Quantum mechanics .pdf

Engineering Physics II (SUBJECT CODE: 303192102) Dr. Swagata Roy & Dr. Mudra Jadav, Assistant Professor Applied Science and Humanities, PIET, Parul University  UNIT- 1 Modern Physics  What to Study in Modern Physics !!! 1. Failures of Classical Mechanics to explain: (a)...

Engineering Physics II (SUBJECT CODE: 303192102) Dr. Swagata Roy & Dr. Mudra Jadav, Assistant Professor Applied Science and Humanities, PIET, Parul University  UNIT- 1 Modern Physics  What to Study in Modern Physics !!! 1. Failures of Classical Mechanics to explain: (a) Stability/Structure of Atom (b) Blackbody Radiation (c) Photoelectric Effect 2. Compton Scattering 3. Wave-Particle Duality 4. Heisenberg’s Uncertainty Principle 5. Wave function and its Physical Significance 6. Operator and Eigen functions 7. Energy and Momentum Operator 8. Schrödinger’s Time Dependent Wave Equation 9. Schrödinger’s Time Independent Wave Equation 10. Particle in 1-D infinite well 11. Quantum Tunnelling 12. Numericals  Introduction Classical mechanics is the branch of physics that deals with the motion of macroscopic objects and forces acting upon them. It provides a comprehensive framework for the understanding how objects move in space and time under the influence of various forces. Phenomena such as motion of large objects (macro scale, > 10-6 m) where the speed involved does not approach the speed of light, can be explained by laws of “classical physics” based on basic laws: i. Newton’s laws of motion Sir Isaac Newton (1643–1727) ii. Newton’s Inverse square law of Gravitation iii. Coulomb’s Inverse square law between electrically charged bodies iv. The law of force on a moving charge in a magnetic field i.e. Lorentz force.  In classical mechanics, it is unconditionally accepted that position, mass, velocity and acceleration etc. of a particle can be measured accurately and simultaneously which is true in day-to-day observations. Domains of Mechanics  Introduction to Quantum Mechanics  Quantum Mechanics is a fundamental theory in physics that describes the behaviour of matter and energy at very small scales such as atom and sub-atomic particles.  In Quantum mechanics, the foundation of principles are purely probabilistic in nature as it is impossible to measure simultaneously the position of momentum of a particle at the sub- microscopic scale.  Introduction to Quantum Mechanics Some basic insights to Quantum Mechanics Max Planck proposed the Quantum theory to explain Blackbody radiation. Einstein applied it to explain the Photo Electric Effect. In the meantime, Einstein’s mass-energy relationship (𝐸 = 𝑚𝑐2 ) had been verified in which the radiation and mass were mutually convertible. Louis de-Broglie extended the idea of dual nature of radiation and matter, when he proposed that matter possesses wave as well as particle characteristics. Pioneers of Quantum Mechanics  Failures of Classical Mechanics  Classical Mechanics describes the motion of macroscopic objects. It provides extremely accurate results as long as the domain of the study is restricted to large objects and the speed involved does not approach the speed of light.  Some unexplainable behaviour (using classical theory) of phenomena at microscopic level gave birth to Quantum Mechanics or it was called as Failure of Classical Physics which could not explain the behaviour of element / matter at microscopic level.  There are three failures of Classical Physics 1. Structure/Stability of an Atom 2. Black Body Radiation 3. Photoelectric effect 1. Structure / Stability of an Atom  1. Structure / Stability of an Atom According to Rutherford, an atom consists of positively charged heavy nucleus surrounded by negatively charged electrons. These negative charged electrons are revolving around the nucleus in circular orbits and they feel strong attractive force by the positively charged heavy nucleus. As a result, they should come closer to the nucleus. However, classical theory of electromagnetic radiation says that whenever a charged particle undergoes accelerated motion, it emits e-m radiation. Thus, an electron being a charged particle must emit energy continuously as its motion is accelerated. Due to this continuous loss of energy, an orbital electron should come closer & closer until it collapses within the nucleus. Hence, an atom would collapse ultimately showing its instability. But we all know, atom is a stable entity. So the stability of an atom couldn't be established by classical physics. Later on Bohr explained the Rutherford model by Old Quantum Theory of atom. BOHR’S OLD QUANTUM THEORY 1. Electrons are allowed to revolve only in certain stable orbits without the emission of radiant energy. These orbits are quantized meaning that they have specific and fixed energies and are known as stationary orbits. 2. Angular momentum of an electron in an allowed orbit is quantized and is an integer multiple of the reduced Planck’s constant i.e. L=nℏ 3. An electron in an allowed orbit doesn’t emit radiant energy. It stays in its orbit without losing energy. However, when an electron jumps from one orbit to another, it absorbs or emit energy equal to the difference in energy between two orbits.  