II. The Formalism of Quantum Mechanics
Grifths chapter 3
A. Dirac Notation
dr1 (r)2 (r)
(single particle case)
Makes mathematical manipulations easier.
Independent of basis.
Bars denote Fourier transforms of .
1. (a) The de Broglie wavelength for any particle is
For the free particle with a periodic wavefunction, the largest wavelength it can
obtain is L. Smaller wavelengths are also allowed, with value L/n, th
Due: Tuesday, October 28, 2014 (in class)
1. (a) Find r2 , x and x2 for the ground state of Hydrogen. Hint: recall that
r2 = x2 + y 2 + z 2 and use the fact that the ground state is spherically symmetric.
(b) Normalize the following
Due: Friday, September 19, 2014 (in class)
1. 1 (x, t) and 2 (x, t) are (non-normalized) one-dimensional wavefunctions that each
separately satisfy the Schrodinger equation.
(a) Prove that the superposition wavefunction 3 (x, t), de
Angular Momentum, Perturbation theory. Due Oct 19,
2012. No Lates will be accepted.
a) Prove the following commutators:
L2 , Lz = 0
[Lx , Ly ] = ihLz ,
where Li is the ith component of the angular momentum operator L = r p and L2 = L L
2D and 3D Harmonic Oscillator. Due Oct 12, 2012.
Consider the following initial states of the symmetric, i.e. x = y , 2D harmonic oscillator,
= (|0 |1 + |1 |0 )
|2 = (|0 |0 + |1 |0 ) .
Calculate x(t) for each state. Comment on any dierence
2 Linear Algebra. Due Sept. 28, 2012.
Consider the matrix,
M = sin cos x + sin sin y + cos z ,
where i are the 2 2 Pauli matrices.
a) Verify that M is hermitian.
b) Find the eigenvalues and normalized eigenvectors of M . Are the eigenvalues
1 Review of QM. Due Sept 21, 2012.
Are the following statements true or false? Explain your reasoning with an explicit example or
1) Every wavefunction must satisfy the time-independent Schroedinger equation.
2) The expectation val
VIII. The Variational
Griffiths Chapter 7
A. The Variational Principle
The variational principle is another
approximation method for quantum
Perturbation theory allowed us to find
approximate wave functions and energies
VII. Time Independent
Griths Chapter 6
A. Nondegenerate Perturbation Theory
Suppose we have solved the time independent
Schrdinger equation for some potential.
H 0 |n0 = En |n0
Therefore we know the eigenvalues En0
VI. IDENTICAL PARTICLES
Griffiths Chapter 5, Shankar Chapter 10
Two particles are said to be identical if they are exact replicas of
each other in every respect.
There should be no experiment that can detect any intrinsic
Grifths Chapter 4
A. Orbital Angular Momentum
The principal quantum number n of the
hydrogen atom gives the energy of the state.
The azimuthal and magnetic quantum numbers l
and m are related to the orbital angular
IV. Solving the Time
Grifths Chapters 2 and 4
A. The Operator Approach to the Harmonic Oscillator
The harmonic oscillator is represented by the force
equation F = -kx.
This gives a potential energy of V(x) = -
III. The Postulates of
Quoted from Bransden and Joachain
A. The Double Slit Experiment
A.1 Postulate #1
To an ensemble of physical systems one can, in
certain cases, associate a wave function or state
function which contains all the inf
Due: Friday, October 3, 2014 (in class)
1. For the quantum Harmonic oscillator:
(a) Find the normalized groundstate 0 (x) and rst-excited state 1 (x) wavefunction
in the position representation. (Hint: begin by applying the operator