PHYS 3143 Quantum Mechanics I

• Lecture 1
  • 1.1 Course policies
  • 1.2 Introduction to QM
  • 1.3 Mathematical prerequisites for the course
  •    Extra
    • 1.3A (Extra): Complex numbers: a full lecture recap
  • 1.4 How to say names of few important (and one unimportant) scientists
  • 1.5 The Original Stern-Gerlach Experiment

• Lecture 2
  • 2.2 Four Experiments
  • 2.3 Experiments 3 & 4
  • 2.4 The Quantum State Vector
  • 2.5 Hilbert Space

• Lecture 3
  • 3.1 The Quantum State Vector
  • 3.2 Hilbert space
  • 3.3 The Quantum State Vector - Bra-ket algebra in z-basis
  •    Extra
    • 3.3A (Extra): bras and kets as complex vectors
  • 3.4 Example: a state in z-basis
  • 3.5 The Quantum State Vector: Expectation value; uncertainty
  • 3.6 Analysis of Experiment 3: |+x> state in z-basis
  • 3.7 Experiment 5: |+y> state in z-basis
  • 3.8 Summary: All states in z-basis

• Lecture 4
  • 4.1 Example: calculating expectation values for given |ψ>.
  • 4.2 Producing particle in the given state |ψ>.
  • 4.3 Example: calculating |ψ> from given expectation values.
  • 4.4 Matrix notation for state vector |ψ>.
  • 4.5 Operators, unitary operators.
  • 4.6 Generator of rotations operator

• Lecture 5
  • 5.1 Operators overview.
  • 5.2 Rotation by a finite angle in terms of ^J_z operator.
  • 5.3 Eigenvalues of ^J_z operator.
  • 5.4 Rotation of the arbitrary state about z axis.
  • 5.5 Hermitian operators correspond to physical experiment.
  •    Extra
    • 5.5A (Extra):Possible outcomes of the experiment represent eigenvalues of the corresponding Hermitian operator.
  • 5.6 Unit operator.
  • 5.7 Matrix representation of operators.

• Lecture 6
  • 6.1 Recap of Lecture 5.
  • 6.2 Example: Representations of an operator in different basis.
  • 6.3 Example: Matrix representation of a rotation operator.
  • 6.4 Bra-ket products and complex conjugation.
  • 6.5 Example: Hermitian operators.
  • 6.6 Switching bases for states.
  • 6.7 Switching bases for operators.
  • 6.8 Example: Eigenvalues of Hermitian operator.
  • 6.9 Example: Eigenstates of Hermitian operator.

HW #1:

• Lecture 7
  • 7.1 The Adjoint of the Product, Expectation Values.
  • 7.2 Sample Test: Problem 1.
  • 7.3 Problem 2: Three Stern-Gerlach apparatuses.
  • 7.4 Problem 3: Operator, eigenvalues, eigenstates.
  •    Extra
    • 7.4A (Extra): What have we learned that a quiz should test?
    • 7.4B (Extra): All basis transformations summarized
    • 7.4C (Extra): Why square roots? A particle in two states at once?
    • 7.4D (Extra): If you measure both, it's not a measurement?

HW #2:

• Lecture 8
  • 8.1 A finite 3D rotation.
  • 8.2 Infinitesimal rotations.
  • 8.3 Commutators.
  • 8.4 Commuting operators.
  • 8.5 Algebra of angular momenta.
  •    Extra
    • 8.6 (Extra): Eigen-values, -vectors workout.

• Lecture 9
  • 9.1 Overview of the previous lecture. Commuting Hermitian operators.
  • 9.2 Eigenstates of angular momentum.
  • 9.3 Raising and lowering operators.
  • 9.4 Eigenvalues of ^J² and ^J_z operators.
  • 9.5 Possible values for j.
  • 9.6 Fermions and bosons. Bose-Einstein condensation. Superfluidity and superconductivity.
  •    Extra
    • 9.6A (Extra): The Quantum Physics of Superconductivity! How does this MAGIC work?

• Lecture 10
  • 10.1 Recap of Lecture 9.
  •    Extra
    • 10.1A (Extra): Who ordered J₊,J_-?
  • 10.2 Example: spin 1/2.
  • 10.3 Example: spin 1.
  • 10.4 Example: spin 3/2.
  • 10.5 Raising, lowering operators ^J₊, ^J_-.
  • 10.6 Example: Calculations for spin 3 particle.
  • 10.7 Uncertainty principle.
  • 10.8 Uncertainty for angular momenta.
  • 10.9 Prequel: Heisenberg uncertainty.