2. Blackbody Radiation An ideal body that absorbs all radiation incident upon it regardless of frequency of radiation is called a Black body. The ability of the body to radiate is closely related to ability to absorb the radiation. Since a body at constant temperature is in thermal equilibrium with its surrounding it must absorb energy from them at the same rate as it emits energy. A closed chamber, a hollow spherical cavity, whose inner surface is coated with platinum black, having a small opening and a projection Fig.1: A spherical cavity blackened opposite to the opening (named as Ferry’s inside and completely closed except a blackbody), acts as a Black body. narrow aperture serves as an ideal black body Light entering the cavity is trapped inside by the multiple reflection from the walls. When heated, the black body emits radiations of all possible wavelengths.  2. Blackbody Radiation In 1897, Lummer and Pringsheim measured the intensities of different wavelength of Blackbody. The energy of radiation emitted from a black body at a temperature T given by Stefan-Boltzmann Law It is given by, 𝐸 𝛼 𝑇4 or, 𝐸 = 𝜎 𝑇 4 Fig 2: The energy distributed in the different 𝜎 = 5.6704 × 10−8 Wm-2K-4 wavelength in the spectrum of a black body radiation. Stefan’s Constant  2. Blackbody Radiation From the spectral distribution following results are considered: 1. At given temperature T, energy density is not distributed uniformly throughout the spectrum. 2. At given temperature T, intensity of the heat radiation increases with increase in wavelength, and at a particular wavelength (𝜆m) its value is maximum, with further increase in 𝜆 the intensity decreases. 3. An increase in temperature causes an increase in the emission intensities for all wavelengths. 4. With increase in temperature, the wavelength 𝜆 𝑚 (𝑚 = 1,2,3, … ) decreases where 𝜆 𝑚 is the wavelength for which the energy emitted is maximum, such that 𝜆 𝑚 T = constant, and is called Wien’s displacement law. 2.8978 × 10−3 𝜆 𝑚 ∙ 𝑇 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 𝜆𝑚 = 𝑚𝐾 0.28978 𝑐𝑚.𝐾 𝑇 The above equation is called as the Wien’s Displacement Law. The area under the curve gives the total energy emitted at a given temperature.  2. Blackbody Radiation Explanations of the Spectra Obtained from Black Body Radiation On the basis of classical physics, a number of attempts were made to explain the observed spectral (energy) distribution of a black body as a function of wavelength. Though, these attempts were not very successful, we shall discuss only three such well known classical laws in this context. 1. Wien's Radiation Formula: In order to explain the observed spectral distribution, Wien first showed that the energy density of radiation of wavelength range between λ and λ + dλ from a cavity (i.e. black body) of temperature T is; Where A and B are constants. LIMITATIONS: Wien's radiation formula explains only the experimental results fairly well (i.e. makes good fitting of experimental curve) for low wavelength region as shown in figure beside.  2. Blackbody Radiation 2. Rayleigh-Jean's Law : Rayleigh and Jeans applied the classical law of equipartition of energy to this electromagnetic black body radiation. They found, the energy density of radiation of wavelength λ and λ + dλ from a black body of temperature T is 𝟖𝝅𝒌𝑻 𝑬𝝀 𝒅𝝀 = 𝟒 𝒅𝝀 𝝀 So, it states that the energy density (Eλ) of a black body radiation (of wavelength λ) is inversely proportional to the fourth power of wavelength (λ) i.e. 𝟏 𝑬𝝀 α 𝟒 𝝀 The above equation describes the Rayleigh-Jeans law  2. Blackbody Radiation The radiation emitted by the atoms is reflected back and forth in cavity to form a system of standing waves for each frequency and when thermal equilibrium is attained the average rate of emission of radiant energy of the atomic oscillators is equal to the rate of absorption of radiant energy by the walls. This energy was calculated for a long wavelength. In the energy spectrum graph, the formula agreed for long wavelength and but goes towards infinity at the shorter wavelength end. This contradiction was known as Ultraviolet Catastrophe. The failure of Rayleigh-Jeans formula gave the first indication of inadequacy of classical physics.  2. Blackbody Radiation (Planck’s Radiation Law) Max Planck was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial contributions to theoretical physics, but his fame as a physicist rests primarily on his role as the originator of Quantum theory, which revolutionized human understanding of atomic and subatomic processes. He is known for Planck's constant, which is of foundational Max Planck (1858–1947) importance for quantum physics.  2. Blackbody Radiation (Planck’s Radiation Law) It is clear that the energy distribution of a black-body could not be explained on the basis of classical concept in which emission or absorption of energy by the atom was supposed to be continuous. Planck realized that the ultraviolet catastrophe occurred because Rayleigh assumed that the standing waves in the cavity of a blackbody consisted of fundamental modes of vibration, each having energy kT. This assumption led to the conclusion that the total energy of the cavity would be infinite. A black-body contains atomic oscillators. The atomic oscillator can not emit or absorb energy continuously, but in the form of discrete units of energy (a tiny packet) called ‘quanta’. An oscillator of frequency (ν) can not have any energy but only discrete values of energy given by, 𝑬 = 𝒏𝒉𝝂 Where 𝒏 is (quantum number of oscillation) = 1,2,3…; 𝒉 is 𝑃𝑙𝑎𝑛𝑐𝑘′ 𝑠 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 6.626 × 10−34 𝐽. 