Physics 48 (Quantum Mechanics) Home Page, Spring 2009

Physics 48: Quantum Mechanics


Welcome to the new semester!


Course Information

Course Catalog Description:

Wave-particle duality and the Heisenberg uncertainty principle. Basic postulates of Quantum Mechanics, wave functions, solutions of the Schroedinger equation for one-dimensional systems and for the hydrogen atom. Three class hours per week. Requisite: Physics 25 and Physics 43 or consent of the instructor.


Times and places:


Physics 25 and 43 or consent of the instructor

Course requirements

Statement of Intellectual Responsibility: particulars for this course

How to get the most from this class:



Required (I've confirmed that it is available at the Jeffrey Amherst bookstore): Additional useful references:

On mechanics in general: Math books:

Mathematica Tutorials

We may use Mathematica in the homework, to obtain numerical solutions to problems that are not analytically solvable and to simplify plotting of results. If you've never used Mathematica before, or haven't used it much, the tutorials will help you get started. They were written by Professor Emeritus Bob Hilborn and revised by Rebecca Erwin '02. If you download the file and save it to the desktop with a .nb suffix in the name, your computer will recognize it as a Mathematica notebook and will start up Mathematica automatically when you double-click on the icon, provided you have Mathematica installed. Mathematica is installed on lots of the college's public machines, including on the computers in the Physics Department computer lab. Alternately, you can pay the $140 or so to buy the student version.

Lecture Schedule
Week Notes Hmwk Other
1. January 26, 2009
Anticipating Quantum Mechanics

Jan 26: The Blackbody Radiation problem I: setting up the problem

Course logistics. Crises in classical physics at the end of the 19th century. Blackbody radiation problem: problem statement, experimental results, work out expression for the energy stored in standing waves of an EM cavity in 1D and 3D.

Jan 28: The Blackbody Radiation problem II: Ultraviolet Catastrophe

Obtain expression for intensity of light as a function of frequency. Raleigh-Jeans formula from classical statistical mechanics (using Boltzmann distribution). Result disagrees with experiment, and is in fact divergent! Planck's suggested solution: energy in each mode is quantized in units of hf. Solution fits experiment well!

Jan 30: More problems for classical physics

Photoelectric effect: classical mechanics can't explain experiments; Einstein's solution: light comes in discrete chunks of energy hf--"photons." Compton effect: take photons seriously as particles--they have energy and momentum; in photon-electron scattering, light wavelength shift successfully predicted by applying (relativistic) conservation of energy and momentum. DeBroglie wavelength: matter behaves as waves, with wavelength inversely proportional to momentum; Davisson and Germer observe interference patterns in electrons. Planetary model of an atom: problem--Nagaoka model supported by Rutherford scattering, but classically unstable. Spectroscopy: hydrogen atom radiates only at particular frequencies, a feature not explained by classical physics.

Read: Griffiths, Chap. 1

Problems: [Due Wednesday 2/4, 11:59 pm]
Chap. 1: 1.3, 1.5, 1.9, 1.14, 1.15, 1.18
2. February 2, 2009
Overview of Modern Quantum Mechanics

Feb 2: Bohr's atom / The Wavefunction in Quantum Mechanics

Wrapping up the historical prelude: Bohr's atom. Bohr postulates angular momentum quantized in multiples of hbar. Interpreting (discrete) energy level differences as photon energies gives correct pattern of spectral lines. Interpret electrons in orbitals as standing DeBroglie matter waves. Wavefunctions in modern quantum mechanics: Schrodinger equation governs the evolution of the wavefunction in QM, as Newton II governs particle trajectory in classical mechanics. But what does the wavefunction describe? Born's statistical interpretation: amplitude square gives probability density for finding particles at a place, time. Interpretations of QM: "realist," "orthodox (Copenhagen)," "agnostic." Bell+experiment decides in favor of Copenhagen. Measurements are funny processes: they collapse the wavefunction. Probability recap: probabilities and expectation values of discrete variables.

