the wave functions, and the quantum numbers 𝑙, 𝑚E , 𝑎𝑛𝑑 𝑚J , remain the same as in the hydrogen atom, meaning the orbital angular momentum states are unchanged. However, the radial wave functions and the energy levels differ because of the new potential function, so the energy now depends on both the principal quantum number (𝒏) and the orbital quantum number (𝒍), unlike hydrogen, where it depends only on 𝑛. The restrictions on the values of the quantum numbers is expressed as: [4.47] For example: In the hydrogen atom, energy levels are determined solely by the principal quantum number 𝑛, regardless of 𝑙. In many-electron atoms, such as helium or carbon, energy levels differ based on both 𝑛 and 𝑙, due to electron-electron interactions and the central-field approximation. Thus, a 2𝑝 electron will have slightly different energy than a 2𝑠 electron, even though they share the same principal quantum number 𝑛 = 2. This approximation serves as a good starting point for understanding atomic structure, though finer effects, like those involving total angular momentum 𝑗, can further refine energy level predictions. The Exclusion Principle The Exclusion Principle, introduced by Wolfgang Pauli in 1925, is crucial for understanding the structure of many-electron atoms. It states that no two electrons in an atom can have the same set of four quantum numbers (𝒏, 𝒍, 𝒎𝒍 , 𝑎𝑛𝑑 𝒎𝒔. In other words, no two electrons can occupy the same quantum-mechanical state. Without the exclusion principle, all electrons in an atom would be expected to occupy the lowest energy state, similar to how the electron in a hydrogen atom resides in its ground state. However, in reality, electrons are distributed across different energy levels and subshells. The exclusion principle explains why elements have distinct chemical properties, despite having a similar number of electrons. For example: Fluorine (Z = 9) tends to gain an electron, making it highly reactive. Neon (Z = 10) is a noble gas, with a completely filled outer shell, and does not react under normal conditions. Sodium (Z = 11) tends to lose an electron, making it reactive like other alkali metals. These differences in behavior are due to the unique electron configurations governed by the exclusion principle. For instance, in fluorine, the electron configuration leaves space for one more electron, making it eager to gain one. Sodium, on the other hand, prefers to lose one electron to achieve a more stable configuration like neon. In practice, this principle works as follow in an atom:  The lowest energy level, n = 1 (the K shell), can hold only two electrons, because only two unique combinations of quantum numbers are possible. Once this shell is filled, additional electrons are forced into higher energy levels (e.g., n = 2, the L shell), leading to different chemical behaviors across elements. The exclusion principle also applies to all spin-½ particles, including protons and neutrons, which helps explain the structure of atomic nuclei. For example, this can be thought of as a classroom rule where each chair can only seat one student. Even if more students want to sit down, they are forced to find other chairs. Similarly, in an atom, only two electrons (with opposite spins) can occupy the same state, forcing other electrons into higher-energy states. This principle is distinct from electrical repulsion. While electrons repel each other due to like charges, the exclusion principle is a quantum-mechanical rule that cannot be overcome, even with added energy. Hence, for this principle: a. No two electrons can share the same quantum state. b. This principle shapes the unique electron configurations and distinct chemical properties of elements. c. The exclusion principle is essential not only for electrons but also for other particles like protons and neutrons. Table 4.7 lists some of the sets of quantum numbers for electron states in an atom. Table 4.7. Quantum States of Electrons in the First Four Shells 4.12. PERIODIC TABLE AND ELECTRON CONFIGURATIONS The structure and chemical behavior of multielectron atoms are primarily determined by the arrangement of electrons, particularly in their outermost shells. The ground-state electron configurations of atoms can be understood using the exclusion principle, where electrons fill the lowest- energy states first. This is a powerful tool that organizes elements based on their properties. The underlying principle behind this organization is the arrangement of electrons in atoms, governed by the Pauli Exclusion Principle. 