Physics 362: Quantum Mechanics I

• These lecture notes are for Physics 362, Quantum Mechanics I, second semester 2024-2025 at the University of Cairo, by Dr. Ahmed Mohamed Hassan.

Evaluation

• Attendance and assignments account for 20% of the final grade.
• The mid-term exam accounts for 20% of the final grade.
• The final exam accounts for 60% of the final grade.
• The total grade is out of 100 marks.

Main Textbook

• Quantum Mechanics by B.H. Bransden and C.J. Joachain is the main textbook.
• Introduction to Quantum Mechanics by David J. Griffiths is a recommended textbook.

Classical Mechanics

• Classical mechanics explains the motion of objects, from macroscopic to astronomical, based on Newton's laws.
• It is limited when dealing with very small scales or extremely high speeds.
• Classical mechanics cannot explain quantum realm phenomena or accurately describe scenarios approaching light speed.
• The limitations led to quantum mechanics and relativity, providing accurate descriptions at microscopic and cosmic levels.

Quantum Mechanics

• Quantum mechanics explores the behavior of matter and energy at the atomic and subatomic level.
• It introduces quantization, where energy and angular momentum are restricted to discrete values.
• Quantum mechanics provides insights into interactions between particles and electromagnetic radiation as a fundamental theory.

Blackbody Radiation

• A blackbody absorbs radiation perfectly.
• It absorbs all incident light without reflecting or transmitting any.
• Heated blackbodies can emit radiation, which led to scientific debate in the late 19th century.
• A spectrum is a graphical representation of emitted light intensity at different wavelengths or frequencies at a specific temperature.
• The energy density (ρ(λ, Τ)) plotted against wavelength (λ) results in a curve.
• The Rayleigh-Jeans formula attempted to explain blackbody radiation using classical theory.
• It states that radiation intensity is directly proportional to temperature (T) and inversely proportional to the fourth power of the wavelength (λ).
• The formula is ρ(λ, Τ) = (2 * c * kB * T) / λ^4, where kB is Boltzmann's constant and c is the speed of light.
• The Rayleigh-Jeans formula predicts infinite intensity as wavelength approaches zero, which is known as the ultraviolet catastrophe.
• Wien's law states that the peak wavelength of light emitted by a blackbody (λ_peak) is inversely proportional to its temperature (T): λ_peak = b / T.
• b is a constant known as Wien's displacement constant.
• Wien's law provided a step forward yet did not perfectly match experimental observations, especially at longer wavelengths.
• In 1900, Max Planck suggested that energy is emitted in discrete packets known as quanta.
• Planck's equation relates the energy E of a quantum to its frequency v: E = h * v, in which h is Planck's constant.
• Planck derived an expression for the energy spectrum that matched experimental data: ρ(ν, Τ) = (8πν^3 / c^3) * (1 / (e^(hv/kT) - 1)).

Einstein and the Photoelectric Effect

• The photoelectric effect demonstrates the particle-like nature of light.
• The experiments by Lenard and Millikan later facilitated the development of Einstein's theory.
• Lenard's experiment explored how electromagnetic waves behave when interacting with matter.
• It was found that the energy is dependent on intensity rather than frequency through varying incident light intensity on a metal surface,
• Classical physics expected electron emission at any frequency if the intensity was sufficient.
• The expectation was that the current generated would depend on intensity and not frequency.
• Millikan's experiment observed that the maximum energy of ejected electrons depended on the incident light's color and wavelength.
• Shorter wavelengths, which are higher frequncies, resulted in photoelectrons with greater kinetic energy.
• A noticeable time delay existed between turning on the light and observing current at low intensities, that indicated the minimum energy is the work function
• The stopping potential, which is the minimum potential to halt the motion of emitted electrons, increased with the intensity of the radiation, regardless of the frequency of light
• In 1905, Einstein explained the photoelectric effect which incorporated the observations of Lenard and Millikan.
• The kinetic energy of emitted electrons depends on the frequency of the incident radiation and not its intensity.
• There is a threshold frequency below which no electrons are emitted for a given metal.
• Electron emission occurs immediately when light shines on the surface, without a time delay.
• According to Einstein's theory, Light is composed of photons, which are discrete packets of energy.
• The energy of a photon is given by Planck's relationship: E = h * v.
• An atom can absorb a whole photon or none at all.
• Part of the energy frees an electron when a photon is absorbed, requiring a fixed energy W, called the work function.
• The remaining energy converts into the kinetic energy of the emitted electron.
• The relationship between the energy of the photon, the work function, and the kinetic energy is: hv = W + KE.
• These assumptions explained the experimental results, solidifying the concept of light as both a wave and a particle.