Feb 4: Normalization of the Wavefunction

More probability: define the variance, and observe that it's zero only if there's no spread. Extend probability defs to continuous variables. Normalization of the wfn: needed for probabilistic interpretation of the wfn. Normalization must also be maintained under time evolution by Schrodinger equation, as we demonstrate explicitly.

Feb 6: The Momentum Operator

Calculate time derivative of expectation value of position, assert that this is the expectation value of velocity. Define expectation of momentum as p=mv. Implies momentum is represented by a derivative operator. Assert that this allows us to calculate expectation of any classical dynamical variable, but substituting classical variables x,p with operators. Solving time-independent Schrodinger equation: outline our approach, using separation of variables.

Read: Griffiths, Chap. 1, start Chap. 2

Problems: [Due Tuesday 2/10, 11:59 pm]
Chap. 2: 1.16, 2.1, 2.2, 2.4, 2.5
Supplemental Reading: The first part of Planck, Photon Statistics, and Bose-Einstein Condensation by Daniel M. Greenberg, et. al. gives you an idea of what Planck did, and did not, believe about quantization.
3. February 9, 2009
Solving the Schrodinger Equation

Feb 9: Separation of Variables and Stationary States

Guess a solution that's a product f(x) g(t), reduce SE to form F(x)=E=G(t). Solve time-dependent ODE in terms of E (easy); can't reduce TISE further without form of potential. Solutions of this form are stationary states: in these states, observables have constant expectation value. Stationary states are energy eigenstates (definite energy values). Linear superpositions of separable solution are also solutions (demonstrate in simple case). Conversely, all solutions can be written as linear superpositions (possibly infinite sum) of stationary states (by completeness relation).

Feb 11: Infinite Square Well I

In stationary state decomposition of general solution to Schrodinger equation, coefficients of stationary states can be fixed by initial conditions. Infinite square well: outside well wfn=0; inside well it's just the SHO ODE (soln: sin and cos). Boundary conditions: wfn vanishes at edge of well. This is equivalent to standing wave condition on a finite string with fixed ends--can only fit integer number of half-wavelengths in the well. Consequence: wavenumber, and hence energy, is quantized. Normalized wfn. Observe: (1) wfns have definite parity symmetry, (2) increasing energy <-> more nodes, (3) energy eigenstates are orthonormal.

Feb 13: Infinte Square Well II / Simple Harmonic Oscillator I

ISW: Energy eigenstates of ISW are complete. More nodes -> wigglier wfn -> more curvature -> larger KE. Energy eigenstates of ISW are orthonormal basis vectors in an infinite dimensional vector space in which functions f(x) are general vectors, integration gives the inner product. Show the correspondence between this vector space and familiar 3-d space vectors. Write stationary states and general solution to full Schrodinger equation (superposition of stationary states). Coeff fixed by initial wfn. Conservation of probability implies sum of (coeff)^2 =1. Energy expectation value is constant in time. SHO: factorization method--try to write Hamiltonian as product of two operators linear in x, p.

Read: Griffiths, Chap. 2

Problems: [Due Tuesday 2/17, 11:59 pm]
Chap. 2: 2.7, 2.10, 2.11, 2.17, 2.38, 2.39
Supplemental reading: The time evolution of wavefunctions in an infinite square well is interesting and pretty, although we won't have much time to spend discussing it. Here are a few articles that tell (and show) you something about it:
4. February 16, 2009
Simple Harmonic Oscillator

Feb 16: Simple Harmonic Oscillator II: factorization of H

Define ladder operators. Expand products of ladder operators. Define commutator of operators, work out canonical commutation relations, commutators of ladder operators. Write H in terms of ladder ops. Demonstrate that ladder operators acting on eigenstate of H give another eigenstate with energy eigenvalue higher or lower by discrete amount. Can operate repeatedly with these ops to get a ladder of states that has no top rung. Bottom rung: normalizable E<0 states don't exist, so there will be a lowest normalizable state.

Feb 18: Simple Harmonic Oscillator III: explicit solutions using ladder operators

Lowest rung of ladder: lowering operator acting on ground state gives zero. Defines a 1st order ODE which we can solve explicitly. Normalize ground state and find its energy. To get excited states, apply raising operator repeatedly. Terminology: define Heisenberg algebra, Heisenberg-Weyl algebra. Define hermitian conjugate of an operator, show that h.c. of d/dx is -d/dx (using integration by parts). x operator is its own h.c. Show that raising and lowering operators are hermitian conjugates of each other.

Feb 20: Simple Harmonic Oscillator IV: more with ladders / series solution

Brief recap of hermitian conjugates. x, p are their own hermitian conjugates. Self-conjugate operators will have real eigenvalues, can represent physical observables. Comlete result for action of raising and lowering operators on energy eigenstates. Work out normalization of energy eigenstates obtained by repeated operation of raising operator. Energy eigenstates are orthonormal. Frobenius method: write TISE in terms of dimensionless variables. Peel off large-x behavior of wfn. Guess form for solution: h[xi] exp[-xi^2/2], plug into TISE, get ODE for h[xi] as new eqn equivalent to TISE. Power series expand h[xi] about xi=0.

Read: Griffiths, Chap. 2

Problems: [Due Tuesday 2/24/09, 11:59 pm]
Chap. 2: 2.12, 2.14, 2.15, 2.42, 2.45, 2.54
Supplemental reading: The Restless Harmonic Oscillator is a simple little paper that considers the small fluctuations of a harmonic oscillator due to thermal, quantum, and gravitational effects. Interesting when considered in the context of modern facilities such as LIGO, and perhaps in the context of Jared Herzberg's colloquium in a few weeks as well.
5. February 23, 2009
More simple potentials in 1D

Feb 23: Simple Harmonic Oscillator V: series solution

Obtain recurrence relation for coefficients in series expansion of h by plugging series expansion rep of h into TISE. Series starts with two base coefficients, a0 and a1, from which all others are determined. h naturally breaks into even and odd series, each determined by one of the base coeffs. Look at large-order of series (and thus the large-xi behavior of h): grows as exp[xi^2], so h diverges as exp[xi^2] at large xi and solution to TISE diverges as exp[xi^2/2] at large xi--not normalizable! Only cure: series must terminate so that the large-order terms never appear.

Feb 25: SHO VI: series solutions / Free particle

SHO: Terminating series for SHO solution gives quantized energies. Wfns now have a polynomial factor proportional to the Hermite polynomials. Write out explicit solutions. Note that wfn is nonzero outside classical turning points. Free particle: SE has same form as ISW, but without the boundary conditions. Write TISE solns as exponentials. Stationary state solution has traveling wave form, with left- and right-movers of same functional form. Condense notation by associating negative k with left-movers. Observe phase velocity is 1/2 classical particle velocity. Observe that the traveling wave stationary states we obtained are not normalizable. Free particles do not have definite energy. General soln to Schrodinger equation is still superposition of stationary states, and we can form wavepacket from these stationary states which are normalizable.

Feb 27: Free particle / Delta function potential

Free particle: Wavepacket--superposition of non-normalizable separable solutions of free particle SE. Determine coefficient function from initial wfn. Coefficient function is the Fourier transform of the t=0 SE. Phase velocity vs. group velocity. Define dispersion relation. Consider a wave packet with a coefficient function sharply peaked in k-space about some k0. Expand w(k) about k0, keep only linear order. Approximate wfn is a phase factor times a traveling wave that moves with group velocity vg=dw/dk(k0), the speed of a classical particle which is twice the phase velocity at the peak. Delta function well: General discussion of bound states and scattering states in classical and quantum mechanics. Delta-well has both types.

Read: Griffiths, Chap. 2

Problems: [Due Tuesday March 3, 11:59 pm]
Chap. 2: 2.20, 2.22, 2.24, 2.26, 2.43, 2.49
Supplemental reading: Completeness of the energy eigenfunctions for the one-dimensional delta-function potential provides just what the title suggests. See also: Completeness of the energy eigenstates for a delta function potential.
6. March 2, 2009
More 1D problems

Mar 2: Delta function well

Bound and scattering states in QM and classical mechanics. Bound state solutions of delta fn well: Schrodinger eqn in x<0, x>0 regions, normalizable solutions. Boundary conditions: Use continuity of psi at x=0. Integrate Schrodinger equation to fix discontinuity in d(psi)/dx. The latter implies energy quantization: single bound state. Normalize bound state wavefunction. Scattering solutions of delta fn well: Write down general positive energy solutions in x<0, x>0 regions. Apply same boundary conditions at x=0 as in bound state calculation.

Mar 4: Delta function well / Finite square well

Scattering solutions of delta fn well: Boundary conditions. Interpret the terms as a scattering experiment: incident, transmitted, and reflected waves. Calculate transmission and reflection coefficients. Finite square well: Bound state solutions--write solutions to the TISE in three regions. Classify solutions as either even or odd parity. Choice to consider an even or odd solution automatically reduces the number of coefficients to fix.

Mar 6: Finite square well

Bound state solutions: consider only the even-parity states (which includes the ground state). Apply boundary conditions. Bound state solutions given by solutions to a transcendental equation. Demonstrate the graphical solution, which shows there's always one such solution regardless well depth. Examine cases of deep, wide well and shallow, narrow well. Cursory discussion of scattering solutions. Sketch of transmission coefficient as a function of energy. Ramsaur-Townsend effect: well is "transparent" at ISW energy levels.

Read: Griffiths, Chap. 2, start Chap. 3

Problems: [Due Tuesday March 9, 11:59 pm]
Chap. 2: 2.28, 2.34, 2.35, 2.47, 2.51, 2.52
Supplemental Reading: Heisenberg's Uncertainty Principle by P. Busch, T. Heinonen, P. Lahti gives more on the uncertainty principle.
7. March 9, 2009
Quantum mechanics in Hilbert space

Mar 9: Linear algebra on Hilbert space

Wave functions are vectors, operators are linear transformations. Language of QM is linear algbera. Recap of vectors, inner products, and linear transformations on finite-dim spaces. On such spaces, inner product exists. In QM, space may be infinite-dim: inner product may not exist. Set of square-integrable (L2) functions is a vector space. Add an inner product and its an inner product space. In fact, it's a complete inner product space (a Hilbert space). Inner product on L2 is finite by Schwarz inequality. Normalization, orthogonality, complete sets of orthonormal functions. Review def of Hermitian conjugate and Hermitian matrix from finite-dim linear algebra, define Hermitian conjugate of linear trans more generally.

Mar 11: Observables and hermitian transformations

Hermitian matrices in finite dimensions: (1) eigenvalues are real, (2) eigenvectors of Hermitian transformation belonging to distinct eigenvalues are othogonal, (3) eigenvectors of a Hermitian transformation span the space. Hermitian operators in QM: that outcomes of measurements are real numbers implies the operators representing observables are Hermitian. Determinate states of an observable are eigenvectors of the corresponding operator.

Mar 13: Eigenfunctions of hermitian operators

Two categories: (1) discrete spectrum: eigenfunctions are normalizable, live in Hilbert space, can represent physical states; (2) continuous spectrum: eigenfunctions no normalizable, do not represent physical states (superpositions of them do). Discrete spectrum (like finite-dim): (1) eigenfunctions have real eigenvalues, (2) eignefunctions belonging to discrete eigenvalues are othogonal (deal with degenerate eigenvalues using Gram-Schmidt), (3) eigenfunctions are complete. Continuous spectrum: eigenfunctions with real eigenvalues are Dirac orthonormalizable and complete. Examples: momentum operator, position operator. Generalized statistical interpretation: measuring an observable is sure to yield an eigenvalue of the corresponding operator. Probability of getting an outcome is equal to the amplitude squared of the inner product of the corresponding eigenfunction with the state (slightly modified for continuous spectrum case).

Read: Griffiths, Chap. 3 and Appendix A

Problems: [Due Tuesday March 24, 11:59 pm]
App. A: A.2, A.5, A.22, A.23, A.26, A.28
Midterm exam 1: Wed. March 11, 7-10 pm. Bring your own equations sheet (1 sheet, 8.5" x 11", both sides ).
8. March 23, 2009
More abstract QM

Mar 23: Uncertainty principle generalized

Generalized statistical interpretation for discrete or continuous case. Derivation of the generalized uncertainty principle. Compatible and incompatible operators.