• Lecture 11
  • 11.1 Orbital and spin angular momenta. Change in notation.
  • 11.2 Pauli spin matrices.
  • 11.3 ^S_n operator for spin-½ particle.
  • 11.4 |+n> state - an eigenstate of ^S_n operator for spin-½ particle.
  • 11.5 ^S_x, ^S_y, and ^S_z operators for spin-1 particle.
  • 11.6 Example: ^S_x, and ^S_z operators for spin-2 particle.

HW #3:

• Lecture 12
  • 12.1 Recap of the previous Lecture: ^S_x, ^S_y, and ^S_z operators for spin-1 particle.
  • 12.2 Eigenstates of ^S_x operator for spin-1 particle.
  • 12.3 Time evolution.
  • 12.4 Schrödinger equation.

• Lecture 13
  • 13.1 Recap of the previous lecture.
  • 13.2 Stationary states.
  • 13.3 Time evolution of expectation values.
  • 13.4 Algorithm for time evolution of the state for time-independent Hamiltonian.
  • 13.5 Example: evolution of spin-½ in magnetic field.
  • 13.6 Example: evolution of expectation values <S_x> and <S_y>.
  •    Extra
    • 13.7 (Extra): Classical Mechanics: precession of the angular momentum due to the gravitational field.

HW #4:

• Lecture 14
  • 14.1 Recap of the previous lecture: time evolution algorithm for time-independent Hamiltonian.
  • 14.2 Recap: spin-½ in magnetic field.
  • 14.3 Time-dependent magnetic field.
  • 14.4 Time-dependent magnetic field, approximate solution.
  • 14.5 Analysis of the approximate solutions.
  • 14.6 Rabi formula, resonance.
  • 14.7 Resonance width, time-energy uncertainty principle.
  •    Extra
    • 14.8 (Extra): Lasers and MRI.

• Lecture 15
  • 15.1A Problem 1: spin-2 particle expectation values.
  • 15.1  Problem 1: spin-3 particle expectation values.
  • 15.2  Problem 2: time evolution of a spin-1 particle.

HW #5:

• Lecture 16
  •      Dirac delta function
    • 16.1 Definition of Dirac delta function.
    •    Extra
      • 16.1A (Extra): Dirac delta functions? Who needs them?
    • 16.2 Step, or Heaviside function.
    •    Extra
      • 16.2A (Extra): Derivatives? Integrate by parts
    • 16.3 Sin(x)/x representation of the Dirac delta function.
    •    Extra
      • 16.3A (Extra): Many roads lead to the delta function
    • 16.4 Exponential representation of the Dirac delta function.
    •    Extra
      • 16.4A (Extra): This is really just Fourier analysis
      • 16.4B (Extra): Discrete Fourier transform
      • 16.4C (Extra): Fourier representation, circular Kronecker delta
  • 16.5 Position operator.
  •    Extra
    • 16.5A (Extra): Position operator looks weird
  • 16.6 Representing the state as a superposition of the position eigenstates.
  • 16.7 Probability amplitude (wave function) in the position representation.
  • 16.8 Transition amplitudes.
  • 16.9 Translation operator.
  •    Extra
    • 16.9A (Extra): Translation by +a, or by -a?
    • 16.9B (Extra): Where did time go?

• Lecture 17
  • 17.1 Recap of Lecture 16.
  • 17.2 Momentum is the generator of translations.
  • 17.3 Position-momentum commutator.
  • 17.4 Energy and momentum operators.
  • 17.5 Uncertainty principle.
  • 17.6 Momentum operator in x-basis.
  • 17.7 Momentum representation.
  • 17.8 Momentum eigenstate in position basis.
  • 17.9 De Broglie wavelength.
  • 17.10 Example: Double slit experiment.

• Lecture 18
  • 18.1 Overview of the previous Lecture. Wave–particle duality.
  • 18.2 Switching between the position and momentum bases. Fourier transform.
  •    Extra
    • 18.3A (Extra): Introductory Physics is the greatest con ever.
  • 18.3 Gaussian integrals.
  • 18.4 Gaussian wave packet in position space.
  • 18.5 Gaussian wave packet in momentum space.
  • 18.6 Time evolution of the Gaussian wave packet in free space.
  • 18.7 Numerical algorithm for evolution of the wave function. Evolution of non-Gaussian wave packets. Evolution with non-constant potential – scattering and tunneling.

• Lecture 19
  • 19.1 Time-dependent Schrodinger equation in position space.
  • 19.2 Time-independent Schrodinger equation in position space.
  • 19.3 Properties of solutions to the Schrodinger equation.
  • 19.4 Example: solution of Schrodinger equation for piecewise potential energy – particle in the finite quantum well.
  • 19.5 Quantum particle in the classically forbidden region. Tunneling.
  • 19.6 Matching wave function.