𝑠 and 𝝂 is frequency.  2. Blackbody Radiation (Planck’s Radiation Law) The atom emits energy only when it passes from a higher energy state to a lower energy state and absorbs 6ℎν energy when it goes from a lower state to a higher energy state. The 5ℎν smallest amount of energy which 4ℎν can be emitted or absorbed by the oscillator is 𝒉𝝂. 3ℎν E In other words, a radiation of 2ℎν frequency 𝝂 is emitted as quantum of energy 𝐄 = 𝒉𝝂. This quantum is ℎν the basic unit of energy and can not be subdivided. The quantum of energy is named as a ‘photon’ (by Einstein).  2. Blackbody Radiation (Planck’s Radiation Law) Using Maxwell-Boltzmann distribution law, Planck derived an empirical formula to explain the experimentally observed distribution of energy in the spectrum of a black body, given by 𝟖𝝅𝒉𝝊𝟑 𝟏 𝑬 𝝊 𝒅𝝊 = 𝟑 𝒉𝝊/𝒌𝑻 𝒅𝝊 𝒄 𝒆 −𝟏 Or, in terms of wavelength 𝝀, it 𝟖𝝅𝒉𝒄 𝟏 can be written as, 𝑬𝝀 𝒅𝝀 = 𝟓 𝒉𝒄/𝝀𝒌𝑻 𝒅𝝀 𝝀 𝒆 −𝟏 This relation agrees and completely fit with the experimental curves. This formula of distribution of energy with wavelengths on the basis of his quantum concept, was deduced by several assumptions called Planck’s quantum postulates.  3. Photoelectric Effect Figure 3 : Albert Einstein received the Nobel Prize in Physics in 1921 for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect, a pivotal step in the development of quantum theory.  3. Photoelectric Effect When electromagnetic radiation of high enough frequency is incident on a metal surface, electrons are emitted from the surface. This phenomenon is called PHOTOELECTRIC EFFECT. The emitted electrons are called photoelectrons. The current produced in the circuit due to emission of electrons from the metal surface by the effect of light is known as photocurrent. This effect was first observed by Heinrich Hertz in 1887.  3. Laws of Photoelectric Effect Laws of Photoelectric emission as explained below: 1). When the frequency of incident radiation is above a certain critical value vo depending on the nature of the material of the plate, the emission of photoelectrons take place. This critical frequency is called threshold frequency.(Threshold frequency is defined as the minimum frequency of the incident light to obtain a photoelectric current from a particular metal surface) 2) The emission of photoelectrons start at the same instant of the light with appropriate frequency falls on the metal surface i.e. the phenomenon is instantaneous. It has been observed that the time lag between the incident photon and emission of photo electron is less than 10-8 s. So, the photoelectric emission is an instantaneous process. 3.) The maximum Kinetic energy of emitted electrons of a particular metal plate is directly proportional to the frequency of the incident light. It is independent of the intensity of incident light. 4) The strength of the photoelectric current is directly proportional to the intensity of the incident light. It is independent of the frequency of incident light.  3. Failure of e-m theory to explain Photoelectric Effect According to Classical theory,  Time is required for transformation of energy of incident photons among the atoms of metal surface. But, practically there is no time lag between the instant of light falls and that of the emission of photoelectrons. Hence it fails to explain why photoelectric emission is almost instantaneous process.  It is seen that with the increase of intensity of incident light, the kinetic energy of emitted electrons cannot be increased but photocurrent is increased. But with the increase in the frequency of incident photons, the K. E. of emitting electrons are increased. This cannot be explained by classical e-m theory.  According to e-m theory, the emission of an electron from a metal surface depends only upon the intensity of incident light and not on the frequency of incident light. But it is seen that if the frequency of incident light is less than the threshold frequency for a particular material, no photoelectric emission is observed (how high the intensity of the radiation may be). So e-m theory again fails to explain the existence of threshold frequency for any material.  3. Quantum Analysis & Einstein's Photoelectric Equation In 1905, Einstein applied the quantum theory of light to explain photoelectric effect. Max Planck assumed in his work of blackbody radiation, that exchange of energy takes place between radiation & matter in quanta of magnitude hv. Einstein extended this idea & considered that: 1.) Light is composed of discrete energy packets called photons that move with velocity of light (c) in free space. The energy of a photon, E = hv where h is the Planck's constant = 6.627 ×10-34 J.s & v = frequency of incident light. 2.) Photoelectric effect is a collision between the incident photons and electrons (free) inside the metal. An incident photon of energy hv is completely absorbed by the electron during collision. The famous Photoelectric equation formulated by Einstein is given by: 𝟏 𝒉𝝂 = 𝒉𝝂𝒐 + 𝒎 𝒗𝟐𝒎𝒂𝒙 𝟐 Wo = hvo is called the Work function of the particular metal and is defined as the minimum energy required to liberate an electron from the metal surface and the corresponding frequency is called the threshold frequency. 