Mar 25: More on the uncertainty principle

How the uncertainty principle affects sequential position and momentum measurements. Minimum uncertainty wavepackets are Gaussian. Energy-time uncertainty principle discussed briefly (distinction from the position-momentum uncertainty principle highlighted). Recap of vectors in Hilbert space, with an emphasis on the geometrical interpretation (basis-independence) of the vector.

Mar 27: Dirac notation / Two-state system

Introduction to Dirac notation: Dirac notation simplifies the change of bases that allow one to talk about the same vector in terms of position, momentum, or energy eigenstate bases. Operators are linear transformations that take one vector into another. Can be represented by matrices with respect to a particular basis. Two-state system: Start with two-dimensional Hilbert space and some basis of vectors. General vector is superposition of these basis vectors. Hamiltonian as 2x2 matrix. How does vector evolve in time from some initial time? Find stationary states (eigenstates of H), add time-dependent phase factors, fix coefficients from initial conditions. An initial state, say one of the original basis states, that's not an eigenstate of H will over time rotate into the other basis state and back. Time scale is set by energy splitting between states. It's a type of beat phenomena.

Read: Griffiths, Chap. 3

Problems: [Due Tuesday March 31, 11:59 pm]
Chap. 3: 3.13, 3.17, 3.27, 3.38, 3.39, 3.40
9. March 30, 2009
Quantum mechanics in 3D

Mar 30: Projection operators / Separation of variables

Projection operators as the "outer product" of a pair of vectors in Hilbert space. Resolutions of unity. Spectral decomposition of a general operator. QM in 3D: Generalizing momentum operators, Schrodinger euqation, normalization conditions, etc. to 3D. Specialize to spherically symmetric potentials. Write TISE in spherical coords and look for solutions separable into radial and angular variables.

Apr 1: The angular equation and the radial equation

Angular equation: Separate angular equation into theta and phi equations by separation of variables. Phi equation is easy to solve. If we demand singlevalued wavefunctions, Phi equation forces separation constant m to be an integer. Theta equation is messier: solution in terms of associated Legendre functions. Normalizability requires separation const. l be a positive integer, m an integer between -l and l (inclusive). Normalized solutions to angular equations are the spherical harmonics. They are orthonormal. Radial equation: Change variable to transform original radial equation into something like a 1D Schrodinger equation, with an effective potential in place of the original potential.

Apr 3: Soving the hydrogen atom

Solve the radial equation for the hydrogen atom: convert to dimensionaless variables, peel off asymptotic behavior at r->0 and infinity, solve remaining equation using series expansion. Find a two-terms recurrence relation. Find series must terminate if resulting solution is to be normalizable. Termination condition gives energy quantization. Write out explicit wfn for ground state (n=1), first excited state (n=2).

Read: Griffiths, Chap. 4

Problems: [Due Tuesday April 7, 11:59 pm]
Chap. 4: 4.1, 4.2, 4.9, 4.16, 4.38, 4.39
10. April 6, 2009
More QM in 3D

Apr 6: Hydrogen atom / Angular momentum

Begin with discussion of exam 1. More hydrogen atom: Degeneracy for general n is n^2. Write out solution to radial equation for general n in terms of associated Laguerre polynomials. Write out full, normalized hydrogen atom wfn for general (n,l,m). Rydberg constant and Rydberg formula. Angular momentum: Define orbital angular momentum operators. Cartesian component angular momentum operators are not compatible. Work out commutation relations. L^2, the squared angular momentum operator, does commute with the cartesian component angular momentum operators. Can find simultaneous eigenstates of both L^2 and one of the components, typically take as L_z.

Apr. 8: Angular momentum using ladder operators

Define ladder operators, show that they raise and lower the L_z eigenvalue by one unit of hbar when they act on an eigenstate of (H, L^2, L_z). Use ladder operators, and the fact that there must be a largest eigenvalue of L_z for a given eigenvalue of L^2, to deduce the spectrum of L^2 and L_z operators. Eigenfunctions will be spherical harmonics.