HW #6:

• Lecture 20
  • 20.1 Recap of the previous lecture.
  • 20.2 Wave function is zero on any finite interval where potential energy is infinite.
  • 20.3 Energy spectrum for the particle in the infinite quantum well.
  • 20.4 Wave functions for the particle in the infinite quantum well.
  •    Extra
    • 20.4A (Extra): Ground state vs. classical intuition. WKB.
  • 20.5 Example 1: rapid change in potential energy.
  • 20.6 Example 2: energy measurement.
  • 20.7 Example 3: time evolution of the wave function.
  • 20.8 Numerical evolution of the state of the particle in the infinite quantum well.
  • 20.9 Example 4: delta function potential energy.

HW #7:

• Lecture 21
  • 21.1 Recap of two previous lectures.
  • 21.2 The importance of the harmonic oscillator. Classical oscillator.
  • 21.3 Hamiltonian for the harmonic oscillator. ^a and ^a⁺ operators and their commutator.
  • 21.4 Hamiltonian for the harmonic oscillator in terms of ^a and ^a⁺ operators.
  • 21.5 Physical meaning of ^a and ^a⁺ operators: lowering and raising operators.
  • 21.6 Spectrum of allowable energies for particle in harmonic potential.
  • 21.7 More on energy spectrum of quantum harmonic oscillator.
  • 21.8 More on the ground state.
  • 21.9 Matrix elements of ^a and ^a⁺ operators.

• Lecture 22
  • 22.1 Recap of the previous lecture.
  • 22.2 Harmonic potential: ground state wave function.
  • 22.3 Harmonic potential: wave functions for excited states.
  • 22.4 Properties of eigenfunctions in harmonic potential.
  • 22.5 Example 1 (harmonic potential): calculation of uncertainties for the particle in state |n>.
  • 22.6 Example 2 (harmonic potential): particle in the superposition of eigenstates. Time evolution and use of ^a and ^a⁺ operators.
  • 22.7 Particle confined in arbitrary potential: numerical calculation of energy spectrum and eigenfunctions. Properties of eigenfunctions.

HW #8:

• Lecture 23
  • 23.1 What will be covered in the test.
  • 23.2 Problem 1. Introduction.
  • 23.3 Interval 2: The well.
  • 23.4 Interval 1: Infinite potential to the left.
  • 23.5 Interval 3: Flat potential to the right.
  • 23.6 Match the left wall, intervals 1 and 2.
  • 23.7 Match the right wall Dirac delta. The ground state energy.
  • 23.8 Problem 1 b) The shape of the ground state wave function.
  • 23.9 Problem 2 a) Harmonic oscillator, a <p> expectation value.
  • 23.10 Problem 2 b) Decode this!
  • 23.11 Problem 2 c), and what problems to review.

HW #9:

• Quiz 3 Problem 1b

• Lecture 24
  • 24.1 Continuous energy spectrum for unconstrained particle.
  • 24.2 The rate of change of the probability density.
  •    Extra
    • 24.2A (Extra): Continuity equation.
  • 24.3 The probability current.
  • 24.4 A potential step.
  • 24.5 Probabilities of transmission and reflection.
  • 24.6 Quantum vs classical over-the-barrier scattering.
  • 24.7 Scattering off step-down potential.

• Lecture 25
  • 25.1 Recap of the previous Lecture.
  •    Extra
    • 25.1A (Extra): Bound states vs. scattering.
    • 25.2A (Extra): Plane wave in space is like eigenstate in time.
  • 25.2 Tunneling through rectangular potential barrier.
  • 25.3 Tunneling examples: numerical simulations.
  • 25.4 Tunneling - underlying mechanism of many physical processes and technologies. Alpha radioactive decay. Tunneling microscopy.
  • 25.5 Over-the-barrier scattering.
  • 25.6 Scattering from delta-function barrier.
  • 25.7 Concluding remarks.

• Lecture 26
  • 26.1 Final practice problem 1 a) Eigenstates.
  • 26.2 Problem 1 a) Time evolution. b) Measurement.
  • 26.3 Alternative way to specify initial state – providing outcome of particular measurement.
  • 26.4 Problem 2 a) Infinite potential well. Wave function normalization.
  • 26.5 Problem 2 b) Time evolution.
  • 26.6 Bonus problem 2 c) Expectation value of the energy.
  • 26.7 Problem 3 a) A harmonic oscillator.
  • 26.8 Bonus problem 3 b) Expectation value in x-representation.
  • 26.9 Bonus problem 3 c) The expectation value integral.
  • 26.10 Problem 4. Step function potential with an attractive delta.
  • 26.11 Problem 4. Left, right potential steps.
  • 26.12 Problem 4. Delta function contribution.
  • 26.13 Problem 4. Matching across delta function.

HW #10: