Study guide for Physics 410 Final Exam

Dec 17, 2010 8 a.m.-10a.m. in P-149

Fine print: although this guide is intended to help you, it is not guaranteed to cover or prepare you for every question on the exam.

The structure of the exam will be very similar to previous exams. There will be emphasis on material covered since Exam II. Open books, open notes, no calculators.

- Interpretation of coordinate-space wavefunction ψ(x) as probability.

- Set up expectation values in one spatial dimension (x).

- Set up expectation values in spherical coordinates (r, θ,φ)

- The form of the momentum operator in coordinate space.

- Commutation relations.

- The uncertainty principle and what it means.

- The time-dependent Schrodinger equation in 1 dimension.

- The time-independent Schrodinger equation in 1 dimension, and how to derive it from the time-dependent Schrodinger equation.

- The time-independent Schrodinger equation as an eigenvalue-eigenfuction equation.

- The solutions (wavefunctions and eigenenergies) for the infinite square well.

- The energy spectrum of the 1-dim harmonic oscillator and how it depends on the principal quantum number N.

- The ground state wavefunction of the 1-dim harmonic oscillator.

- Nodal structure of one-dimension wavefunctions.

- Scattering states: transmission and reflection coefficients (derived by matching wfns at boundaries).

- Orthogonality of states and superposition of states.

- Expansion of a vector in an arbitrary basis; how to find the expansion coefficients.

For example, given an arbitrary function f(x) defined on an interval 0 < x < a, do you know how to expand in terms of the eigenstaets of the infinite square well?

- Properties of a Hermitian matrix (operator)

- similarity and unitary transformations

Do you know how to recognize a unitary matrix? When are you guaranteed a unitary transformation? Do you know the difference between a unitary matrix and a Hermitian matrix?

(over)

- “matrix elements” of an operator in a specified basis.

- General solutions of the time-dependent equation using the solutions of the time-independent equation.

- General statistical interpretation; “collapse” of the wavefunction and the probability to find particle in a measured eigenstate.

- examples in 2x2 matrices; finding eigenvalues; eigenvectors of a 2x2 matrix.

- Quantum numbers for orbital angular momentum, l and m: how to interpret them, and what values they can have.

- Spherical harmonics.

- the “radial Schrodinger equation”

- The Coulomb potential and the hydrogen atom:

Principal quantum number N and its relation to l, n (the radial nodal quantum number)

Cartoons of radial wfns

Dependence of bound state energies on N.

The Bohr radius.

Radial wavefunctions for the hydrogen atom.

- Extra credit: Variational theory