3. Quantum Analysis & Einstein's Photoelectric Equation  If the energy (hv) of incident photon is greater than that of work function (Wo = hvo) of the metal, the electrons are emitted from the metal surface.  Each collision of incident photons, produces a photoelectron. So with the increase of intensity of incident light, an increase in the number of photons incident on metal surface is obtained. Thus greater number of photons incident on the metal surface, the greater will be the number of photoelectrons emitted. This implies, the photoelectric current is proportional to the intensity of incident photons.  As the work function is constant for a particular metal, the kinetic energy of photoelectrons is proportional to the frequency (v) of the incident photon.  In photoelectric effect, there is a collision between photon of the incident light and electron of the metal. So, it should be an instantaneous phenomenon as time of collision between them is in the order of 10-8 s. Thereby, Einstein's quantum theory satisfactorily explained Photoelectric Effect.  Compton Effect Arthur Holly Compton was an American scientist who won the Nobel Prize in Physics in 1927 for his discovery of the Compton effect.  Compton Effect Compton Effect gives direct and conclusive evidence in support of the particle nature electromagnetic radiation. When a monochromatic beam of X-rays (or the electromagnetic radiation of short wavelengths) of wavelength 𝝀 is allowed to incident on scattering materials, the beam scattered contains the radiation of longer wavelength 𝝀 ′. The difference between 𝝀 and 𝝀 ′ i.e., 𝝀 ′ − 𝝀 is known as Compton shift and the effect is called as a Compton Effect. The Compton shift does not depend on the wavelength of incident radiation and nature of scattering materials. It depends on the scattering angle only. 𝒉 ∴ 𝝀′ − 𝜆 = (1-cosϴ ) 𝒎𝒄 The above equation is called as the Compton Shift in X-ray photon’s wavelength.  Compton Effect Figure 4 represents the scenario of before and after collision of an e-m radiation of low wavelength with electron. Before Collision: The energy of incident radiation be given 𝒉𝒄 b y 𝑬 = 𝒉𝝊 = 𝝀. Figure 4 : As the wave is in motion, the momentum 𝒉𝝊 𝒉 will be 𝒑 = 𝒄 = 𝝀 As the electron initially is at rest, its energy be E0 and its momentum be 𝒑𝟎 (zero). After Collision: The electron is recoiled at an angle ϕ and the energy and momentum changes to 𝐸𝑒 and 𝑝𝑒. The radiation is scattered at an angle θ with the energy ’’ 𝒉𝒄 𝒉𝝊′ 𝒉 𝑬′ = 𝒉𝝊′ = and momentum 𝒑′ = =. 𝝀′ 𝒄 𝝀′ Note: The scattered wave will have longer wavelength.  Compton Effect 1. Compton Shift Δλ = λ’- λ 2. COMPTON WAVELENGTH: Since, the maximum value of cos θ = 1, the wavelength of the scattered photon is always greater than the incident photon. This change is independent of the wavelength λ of the incident photon and the quantity h/moc (λc) is known as COMPTON WAVELENGTH of the scattering particle. CASE: A θ = 0 : then λ’-λ=0 , then no scattering along the direction of the incident photon. Case: B θ = π/2 : then λ’-λ= h/moc (= λc) COMPTON WAVELENGTH = 0.02427 Å Case: C θ = π : then λ’-λ= 2h/moc i.e. wavelength will be twice of the Compton wavelength λc and this is the maximum possible shift = 0.04854 Å  Compton Effect Compton Effect is an important milestone in development of Modern Physics. Compton Effect proves the following: 1. It proves the particle nature of the electromagnetic radiation. 2. It verifies the Planck’s Quantum hypothesis. 3. A precise quantum of energy ℎυ can be assigned to a photon and a precise 𝒉𝝊 𝒉 quantum of momentum 𝒄 = 𝝀 can be assigned to a photon. So not only energy but momentum of electromagnetic radiation is quantized. 𝒎𝟎 4. It provides indirect verification of the relation, 𝒎 = and 𝐸 = 𝑚𝑐 2 𝒗𝟐 𝟏− 𝟐 𝒄 as these relations can be used in delivering the expression for Compton Effect.  Various types of Waves 𝒉 𝒉𝒄 v λ= λ= λ= 𝒑 𝑬 𝒇  Wave-Particle Duality In 1924, de Broglie postulated the wave nature of electrons and suggested that all matter has wave properties. This concept is known as the de Broglie hypothesis. de Broglie won the Nobel Prize for Physics in 1929, after the wave-like behaviour of matter was first experimentally demonstrated in 1927. Wave-Particle Duality Proof of Wave-Particle Duality  Wave-Particle Duality 𝒉 𝒉 λ= = (5) 𝒑 𝒎𝒗 Equation (5) represents general de-Broglie equation that can be applied for particles and waves. 𝜆 is called de-Broglie wavelength. De-Broglie proposed that the matter possess wave as well as particle characteristics. He suggested that a moving particle is associated with a wave, called de-Broglie wave or matter wave. The wavelength of such wave is given by relation (5). Wave-Particle Duality Wave-Particle Duality  Heisenberg’s Uncertainty Principle Werner Heisenberg was a German theoretical physicist and one of the main pioneers of the theory of Quantum Mechanics. He is known for the Uncertainty principle, which he published in 1927. Heisenberg was awarded the 1932 Nobel Prize in Physics for the development of “Quantum Mechanics”.  Heisenberg’s Uncertainty Principle In Classical mechanics, We can determine the position (x) and momentum (p) of macroscopic bodies simultaneously with perfect or same accuracy from its initial position, momentum and the forces acting upon it. However, in Quantum mechanics, Each moving subatomic particle like electron, proton etc. is associated with a wave packet that is extending throughout a region of space. Thus, when a sub–atomic particle is in motion, its position can be anywhere within the wave packet. Hence, there will be an uncertainty in specifying the position of the particle. At the same time, a wave packet consists of a range of wavelengths. Thus from de Broglie relation (p=h/λ), there will be an uncertainty in measurement of momentum of a sub-atomic particle. Therefore, the momentum and position of a moving sub-atomic particle cannot be measured simultaneously with perfect accuracy.  Heisenberg’s Uncertainty Principle On the basis of these considerations, Werner Heisenberg, in 1927, enunciated the “Uncertainty Principle”. Statement of Uncertainty Principle: It is impossible to determine both position(x) and momentum(p) of a quantum mechanical particle simultaneously with perfect precision (accuracy). Mathematically, Heisenberg Uncertainty Principle states that “the product of uncertainty in the simultaneous measurement of the position and momentum of a particle is equal to or greater than the ћ/2 (=h/4π)”, where h is the Planck's constant, i.e. Or, Here, for motion along X-axis, ∆x is the uncertainty in determination of position and ∆p is the uncertainty in determination of corresponding 𝒉 momentum of the particle. Here ћ = 𝟐𝝅, is known as the reduced Planck’s constant. This is the position-momentum Heisenberg Uncertainty relation.  Heisenberg’s Uncertainty Principle Similarly, we can also write, ∆𝐸 ∙ ∆𝑡 ≥ ћ/2 or h/4π ; ∆𝐽 ∙ ∆𝜃 ≥ ћ/2 or h/4π Where ∆𝐸 & ∆𝑡 determine the uncertainties in energy & time respectively. Again, ∆𝐽 & ∆𝜃 are uncertainties in measurement of angular momentum J and angular displacement θ, respectively.  Wave Function From the analysis of electromagnetic waves, sound waves and other such waves, it has been observed that the waves are characterized by certain definite properties. In case of electromagnetic waves, electric and magnetic field vary periodically. In a similar way, in matter wave the quantity that varies is called the wave function denoted by Ψ. 1. WAVE FUNCTION The space-time behaviour of each moving quantum mechanical particle can be described by a function. This function is known as wave function and is generally denoted by Ѱ (x, t) in 1-D or Ѱ (r, t) in 3-D. This gives the idea of probability of finding a particle about a given position (where, r is the position vector of the particle). The magnitude of Ѱ (r, t) is large in the regions where the probability of finding the particle is high and is small in the regions where the probability of finding is low. The wave function Ψ ( r , t ) gives the complete knowledge of behaviour of particle and Ψ r gives the stationary state which is independent of time.  Significance of Wave Function WELL BEHAVED FUNCTIONS Wave functions with all these properties can yield physically meaningful results when used in calculations, so only such well-behaved wave functions are admissible as the mathematical representation of real bodies. To summarize,  Ψ ( r, t) gives the idea of probability of finding a particle about a position.  Ѱ ( r, t) and their first derivatives must be continuous and single valued everywhere.  Ψ must be normalised which means that Ψ must go to 0 as x → ±∞, y → ±∞ ,z → ±∞ in order that ‫| ׬‬Ψ |2 dV must be finite over all space.  The magnitude of Ѱ ( r, t) is large in the regions where the probability of finding the particle is high and is small in the regions where the probability of finding is low.  The product of normalized wave function Ѱ and its complex conjugate Ѱ* gives the probability of finding the particle per unit volume of a given space at a particular time.  Wave Function PROBABILITY DENSITY Probability density of a particle is the probability of finding the particle per unit volume of a given space at a particular time. It is generally expressed as the product of normalized wave function Ѱ and its complex conjugate Ѱ*. So, probability density P = 𝛙 ( r, t)* 𝛙 ( r, t) = 𝛙 (r, t) 2 or, 𝛙 ∗𝛙 = 𝛙 𝟐  If we consider a small volume element dV, the probability of finding the particle existing within this volume dV is given by, 𝛙 ∗ 𝛙  Since the particle if existing somewhere in space, the total probability P to find the particle in space must be equal to 1. +∞ Thereby we can write, P = ‫׬‬−∞ 𝚿 ∗ 𝚿 ⅆ𝐕 = 𝟏 (where dV is the volume element)  Normalisation of wave Function Normalised Wave function +∞ A wave function is said to be normalised if ‫׬‬−∞ 𝛙 𝟐 ⅆ𝐕 = 𝟏 The process of integrating ψ 2 over all space to give unity is called normalisation. Note: The probability that the electron or a particle located somewhere must be unity. 𝒙 P = ‫ 𝟐 𝐧𝛙 𝟐 𝒙׬‬ⅆ𝒙 = 𝟏 𝟏 Above equation shows the probability of finding any particle between 𝑥1 and 𝑥2 in nth state. Note: Here, we consider wave function in 1D, i.e., ψ(x,t).  Operators in Quantum Mechanics  An operator is a mathematical rule that changes a given function into a new function. If A is an operator, it is generally represented by Â.  