Bohr's Model of the Hydrogen Atom: Atomic Spectra

• Niels Bohr's model, developed in 1913, addressed the atomic paradox by introducing quantization into classical mechanics.
• The model incorporated Planck's quantization and Einstein's photon theory.
• Classical electromagnetism predicted that an electron orbiting the nucleus in hydrogen would continuously emit light.
• Bohr incorporated quantization and the photon theory into the classical mechanics description of the atom.
• He stated that an electron would not emit radiation in a stationary state, yet it would emit or absorb a photon when transitioning to a different orbit.
• The energy difference between the initial and final orbits, |∆Ε| or |Ef – E_i|, relates to the frequency (v) or wavelength (λ) of the emitted or absorbed photon.
• The equation is |ΔΕ| = |Ef – Ei| = h * v = h * c / λ.
• Bohr's model introduced quantized energies for a single electron in the hydrogen atom.
• The energies were given by E_n = Constant / n^2
• |ΔΕ| = Constant * (1/n_i^2 - 1/n_f^2) = h * c / λ
• The energy expression can also be expressed for hydrogen-like atoms in the equation E_n = -k * Z^2, where Z is the atomic number.
• Bohr's model provided a framework for understanding atomic spectra, while it also explained the discrete energy differences.

de Broglie and Matter Waves

• In 1924, Louis de Broglie stated that material particles, such as electrons, have wave-like properties similar to radiation.
• De Broglie postulated that material particles also exhibit wave-like properties.
• This idea challenged the traditional notion of particles as discrete entities.
• For free material particles, De Broglie associated a frequency (v) and wavelength (λ) with their wave-like behavior.
• The quantities relate to the particle's energy (E) and momentum (p) by the equations ν = E / h and λ = h / p.
• For non-relativistic particles with mass m and velocity v, the de Broglie wavelength is λ = h / (m * v).
• The double-slit experiment is concrete evidence for wave-particle duality.
• Matter and energy, such as light, can exhibit wave-like and particle-like characteristics.
• This revolutionized understanding of the particles, as the de Broglie wavelength provided a quantitative measure of the wave-like, and established a fundamental principle in quantum mechanics.

Schrödinger's Wave Equation and Wave Function

• In 1926, Erwin Schrödinger formulated a wave equation that accurately calculated the energy levels of electrons in atoms.
• Schrödinger incorporated Louis de Broglie's hypothesis that particles possess wave-like properties.
• De Broglie's postulate related a particle's frequency (v) and wavelength (λ) to its energy (E) and momentum (p) using the equation v = E/h and λ = h/p. in which h is Planck's constant
• Schrödinger drew inspiration from optics' mathematical formulation, in which the propagation of light rays could be derived from wave motion.
• The equation to derive was Ψ(x,t) = A cos(kx – wt), in which Ψ is the wave function, x is position, t is time, A is amplitude, k is the wave number, and w is the angular frequency.
• Schrödinger looked at the conservation of energy, with the particle's kinetic energy (Κ.Ε.), potential energy (V), and total energy (E).
• This derived an equation related to the energy of the particle, in which the Schrödinger equation is (iћ) * (∂Ψ(x, t) / ∂t) = (-ћ^2 / (2m)) * (∂^2Ψ(x, t) / ∂x^2) + V(x)Ψ(x,t).
• ћ is the reduced Planck's constant and m is the mass of the particle.
• The solution to the Schrödinger equation, Ψ (x,t), is the wave function.
• The wave function contains motion of a matter wave, all information about a physical system, and to determine various dynamical variables.
• The modulus squared of the wave function, |Ψ(x, t)|^2, represents the probability density as a particular position.
• The integral of |Ψ(x,t)|^2 over a given interval provides the probability of finding the particle within that interval.
• Influence of external factors like potential energy can depend on wave function and associated matter waves.

Derivation of the Time-Dependent Schrödinger Equation

• This derivation starts with a complex plane wave.
• This considers a complex plane wave Ψ(x,t) = Aei(kx-wt).
• By differentiating Ψ(x, t) = Aei(kx-wt) with respect to t in that gives the equation ∂Ψ(x, t) / ∂t = -iωΨ(x, t).

Derivation of the Time-Independent Schrödinger Equation:

• This starts with the time-dependent Schrödinger equation.
• It applies the separation of variables by assuming that the wave function expresses itself as the product of a spatial part and a temporal part.

Postulates of Quantum Mechanics:

• The Schrödinger equation describes the behavior of quantum systems.
• It consists of time-dependent and time-independent versions.
• The time-dependent Schrödinger equation describes the evolution of the wave function Ψ(x,t) with respect to time.
• The wave function represents a quantum system and contains information about the probability distribution of a particle at a particular position.
• The time-independent Schrödinger equation is derived from separation of the time. It makes spatial part of the wave function, in this equation: [-ћ²/ (2m)] * (d²ψ(x)/ dx²) + V(x) ψ(x) = E ψ(x).
• ψ(x) represents the spatial part of the wave function.
• V(x) is the potential energy function.
• E is the total energy of the system.
• The Hamiltonian operator H operates on the wave function ψ(x) to produce the energy E. also called the eigenvalue equation.
• The wave function ψ(x) is called an eigenfunction (eigenvector), and the resulting numerical value is called the eigenvalue.

Superposition Principle:

• The Superposition Principle is a fundamental concept in quantum mechanics.
• It states that for a quantum system, its wave function can be an eigenfunction of the operator with corresponding eigenvalue E (Hψ(x) = Εψ(x))
• Superposition states that multiple wave functions ψ₀(x), ψ₁(x), ψ₂(x) can all be valid solutions of the Schrödinger equation.
• Superposition allows a quantum system to exist in multiple states at the same time until it is measured or observed, which differs from definite states in classical systems.
• The double-slit experiment is an example of how particles produce an interference pattern, suggesting they exist in a superposition of states.
• For a quantum system, the wave function Φ(x) expresses itself as a linear combination of multiple wave functions ψₙ(x), being multiplied by coefficients cₙ.
• The equation represents superposition of states in which coefficients are the probability amplitudes that determines state
• The superstition principle then shows physicals predict behavior

Probability Density:

• Probability density provides information on the likelihood of finding a particle in a specific location and is related to the wave function through the equation: ρ(x,t) = |Ψ(x,t)|².
• The probability density is also equal to the absolute square of the wave functions: |Ψ(x, t)|².
• The probability of finding the particle in an interval range is [x, x + dx].
• The wave function (x,t) indicates the product of spatial and temporal wave functions (ψ(x) *(t)).
• (t) = e^(-iEt/h), where it is squared against the probability it remains solid, giving a location for constant time.
• It must be specified that every point or location to predict it must be normalized and has value of 1.
• There are certain quantum, with certain densities, make it from a classical view

The Uncertainty Principle:

In classical mechanics, the state of a particle is completely determined by its position x(t) and momentum p(t).

• Quantum mechanics introduces the wave function Ψ(x) to describe the particle's behavior.
• The uncertainty principle says the position and momentum of a particle cannot be simultaneously measured.
• There is a trade-off between the measurements.
• Mathematically: ∆x · ∆p ≥ ħ/2, where ∆x is the uncertainty in position, ∆p is the uncertainty in momentum, and ħ is the reduced Planck's constant.
• the uncertainty principle is not due to technological limitations but is a fundamental property.
• The uncertainty principle and accurately measure the position is not possible, the same with momentum

Conditions for Validate Wave function

• Finite: The wave function must be finite for all values of x.
• Single-valued: The wave function should have a unique value for each value of x.
• Continuous and its derivatives must be continuous: The wave function and its derivatives should be continuous to ensure smooth behavior.
• Vanishes at endpoints: The wave function should approach zero as x approaches infinity or negative infinity, and must be a neglible probability finding.
• Normalized: The wave function must be normalized to a total probability equal to one.
• There exist accepatable and non-accpetable functions

Observables and Operators

-In postate system have corresponding linear and Hermitian operato

• LINEAR OPERATORS: A linear operator satisfies the following property: Â(cψ + dφ) = cÂψ + dÂφ. Ex Position operator (X), Derivative + dx second deivative , (d2 , non sin log square root HERMITIAN OPERATORS: -A Hermitian operator satisfies the following property: [ ¢(Â¥)dr = [ ¢ ¢dt Ex Position operator (x), Potential energy operator (V(x)), Imaginary, Non Hermitian
• Imagninary unit time, deritvaitv d , imaginary units time

Postulate 3: Eigenvalue Equation

• The eigenvalue equation is a principle for eigenfucntions: Â¥(x) = a¥(x),  for observable, V associated observale, a can be mesured -The energy states how eigenficients act the eigen states

In other words

• The postulate states that every observable quantity follows an operator line orheretial
• The examples illustrate the operators such as Quant operator with corresponding math
• The eigneval use eigenfunctions to solce
• The higher states we get different elign values or values

Postulate 4: Expectation values

1. Normalized Wave Function: For that the average value (expectation value) comes about through to the operator Â
   : (A)= [ ¢* ¢ dt Non-Normalized Wave Function then the average value is givin
   : (A) = [[ ¢* ¢ dr]/[¢*¢dr] calculate any expectation value, it is necessary to know the wave function ψ(x) that describes the state of the system. example that we get values for normal and not norm

Postulate 5: Time evolution

• In quantum mechanics, the time evolution of a system is governed by the Schrödinger equation. This postulate states that the wave function, Ψ(x, t), representing the state of the system, changes with time according to the following equation . iħ ƏΨ(x, t)/ ∂t =. -H2 2m Ə2Ψ(x, t) Əx2 + V(x)Ψ(x,t) —or——iħ ƏΨ(x,t)/ ∂t = HY(x, t)
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