Physics 48 (Quantum Mechanics) Home Page, Spring 2010

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 ordered it through Amherst Bookstore, but they couldn't get it. Try online retailers): Additional useful references:

Other books on quantum 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 25, 2010
Anticipating Quantum Mechanics

Mon Jan 25 (11 am-12 pm): 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.

Wed Jan 27 (11 am -12 pm): 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!

Th Jan 28: 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. 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

Read: Griffiths, Chap. 1

Problems: none this week
2. February 1, 2010
Overview of Modern Quantum Mechanics

Feb 2: The Wavefunction and its normalization

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. 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

Feb 4: Normalization of the wavefunction / Operators and expectation values

Demonstrate that normalization is maintained under time evolution by Schrodinger equation: Schrodinger equation evolution conserves total probability. 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. 2

Problems: Due Thurs, 2/11/10 at 10 am.

Chap 1: 1.3, 1.5, 1.7, 1.9, 1.14, 1.15, 1.16, 1.18
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 8, 2010
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). In stationary state decomposition of general solution to Schrodinger equation, coefficients of stationary states can be fixed by initial conditions.

Feb 11: 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. 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.

Read: Griffiths, Chap. 2, first half

Problems: [Due Wed. Feb. 17, 11:59 pm]

Chap. 2, 2.1, 2.2, 2.4, 2.5, 2.7, 2.10, 2.11, 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 15, 2010
Simple Harmonic Oscillator

Feb 16: ISW / Simple Harmonic Oscillator I: factorization of H

ISW: 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. 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 II: 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. Define hermitian conjugate of an operator, show that x operator is its own h.c., h.c. of d/dx is -d/dx (using integration by parts), h.c. of a complex number is the complex conjugate, and h.c. of p is p. Show that raising and lowering operators are hermitian conjugates of each other. Self-conjugate operators will be shownn to 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.

Read: Griffiths, Chap. 2

[Due Wed. February 24, 11:59 pm]

Chap. 2: 2.12, 2.14, 2.15, 2.17, 2.36, 2.41, 2.42, 2.45
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.
5. February 22, 2010
More simple potentials in 1D

Feb 23: Simple Harmonic Oscillator III: series solution [Jagu guest lecture]

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. 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. 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.

Feb 25: Free particle

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. 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.

Read: Griffiths, Chap. 2

Problems: [Due Wed. March 10, 11:59 pm]
Chap. 2: 2.20, 2.22, 2.24, 2.26, 2.34, 2.35, 2.43, 2.47, 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 1, 2010
More 1D problems

Mar 2: Delta function well

General discussion of bound states and scattering states in classical and quantum mechanics. Delta-well has both types. 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. Interpret the terms as a scattering experiment: incident, transmitted, and reflected waves.

Mar 4: Finite square well

Scattering solutions of delta fn well: Complete calculation of 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. 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.

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

Problems: carried over from last week
Midterm exam 1: Wed. March 3, 7-10 pm. Bring your own equations sheet (1 sheet, 8.5" x 11", both sides ).

Supplemental Reading: Heisenberg's Uncertainty Principle by P. Busch, T. Heinonen, P. Lahti gives more on the uncertainty principle.
7. March 8, 2010
Quantum mechanics in Hilbert space

Mar 9: Finite square well / Linear algebra on Hilbert space

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. Abstract quantum mechanics: 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.

Mar 11: Observables and hermitian transformations

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. 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.

Read: Griffiths, Chap. 3 and Appendix A

Problems: [Due Fri. March 26, 11:59 pm]
Chap. 2: 2.51, 2.52, 3.2; App. A, A.2, A.5, A.22, A.23, A.26, A.28
Supplemental reading:

R. Blumel presents a series solution to the finite square well in Analytical solution of the finite quantum square-well problem.

Griffiths goes into more detail about the square well in Exact and approximate energy spectrum for the finite square well and related potentials.

The square well is used by Sprung, et. al. to illustrate the use of the S-matrix in scattering in Poles, bound states, and resonances illustrated by the square well potential.