Mathematical operations in algebra and calculus like adding, subtracting, multiplying, dividing, finding the square root, differentiation or integration are 𝒅 represented by symbols like +, -,×, ÷, √, , ‫ ׬‬can be considered as operators. 𝒅𝒕 Example: If  is an operator and stands for d/dx (say), Then, when it operates on a function x2, it gives d/dx (x2) = 2x In Quantum Mechanics, each dynamical variable such as position (x), momentum (p), energy (E) are represented by linear operators. In quantum mechanics, an operator when applied to a wave function gives the corresponding observable quantity of the system. Example of quantum operators: the linear momentum operator 𝐩 ෝ when applied to the wave function ψ gives the corresponding observable quantity linear momentum 𝐩 and the total energy operator 𝐇 ෡ when applied to ψ gives the corresponding observable quantity ‘total energy’ E.  Eigen Function and Eigen Values When an operator acting on a function always produce the Problem 1: Why sin2x is not an eigen function same function multiplied by a 𝒅𝟐 of operator  = ( 𝟐)? 𝒅𝒙 constant factor, the function is Solution : called an Eigen function and the  f(x) = c f(x) = c = constant, constant is known as Eigen value then f(x) is an eigen function and its eigen value is of the given operator. c. Here, f(x) = sin2x 𝑑2  f(x) = 2 (sin2x ) If  is an operator that operates 𝑑𝑥 = - 2 sin x + 2 cos2x 2 on a given function f(x), then, = 2 – 4 sin2x  f(x) = c f(x) So, sin2x is not an eigen function. (c is a constant). Problem 2: Which one of the following This equation is called Eigen functions are eigen functions of the operator 𝒅𝟐 value equation, the constant c is ? Calculate also the eigen values where 𝒅𝒙𝟐 known as Eigen value and the appropriate. function is known as Eigen (i) Cos x (ii) sin 2x (iii) e4x (iv) (x3 + 2x +5) function of the corresponding operator Â. Operators in Quantum Mechanics Momentum Operator Operators in Quantum Mechanics Energy Operator  Schrödinger’s Wave Equation E. Schrödinger, was a Nobel Prize–winning Austrian Physicist who developed fundamental results in Quantum theory. In particular, he is recognized for postulating the Schrödinger equation, an equation that provides a way to calculate the wave function of a system and how it changes dynamically with time.  Schrödinger’s Wave Equation The de-Broglie hypothesis states that a wave is associated with a particle during its motion. It should be very clear that what type of wave is associated with the motion of the particle and what type of mechanics is required for the formulation for such waves. Erwin Schrödinger worked extensively on wave mechanics, which used to deal with the matter wave. He gave two very important equations for motion of matter waves. 1. Time Dependent Schrödinger Equation (The potential energy of a particle depends on time). 2. Time Independent Schrödinger Equation (The potential energy of a particle does not depend on time).  Schrödinger’s Wave Equation The Schrödinger equations may have many solutions out of these solutions; some are imaginary, which have no significance. The solutions which have significance for certain value are called the Eigen values. In an atom, Eigen values correspond to the energy values that are associated with different orbitals of atom. The solution of the wave equation for this definite value of E gives the corresponding value of wave function Ψ known as Eigen functions. Only these Eigen functions have a physical significance which satisfies following conditions 1. They must be single valued function. 2. They should be finite. 3. They should be continuous throughout the entire space under consideration. Time Dependent Schrödinger’s Wave Equation Case 1: FREE PARTICLE Let us consider a free particle moving along x-direction. It can be described by a wave function, 𝛹 𝑥, 𝑡 = 𝐴𝑒 𝑖 𝑘𝑥−𝜔𝑡 … ….. (1) Where 𝜔 = 2𝜋ν = Angular frequency 2𝜋 𝑘 = wave vector or propagation constant or wave number; 𝑘 = λ The total energy of a free particle (of mass m) moving in x-direction 𝐸 = 𝐾. 𝐸. + 𝑃. 𝐸. 𝑝2 𝐸= … … … (2) (P.E. = 0 for free particle) 2𝑚 According to quantum theory, ℎ Energy 𝐸 = ℎ𝜈 = 2𝜋𝜈 = ℏ𝜔 … ….. (3) 2𝜋 ℎ ℎ 2𝜋 Momentum 𝑝 = = = ℏ𝑘 … ….. (4) 𝜆 2𝜋 𝜆 Differentiate equation (1) with respect to t, 𝜕𝛹(𝑥, 𝑡) = −𝑖𝜔 𝐴𝑒 𝑖 𝑘𝑥−𝜔𝑡 𝜕𝑡 𝜕𝛹(𝑥, 𝑡) = −𝑖𝜔 𝛹 𝑥, 𝑡 (𝑓𝑟𝑜𝑚 𝑒𝑞. 1 ) 𝜕𝑡 Time Dependent Schrödinger’s Wave Equation Multiply with 𝑖ℏ on both sides, 𝜕𝛹(𝑥, 𝑡) 𝑖ℏ = −𝑖 2 ℏ𝜔 𝛹 𝑥, 𝑡 = ℏ𝜔𝛹 𝑥, 𝑡 (𝑎𝑠 − 𝑖 2 = 1) 𝜕𝑡 Substitute equation (3) 𝜕𝛹(𝑥, 𝑡) 𝑖ℏ = 𝐸 𝛹 𝑥, 𝑡 … …. (5) 𝜕𝑡 Now differentiating equation (1) with respect to 𝑥 twice, 𝜕𝛹(𝑥, 𝑡) = 𝑖𝑘 𝐴𝑒 𝑖 𝑘𝑥−𝜔𝑡 𝜕𝑥 2 𝜕 𝛹(𝑥, 𝑡) 2 = 𝑖𝑘 2 𝐴𝑒 𝑖 𝑘𝑥−𝜔𝑡 𝜕𝑥 2 𝜕 𝛹(𝑥, 𝑡) 2 = −𝑘 2 𝛹 𝑥, 𝑡 (𝑎𝑠 𝑖 2 = −1 𝑎𝑛𝑑 𝑓𝑟𝑜𝑚 𝑒𝑞. 1 ) 𝜕𝑥 −ℏ2 Multiply with both sides 2𝑚 −ℏ2 𝜕 2 Ψ(x, t) ℏ2 k 2 = Ψ x, t 2m 𝜕x 2 2m Time Dependent Schrödinger’s Wave Equation Substitute equation (4) −ℏ2 𝜕 2 𝛹(𝑥, 𝑡) 𝑝2 2 = 𝛹 𝑥, 𝑡 ……. 6 2𝑚 𝜕𝑥 2𝑚 Substitute equation (2) in above equation −ℏ2 𝜕 2 Ψ(𝑥, 𝑡) 2 = 𝐸 Ψ 𝑥, 𝑡 … …. 7 2𝑚 𝜕𝑥 Comparing equation (5) and (7),we can write −ℏ𝟐 𝝏𝟐 𝚿(𝐱, 𝐭) 𝛛𝚿(𝐱, 𝐭) = 𝐢ℏ … 𝟖 𝟐𝒎 𝛛𝒙𝟐 𝛛𝒕 This is one-dimensional (1D) time dependent Schrodinger wave equation for a free particle of mass m.  Time Dependent Schrödinger’s Wave Equation Case 2: PARTICLE UNDER AN EXTERNAL FORCE FIELD When a particle is moving along x-direction under an external force field, total energy can be written as, 𝐸 = 𝐾. 