1. Electron Configuration Basics: The arrangement of electrons in an atom's orbitals determines its properties.  §Electrons are added to an atom one at a time, filling the lowest energy levels first (smallest values of 𝑛 and 𝑙). § The outermost electrons (valence electrons) are crucial for determining chemical properties. § The following are the principles governing this. i. Aufbau Principle: Electrons fill orbitals starting from the lowest energy level and moving to higher ones. ii. Pauli Exclusion Principle: No two electrons can have the same set of quantum numbers. iii. Hund's Rule: Orbitals within a subshell are filled with single electrons before pairing. 2. Examples of Electron Configurations: o Hydrogen (Z=1): Configuration is 1𝑠(. It has one electron in the lowest energy state. o Helium (Z=2): Configuration is 1𝑠 *. Both electrons occupy the 1𝑠 state with opposite spins, resulting in a filled shell, making it stable (noble gas). o Lithium (Z=3): Configuration is 1𝑠 * 2𝑠(. The outer 2𝑠 electron is less tightly bound, making lithium reactive (alkali metal). o Beryllium (Z=4): Configuration is 1𝑠 * 2𝑠 *. It has two valence electrons, leading to a valence of +2 (alkaline earth metal). o Neon (Z=10): Configuration is 1𝑠 * 2𝑠 * 2𝑝@. All shells are filled, making it very stable (noble gas). o Sodium (Z=11): Configuration is 1𝑠 * 2𝑠 * 2𝑝@ 3𝑠(. Similar to lithium but with one electron in the next higher shell, also reactive (alkali metal). o Fluorine (Z=9): Configuration is 1𝑠 * 2𝑠 * 2𝑝R. With one electron short of a filled outer shell, it is very reactive and typically forms compounds with a valence of -1 (halogen). Figure 4.24: Schematic charge representation in a lithium atom 3. Periodic Trends: Periods: Horizontal rows in the periodic table. Elements in a period have the same number of electron shells.  Groups: Vertical columns in the periodic table. Elements in a group have similar electron configurations in their outermost shell, leading to similar chemical properties. o Elements in the same column (group) of the periodic table share similar outer electron configurations, leading to similar chemical properties. o Noble Gases (Group 18): Have completely filled outer shells (e.g., helium, neon). They are generally unreactive. o Alkali Metals (Group 1): Have a configuration of noble gas plus one electron (e.g., lithium, sodium). Have one valence electron in an s-orbital. They are highly reactive metals. o Alkaline Earth Metals (Group 2): Have a configuration of noble gas plus two electrons (e.g., beryllium, magnesium). Have two valence electrons in an s-orbital. They are also reactive metals, but less so than alkali metals. o Halogens(Group 17): Have a configuration of noble gas minus one electron (e.g., fluorine, chlorine). Have seven valence electrons. They are highly reactive nonmetals. 4. Shell Filling Complications: The 3d and 4s subshells have similar energy levels, affecting the order in which they are filled. For example, potassium (𝑍 = 19) has its additional electron in the 4s subshell instead of the 3d subshell. For example: o Argon (Z=18): Configuration is 1𝑠 * 2𝑠 * 2𝑝@ 3𝑠 * 3𝑝@ (all filled). o Potassium (Z=19): Configuration is 1𝑠 * 2𝑠 * 2𝑝@ 3𝑠 * 3𝑝@ 4𝑠( ; the extra electron fills the 4𝑠 subshell. 