Apr 10: Orbital angular momentum eigenfunctions and Spin

Orbital angular momentum: Write out the angular momentum operator in its position space representation. Use this to write out position space representation of L^2 and L_z eigenvalue equations. These are identical to angular equations encountered in 3D spherical potential problem. Solutions were spherical harmonics, so the spherical harmonics are eigenfunctions of L^2 and L_z. Note that algebraic approach permits l and m to take on half-integer values, while discussion of orbital angular momentum based on single-valuedness of wfn did not. Spin: Electron has an intrinsic angular momentum about its center of mass, apparently not related to motion of charge and of a fixed total S^2 value for all electrons. Algebraic theory of spin like that of orbital angular momentum, but the eigenvectors aren't functions in position space so aren't representable as spherical harmonics--hence no reason to exclude half-integer values. Electron is spin-1/2: two-dimensional Hilbert space. Work out S^2, S_z, S_+/- in terms of Pauli matrices.

Read: Griffiths, Chap. 4

Problems: [Due Tuesday April 14, 11:59 pm]
Chap. 4: 4.19, 4.20, 4.28, 4.30, 4.53, 4.56
11. April 13, 2009
Angular momentum / Identical particles

Apr 13: More on spin

Results of measurements of S_z and S_x on a general state. Electron in a magnetic field: Hamiltonian, time evolution of general state, expectation of components of spin operator. Stern-Gerlach experiment.

Apr 15: Addition of angular momenta / identical particles

For system of two (distinguishable) spin-1/2 particles: write basis for the two-particle Hilbert space as product of single-particle basis states (Hilbert space is 4-dimensional). Calculate total S_z for system in these basis states: looks like a collection of s=1 and s=0 states. m=0 basis states are not eigenstates of S^2. Construct s=1 (triplet) states by acting on m=1 basis state with lowering operator; remaining orthonormal combination is s=0 (singlet) state. Check explicitly, by direct calculation, that states constructed this way are eigenstates of S^2. State general result for allowed values of s, m when adding angular momenta of two particles. Eigenfunctions of total S, S_z written as sum of products of single-particle basis states using Clebsch-Gordon coefficients. Identical particles: Generalize Schrodinger equation, generalized statistical interpretation, and normalization of wfn to two particle systems.

Apr 17: Systems of two identical particles

The world allows particles to be fundamentally indistinguishable. Indistinguishability is implemented in the multiparticle wfn by (1) symmetrizing (bosons) or (2) antisymmetrizing (fermions) products of distinguishable single-particle wfns. Fermions are half-integer spin, bosons are integer spin. Identical fermions cannot occupy same single-particle state (Pauli exclusion principle). Formulate behavior of wavefunction under exchange using the exchange operator, which commutes with Hamiltonian, and invoke symmetrization postulate. Exchange forces: calculate mean square distance between distinguishable particles, fermion, and bosons.

Read: Griffiths, Chap. 5

Problems: [Due Tuesday April 21, 11:59 pm]
Chap. 4: 4.33, 4.36, 4.51, Chap 5: 5.5, 5.13, 5.32
12. April 20, 2009
Identical particles

Apr 20: Atoms

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Apr 22: Solids: T=0 Fermi gas

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Apr 24: Solids: Bloch model

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Read: Griffiths, Chap. 5 and 6

Problems: [Due Thursday April 30, 11:59 pm]
Chap. 5: 5.16, 5.20, 5.29, 5.35, 5.36
13. April 27, 2009
Quantum statistical mechanics

Apr 27: Bloch model / quantum statistical mechanics

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Apr 29: Enumerating states of many-particle systems

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May 1: Quantum statistical mechanics / Non-degenerate perturbation theory

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Read: Griffiths, Chap. 6

Problems: [Due Friday May 8, 5:00 pm]
Chap. 6: 6.1, 6.5, 6.18, 6.23, 6.31, 6.36
14. May 4, 2009
Approximation methods: time-independent perturbation theory

May 4: Non-degenerate perturbation theory / degenerate perturbation theory

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May 6: Degenerate perturbation theory / fine structure of hydrogen atom

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May 8: Fine structure in hydrogen atom spectroscopy

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Read: none

Problems: none