𝐸. + 𝑃. 𝐸. 𝑝2 𝐸= + 𝑉 𝑥, 𝑡 … … (9) 2𝑚 Multiply above equation with wave function Ψ 𝑥, 𝑡 , 𝑝2 𝐸Ψ 𝑥, 𝑡 = Ψ 𝑥, 𝑡 + 𝑉 𝑥, 𝑡 Ψ 𝑥, 𝑡 … … … (10) 2𝑚 Substitute equation equation (6) into (10), −ℏ2 𝜕 2 Ψ(𝑥, 𝑡) 𝐸Ψ 𝑥, 𝑡 = + 𝑉 𝑥, 𝑡 Ψ 𝑥, 𝑡 ….. (11) 2𝑚 𝜕𝑥 2 Compare equation (5) and (11), −ℏ2 𝜕 2 Ψ(𝑥, 𝑡) 𝜕Ψ(𝑥, 𝑡) + 𝑉 𝑥, 𝑡 Ψ 𝑥, 𝑡 = 𝑖ℏ 2𝑚 𝜕𝑥 2 𝜕𝑡 −ℏ𝟐 𝝏𝟐 𝝏𝜳(𝒙, 𝒕) + 𝑽 𝒙, 𝒕 𝜳 𝒙, 𝒕 = 𝒊ℏ 𝟐𝒎 𝝏𝒙𝟐 𝝏𝒕 This is the one-dimensional (1D) time dependent Schrodinger wave equation for a particle with mass m.  Time Dependant Schrödinger’s Wave Equation −ℏ𝟐 𝟐 𝝏𝜳(𝒓, 𝒕) 𝛁 + 𝑽 𝒓, 𝒕 𝜳 𝒓, 𝒕 = 𝒊ℏ 𝟐𝒎 𝝏𝒕 This is the three-dimensional (3D) time dependent Schrodinger wave equation. 𝜕2 𝜕2 𝜕2 −ℏ𝟐 𝟐 𝑤ℎ𝑒𝑟𝑒 𝛻2 = + + ෡= and, Hamiltonian operato𝑟 = 𝐻 𝛁 + 𝑽 𝒓, 𝒕 𝜕𝑥 2 𝜕𝑦 2 𝜕𝑧 2 𝟐𝒎 Time Independent Schrödinger’s Wave Equation Let us first write (time dependent) Schrodinger wave equation for a particle moving in X-direction, −ℏ2 𝜕 2 𝜕𝛹(𝑥, 𝑡) + 𝑉 𝑥, 𝑡 𝛹 𝑥, 𝑡 = 𝑖ℏ ….. (1) 2𝑚 𝜕𝑥 2 𝜕𝑡 In some particular cases, the potential energy of a moving particle does not depend explicitly on time. Consider a particle moving in x-direction and the potential energy is a function of position only, i.e., 𝑉=𝑉 𝑥 …… 2 One can write the wave function for this case as a product of two functions, known as method of separation of variables, Ψ 𝑥, 𝑡 = Ψ 𝑥 𝑓 𝑡 …… 3 Here the wave function is the product of two separate functions Ψ 𝑥 and 𝑓 𝑡. Ψ is only function of 𝑥 and 𝑓 is only function of 𝑡.  Time Independent Schrödinger’s Wave Equation Substituting equation (2) and (3) into (1), −ℏ2 𝜕 2 𝜕 +𝑉 𝑥 𝛹 𝑥)𝑓(𝑡 = 𝑖ℏ 𝛹 𝑥 𝑓(𝑡) 2𝑚 𝜕𝑥 2 𝜕𝑡 −ℏ2 𝑑2 Ψ 𝑥 𝑑𝑓(𝑡) 𝑓 𝑡 + 𝑉 𝑥 𝛹 𝑥)𝑓(𝑡 = 𝑖ℏ 𝛹 𝑥 2𝑚 𝑑𝑥 2 𝑑𝑡 Note: Here 𝜕 is replaced by 𝑑. After separation of variables, we are dealing with functions of single variables (𝛹 𝑥 𝑎𝑛𝑑 𝑓(𝑡)) only. Hence, the partial derivatives are replaced by total derivatives. Now, dividing the above equation by 𝛹 𝑥)𝑓(𝑡 on both sides, we get −ℏ2 1 𝑑2 Ψ 𝑥 1 𝑑𝑓(𝑡) + 𝑉 𝑥 = 𝑖ℏ 2𝑚 Ψ(𝑥) 𝑑𝑥 2 𝑓(𝑡) 𝑑𝑡 The L.H.S of above equation is a function of 𝒙 only and R.H.S. is a function of 𝒕 only. This is only possible when they are separately equal to a constant and it is equal to the total energy 𝑬.  Time Independent Schrödinger’s Wave Equation Therefore, −ℏ2 1 𝑑2 Ψ 𝑥 1 𝑑𝑓(𝑡) + 𝑉 𝑥 = 𝑖ℏ = 𝐸 ….. (4) 2𝑚 Ψ(𝑥) 𝑑𝑥 2 𝑓(𝑡) 𝑑𝑡 From equation (4), one can write −ℏ2 1 𝑑2 Ψ 𝑥 + 𝑉 𝑥 = 𝐸 ….. (5𝑎) 2𝑚 Ψ(𝑥) 𝑑𝑥 2 1 𝑑𝑓(𝑡) 𝑖ℏ =𝐸 ….. (5𝑏) 𝑓(𝑡) 𝑑𝑡 We can rewrite equation (5a) (multiplying with Ψ(𝑥)) −ℏ2 𝑑2 Ψ 𝑥 +𝑉 𝑥 Ψ 𝑥 =𝐸Ψ 𝑥 2𝑚 𝑑𝑥 2 −ℏ𝟐 𝒅𝟐 +𝑽 𝒙 𝜳 𝒙 =𝑬𝜳 𝒙 ….. 𝟔 𝟐𝒎 𝒅𝒙𝟐 This is one dimensional (1D) time independent Schrodinger wave equation  Time Independant Schrödinger’s wave Equation −ℏ𝟐 𝟐 𝛁 +𝑽 𝒓 𝜳 𝒓 =𝑬𝜳 𝒓 ….. (𝟕) 𝟐𝒎 This is three dimensional (3D) time independent Schrodinger wave equation 𝜕2 𝜕2 𝜕2 Where, 𝛻 2 = + + , and 𝜳 𝒓 ≡ 𝜳 𝒙, 𝒚, 𝒛 𝜕𝑥 2 𝜕𝑦 2 𝜕𝑧 2 −ℏ𝟐 ෡= 𝐇𝐚𝐦𝐢𝐥𝐭𝐨𝐧𝐢𝐚𝐧 𝐨𝐩𝐞𝐫𝐚𝐭𝐨𝐫 = 𝐻 𝛁𝟐 + 𝑽 𝒓 𝟐𝒎 Equation (7) can be re-written as, ෡ 𝒓 =𝑬𝜳 𝒓 𝑯𝜳 The above is the Eigen value equation, 𝛹 𝑟 is Eigen function, 𝐸 is Eigen value. Hamiltonian operator, operating on the wave function Ψ produces the same function Ψ multiplied by the total energy 𝐸. The Eigen value 𝐸 represents the value of total energy of a system.  Particle in a One Dimensional Infinite Well/Potential Box Consider a free particle of mass ‘𝑚’ placed inside a one-dimensional box of infinite height and width 𝑎. The motion of the particle is restricted by the walls of the box. The particle is bouncing back and forth between the walls of the box at 𝑥 = 0 and 𝑥 = 𝑎. (motion is restricted in x-direction). Particle is free, i.e. potential energy V of the particle inside the box is zero. The walls of the box are non-penetrable. V(x) V(x) This means that outside the box the potential energy is so large that the total energy of the particle can never exceed it. The particle is trapped inside the box. The P.E. of the particle on/outside the walls is infinite. The particle cannot escape from the box, V = 0 for 0 < 𝑥 < 𝑎 Fig 5. One-dimensional box V = ∞ for 0 ≥ 𝑥 ≥ 𝑎 (Infinite Potential Well) Particle in a One Dimensional Infinite Well/Potential Box Since the particle cannot be present outside the box, i.e. |Ψ |2 = 0 for 0 ≥ 𝑥 ≥ 𝑎 The Schrödinger one – dimensional time independent equation is −ℏ𝟐 𝒅𝟐 + 𝑽 𝒙 𝝍 𝒙 = 𝑬 𝝍(𝒙) 𝟐𝒎 𝒅𝒙𝟐 −ℏ𝟐 𝒅𝟐 𝝍(𝒙) + 𝑽 𝒙 𝝍 𝒙 = 𝑬𝝍 𝒙 … …. (𝟏) 𝟐𝒎 𝒅𝒙𝟐 Solution : 𝒅𝟐 𝝍(𝒙) 𝟐𝒎𝑬 For freely moving particle 𝑽 𝒙 = 𝟎, so + 𝟐 𝝍 𝒙 = 𝟎 ….. (𝟐) 𝒅𝒙𝟐 ℏ 𝟐𝒎𝑬 Taking, = 𝒌𝟐 … … (𝟑) ℏ𝟐 𝟐𝝅 where 𝑘 is known as wave vector, 𝒌 = 𝝀 Particle in a One Dimensional Infinite Well/Potential Box Equation (2) becomes, 𝑑2 ψ(𝑥) 2 + 𝑘 2 ψ 𝑥 = 0 ….. (4) 𝑑𝑥 Solution of eq. (4) can be given by, Ψ 𝑥 = 𝐴 sin 𝑘𝑥 + 𝐵 cos 𝑘𝑥 … …. (5) where 𝐴, 𝐵 𝑎𝑛𝑑 𝑘 are unknown quantities and to calculate them it is necessary to construct boundary conditions. The boundary conditions are (i) When 𝑥 = 0, Ψ 𝑥 = 0 (ii) When 𝑥 = 𝑎, Ψ 𝑥 = 0 Putting Boundary Conditions (i) in eq. (5), we get Ψ 𝑥 = 0 = 𝐴 sin 0 + 𝐵 cos 0 𝐵 = 0 ….. (6) Putting Boundary Conditions (ii) in eq. (5), we get Ψ 𝑥 = 0 = 𝐴 sin 𝑘𝑎 + 𝐵 cos 𝑘𝑎 But from eq. (6) put 𝐵 = 0 𝐴 sin 𝑘𝑎 = 0 But 𝐴 ≠ 0, so sin 𝑘𝑎 = 0 𝑘𝑎 = 𝑛𝜋 𝒏𝝅 or, 𝒌= ….. 7 𝑤ℎ𝑒𝑟𝑒 𝑛 = 1,2,3 … 𝒂  Particle in a One Dimensional Infinite Well/Potential Box Substituting equation (7) in equation (3) 2𝑚𝐸 𝑛2 𝜋 2 = 2 ℏ2 𝑎 𝑛 𝜋 ℏ2 2 2 ℎ E= 2 (ℏ = ) 𝑎 2𝑚 2𝜋 𝑛 2 ℎ2 𝐸= 8𝑚𝑎2 𝒏𝟐 𝒉𝟐 In general, E= ….. 