5. Transition and Rare Earth Elements: o Transition metals (𝑍 = 21 𝑡𝑜 𝑍 = 30) fill 3𝑑 𝑎𝑛𝑑 4𝑠 subshells, showing similar properties (e.g., scandium to zinc). o Rare earth elements (𝑍 = 57 𝑡𝑜 𝑍 = 71) typically have one or two electrons in the 6𝑠 subshell and partially filled 4𝑓 subshells, displaying closely related chemical properties. The arrangement of electrons in an atom not only determines its stability but also its reactivity and the types of compounds it can form. Understanding electron configurations helps explain the regularities observed in the periodic table and the chemical behaviors of different elements it also provides insight into the stability, reactivity, and classification of elements. Figure 4.25: Bone cancer radiotherapy One application of Electron Configurations is in Bone Cancer Radiotherapy The orange spots in this colored x-ray image are bone cancer tumors. One method of treating bone cancer is to inject a radioactive isotope of strontium ( ST ⬚𝑆𝑟) into a patient’s vein. Strontium is chemically similar to calcium because in both atoms the two outer electrons are in an s state (the structures are 1𝑠 * 2𝑠 * 2𝑝@ 3𝑠 * 3𝑝@ 4𝑠 * 3𝑑(5 4𝑝@ 5𝑠 * for strontium and 1𝑠 * 2𝑠 * 2𝑝@ 3𝑠 * 3𝑝@ 4𝑠 * for calcium). Hence the strontium is readily taken up by the tumors, where calcium turnover is more rapid than in healthy bone. Radiation from the strontium helps destroy the tumors. Screening and Effective Nuclear Charge Screening is a phenomenon in multi-electron atoms where inner electrons shield outer electrons from the full nuclear charge. This shielding reduces the effective nuclear charge (𝑍GII )experienced by the outer electrons. The screening effect significantly influences the energy levels of electrons in multielectron atoms, such as sodium (𝑍 = 11). § Effective Nuclear Charge (Zeff): The net positive charge experienced by an electron. § Shielding: Inner electrons shield outer electrons from the full nuclear charge. § Energy Levels: The energy levels of an electron depend on both the principal quantum number (n) and the orbital angular momentum quantum number (l). Factors Affecting Screening: Orbital Shape: The shape of the orbital influences the degree of shielding. For example, s- orbitals penetrate closer to the nucleus than p-orbitals, resulting in less effective shielding for s- electrons. Electron Configuration: The arrangement of electrons in an atom affects the screening effect. Filled shells and subshells provide more effective shielding than partially filled ones. Consequences of Screening: Energy Level Splitting: Screening causes different energy levels for electrons in the same shell but with different orbital angular momentum quantum numbers. Periodic Trends: Screening explains trends in ionization energy, atomic size, and electronegativity across the periodic table. By understanding the concept of screening and effective nuclear charge, we can better explain the properties of atoms and molecules. 1. Energy Levels in Sodium: o Sodium has 11 electrons filling its K and L shells. o The energy levels of the remaining (11th) electron are observed to be: § 3s: −5.138 eV (most negative, ground state) § 3p: −3.035 eV § 3d: −1.521 eV § 4s: −1.947 eV o Notably, the 4s state is lower in energy than the 3d state, despite having a larger principal quantum number n. 2. Screening: o When the outer electron is considered, it experiences the charge of the nucleus screened by the inner electrons. o The effective charge (𝑍GII ) felt by the 11th electron is reduced because 10 inner electrons shield the 11 protons, leaving an effective charge of approximately +1 (instead of +11). o Thus, the 11th electron behaves as if it were in a hydrogen-like atom with 𝑍GII ≈ 1. 3. Energy Level Calculation: o The energy of an electron in a screened atom can be approximated by the hydrogen-like GU formula: 𝐸; = −113.6 ! ! ; V%&&  o For the 3d state, the screening reduces the effective charge to about +2, leading to: ≈2((K.@ GU 𝐸K> K!.*! ≈ −1.51 𝑒𝑉 o This is close to the experimentally measured value of −1.521 𝑒𝑉. 4. Radial Probability Functions: o The distribution of the 3d state places the electron predominantly outside the filled K and L shells, allowing it to feel a net charge of approximately 𝑍GII = 2. o For the 3p state, its probability distribution indicates that it can spend more time near the nucleus and within the K and L shells, leading to a greater effective charge than 1 but less than 2, hence its higher energy level of −3.035 𝑒𝑉.