8. March 22, 2010
More abstract QM

Mar 23: Hermitian operators: eigenfunctions and eigenvalues

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). Wavefunction collapses into eigenstate after measurement. Can expand general state in terms of eigenfunctions of any hermitian operators. Probability of measuring particular eigenvalue corresponds to expansion coefficient squared.

Mar 25: Uncertainty principle

Expanding wavefunctions in terms of eigenfunctions of a general hermitian operator. Derivation of the generalized uncertainty principle. Compatible and incompatible operators. 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).

Read: Griffiths,

Problems: [due Friday April 2, 11:59 pm]
Chap. 3: 3.6, 3.12, 3.13, 3.17, 3.27, 3.38, 3.39, 3.40
Supplemental reading:

9. March 29, 2010
Dirac notation / two state systems

Mar 30: Dirac notation / two state systems

Recap of vectors in Hilbert space, with an emphasis on the geometrical interpretation (basis-independence) of the vector. 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.

Apr 2: External review committee visits: class cancelled

Read: Griffiths, Chap. 4

Problems: [due Friday April 9, 11:59 pm]
Chap. 4: 4.1, 4.2, 4.5, 4.9, 4.14, 4.16, 4.38, 4.39
Supplemental reading:

10. April 5, 2010
QM in 3D

Apr 6: Projection operators / QM in 3D: 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. 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).

Apr 8: Hydrogen atom

Angular equation: 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. 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.

Read: Griffiths, Chap. 4

Problems: [due Friday April 23, 11:59 pm]
Chap. 4: 4.19, 4.20, 4.28, 4.30, 4.36, 4.51, 4.53, 4.56
11. April 12, 2010
Angular momentum and spin

Apr 13: Hydrogen atom / Angular momentum

Hydrogen atom:Write out wavefunction for first excited state (n=2). 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. 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 relate eigenvalue of L^2 to the largest L_z eigenvalue (top rung of ladder).

Apr 15: Angular momentum: orbital and spin

Use similar argument to relate lowest L_z eigenvalue (bottom rung of ladder) to eigenvalue of L^2. With these, we can deduce the full spectrum of the angular momentum operators L^2 and L_z. Looks like too many states--half-integer m states appear that we did not permit in our explicit solution of TISE because of single-valued wavefunction. 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. Results of measurements of S_z on a general state.

Read: Griffiths, Chap. 4

Problems: [due Friday April 23, 11:59 pm]

(problems rolled over from last week)
Midterm exam 2: Wed. Apr. 14, 7-10 pm. Bring your own equations sheet (1 sheet, 8.5" x 11", both sides ).
12. April 19, 2010
Angular momentum II

Apr 20: Experiments involving angular momentum

Results of measurements of S_z and S_x on a general state (demonstrating that we have not "privileged" S_z by choosing that as one of the operators we're diagonalizing in the basis we've chosen). Electron in a magnetic field: Hamiltonian, time evolution of general state, expectation of components of spin operator. Stern-Gerlach experiment. addition of angular momentum: 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.

Apr 22: Addition of angular momentum / identical particles

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. The world allows particles to be fundamentally indistinguishable in a way that has no counterpart in classical mechanics.

Read: Griffiths, Chap. 5

Problems: [Due Friday April 30, 11:59 pm]
Chap. 5: 5.1, 5.5, 5.13, 5.16, 5.20, 5.32, 5.35, 5.36
13. April 26, 2010
Identical particles

Apr 27: Systems of two identical particles / atoms

lecture contents

Apr 29: Solids / quantum statistical mechanics

lecture contents

Read: Griffiths, Chap.

Problems: [Due ]
14. May 3, 2010
Approximation methods: time-independent perturbation theory

May 3: Non-degenerate perturbation theory / degenerate perturbation theory

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May 5: Degenerate perturbation theory / fine structure of hydrogen atom

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May 7: Fine structure in hydrogen atom spectroscopy

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Read: none

Problems: none