𝟖 𝑤ℎ𝑒𝑟𝑒 n = 1,2,3,4…. 𝟖𝒎𝒂𝟐 Equation (8) represents Eigen values of energy for different energy levels. This indicates discrete energy levels in quantum mechanics. Wave Function can be written as, (from equation (5) and (6)) 𝜳 𝒙 = 𝑨 𝒔𝒊𝒏 𝒌𝒙 Substitute equation (7), 𝒏𝝅𝒙 𝜳 𝒙 = 𝑨 𝒔𝒊𝒏 𝒂 … …. (𝟗) Particle in a One Dimensional Infinite Well/Potential Box Let us find the value of A, if a particle is definitely present inside the box 𝒂 𝑷 = න 𝜳(𝒙)𝟐 𝒅𝒙 = 𝟏 𝟎 𝒂 𝒏𝝅𝒙 𝑷=න 𝑨𝟐 𝒔𝒊𝒏𝟐 𝒅𝒙 = 𝟏 …. 𝒇𝒓𝒐𝒎 𝒆𝒒. 𝟗 𝟎 𝒂 𝒂 𝒏𝝅𝒙 𝟏 Or, ‫𝟐𝒏𝒊𝒔 𝟎׬‬ 𝒅𝒙 = 𝒂 𝑨𝟐 𝟐𝒏𝝅𝒙 𝒂 𝟏−𝐜𝐨𝐬 𝒂 𝟏 Or, ‫𝟎׬‬ 𝒅𝒙 = 𝟐 𝑨𝟐 𝒂 𝟐𝒏𝝅𝒙 𝟐 Or, ‫ 𝟏 𝟎׬‬− 𝒄𝒐𝒔 𝒂 𝒅𝒙 = 𝑨𝟐 𝟐𝒏𝝅𝒙 𝒂 𝐬𝐢𝐧 𝒂 𝒂 𝟐 Or, 𝒙 − = 𝟐 𝟎 𝟐𝒏𝝅Τ 𝒂 𝟎 𝑨 𝒔𝒊𝒏 𝟐𝒏𝝅−𝒔𝒊𝒏 𝟎 𝟐 Or, 𝒂 − 𝟎 − 𝟐𝒏𝝅Τ = 𝒂 𝑨𝟐 𝟐 Thereby, 𝒂= 𝟐 𝑨 𝟐 𝑨= ….. (𝟏𝟎) 𝒂  Particle in a One Dimensional Infinite Well/Potential Box Substituting equation (10) into Eq. (9),we have 𝟐 𝒏𝝅𝒙 𝝍𝒏 (𝒙)= 𝒂 𝐬𝐢𝐧 𝒂 ….. (𝟏𝟏) The wave function in 𝑛 𝑡ℎ energy level is given in Equation (11). Therefore the particle in the box can have discrete values of energies. These values are quantized. Note that according to classical mechanics, the particle in a box can have any energy value from 0 to infinity, but according to quantum mechanics only discrete values of energy are permissible. The particle cannot have zero energy. i.e. momentum and energy are quantized. The Ψ1 Ψ2 , Ψ3 are normalized wave functions, given by equation (11). Each E𝑛 value is known as Eigen value and the corresponding wave function is known as Eigen function. The wave function Ψ 1 has two nodes at 𝑥 = 0 and 𝑥 = 𝑎 The wave function Ψ 2 has three nodes at 𝑥 = 0, 𝑥 = 𝑎/2 and 𝑥 = 𝑎 The wave function Ψ 3 has four nodes at 𝑥 = 0,𝑥 = 𝑎/3 ,𝑥 = 2𝑎/3 and 𝑥 = 𝑎 The wave function 𝚿 𝒏 has (𝐧 + 𝟏) nodes with boundary conditions. A node is the position of zero displacement. Particle in a One Dimensional Infinite Well/Potential Box  Quantum Tunnelling QUANTUM TUNNELING: Quantum tunnelling is a fundamental concept in quantum mechanics, describing the ability of particles to penetrate energy barriers that classical physics would deem impenetrable. In classical physics, particles encountering barriers with energy higher than their own kinetic energy are unable to pass through. However, at the quantum level, particles exhibit both particle-like and wave-like characteristics. This wave-particle duality enables them to exhibit tunnelling.  In Quantum physics, particle are described by waves (matter waves) as per de-Broglie hypothesis.  When such particle of sufficiently less mass impinges on the barrier of finite potential, the solution of Schrodinger equation exists on the other side of the barrier even if energy of particle is less than barrier height. This gives a small probability that the particle may penetrate the barrier.  Quantum Tunnelling  However, the probability exponentially decreases as the height or width of the barrier increases or mass of the particle increases.  Electrons can tunnel through the barrier of thickness 1-3 nm whereas proton can tunnel through the barrier of thickness less than 0.1nm.  Tunnelling plays an important role in nuclear fusion in stars, radioactivity, Tunnel Diode, Scanning Tunnelling Microscopy, electron tunnelling in photosynthesis, proton tunnelling in DNA mutation etc. Quantum Tunnelling Nuclear Fusion in STARS Quantum tunnelling plays a crucial role in nuclear fusion within stars. Protons need to overcome the electrostatic repulsion barrier to fuse into helium nuclei, and quantum tunnelling enables this process, facilitating the energy production in stars, which is the ultimate energy source for the thriving of life on earth.  Quantum Tunnelling Radioactive Alpha decay Protons and neutrons are bound inside nuclei, that means energy must be supplied The answer was supplied in 1928 by to break them away. the Russian physicist George In a nucleus, there is an attractive nuclear potential. Gamow. The α particle tunnels Protons and neutrons have kinetic energy, but it is about 8 MeV less than that needed to get out. That through a region of space it is is, they are bound by an average of 8 MeV per forbidden to be in, and it comes out nucleon. In α decay, two protons and two neutrons of the side of the nucleus. This is spontaneously break away, yet the protons and known as Quantum mechanical neutrons do not have enough kinetic energy to get Tunnelling. over the rim. So how does the α-particle get out?  Quantum Tunnelling Scanning Tunnelling Microscope Tunnel Diode  Quantum Tunnelling Potential barrier of finite height and width: Consider a particle of mass m incident on a potential barrier of height V0 and width L. The energy of the particle is E and EL  Quantum Tunnelling From quantum mechanical calculations, the transmission probability T of a moving particle (wave) to cross the barrier is dependent on both (i) total energy of the particle, E (ii) The width of the barrier, L and is approximately given by the equation below: 𝟏𝟔𝑬 𝑬 T≈ 𝟏− 𝒆−𝟐𝜶𝑳 𝑽𝟎 𝑽𝟎 Thus there is higher probability of tunnelling out of the barrier when both 𝛂 𝐚𝐧ⅆ 𝐋 𝐚𝐫𝐞 𝐬𝐦𝐚𝐥𝐥. 𝟐𝒎 Here V0 is the height of the barrier and α = 𝟐 𝑽𝟎 − 𝑬 ћ  Books For Reference  Quantum Mechanics (A Textbook for Undergraduates) MAHESH C. JAIN  A Textbook of Engineering Physics by M N AVADHANULU, PG KSHIRSAGAR, TVS ARUN MURTHY