[6 pts] Calculate the product xp of the position and momentum uncertainties for the ground state
0 and for the rst excited state 1 of the harmonic oscillator. It may be useful to recall the denitions
of the position and momentum operators in terms of ^a and ^a+:
(^a + ^a+); ^p = i
(^a ^a+): (1)
2 What is Momentum Anyway?
[10 pts] Prove that
Hint: it is helpful to recall the following formulas:
@x2 + V (x) ; (2)
h^ Qi =
(x; t)^Q (x; t)dx; (3)
^p = ih
3 Double Delta Potential
[14 pts] Consider a particle of mass m in a potential given by
V (x) = [(x + a) + (x a)]; ; a > 0:
This system has two bound states for very large a, but only a single bound state for very small a.
- (2 pts) The stationary states in this potential may be taken to be either even or odd. Explain
why in one sentence. (Hint: either look at the form of V (x) or consider the parity operator.)
- (4 pts) Sketch the wave functions of the two bound states for very large a. Be sure to label
which of the two is the ground state and which is the excited state. Then sketch the wave
function of the single bound state for very small a.
For arbitrary a > 0, the bound state energies E are determined by the following transcendental
equation for the variable =
m 1; (x) even;
m ; (x) odd:
- (3 pts) Find the energies of the even and odd bound states in the limit as a goes to innity.
- (3 pts) Find the energy of the single bound state in the limit as a goes to zero.
- (2 pts) Estimate the value of a at which the system goes from having one bound states to two.
You are NOT expected to solve for a exactly.
4 Half Harmonic oscillator
[8 pts] Consider the half-harmonic oscillator potential (which represents, for example, a string which
can be stretched but not compressed), given by the potential
V (x) =
2m!2×2 for x > 0;
1 for x 0:
- (4 pts) What are the allowed energies? Explain your reasoning.
- (4 pts) What is the wave function for the ground state of the half-harmonic oscillator, written
as a function of x? Make sure your solution is normalized.
5 Two-Level System
[14 pts] Consider the photons and polarizers described in class, which is an instance of a two-level
system. Recall that the elements of the polarization vector correspond to probability amplitudes,
similar to the expansion coecients cn of wave functions.
- (3 pts) For a photon incident on a linear polarizer aligned in the ^x direction (meaning that it
transmits photons in the jxi-polarized state), what is the transmission probability if the initial
polarization vector of that photon is ~E = Ex^x + Ey ^y?
- (3 pts) A photon prepared in the jxi state is incident on a linear polarizer oriented at angle
relative to ^x. Solve for and sketch a graph of the probability of transmission (i.e. the probability
that the photon makes it through the polarizer) as a function of .
- (4 pts) As we showed in class, using the fjxi; jyig basis, an `x-polarizer’ transmits photons
polarized in the x-direction while re
ecting photons polarized in the y-direction. It makes a
measurement of the operator (expressed in the fjxi; jyig basis)
A polarizer rotated at +45 degrees relative to the ^x-axis measured the operator
What is the operator corresponding to a polarizer oriented at +30 degrees relative to the ^x-axis?
- (4 pts) Consider a particle prepared in the right-hand polarized state, j i = jRi = p1
(jxi + ijyi).
For times t < 0, the Hamiltonian for this particle is 0: in other words, it does not interact with
anything and nothing changes about its state. For times t 0, the following Hamiltonian is
; 0: (8)
Under the time evolution governed by ^H, is there ever a time t when the state becomes
[a] j (t)i = jLi = p1
(jxi ijyi)? Explain your reasoning. If yes, what is the rst time t
when this occurs?
[b] j (t)i = jxi? Explain your reasoning. If yes, what is the rst time t when this occurs?
6 Rigged Hilbert Space
[8 pts] Consider complex-valued functions f; g of a real variable x, with inner product dened as
1 f(x)g(x)dx. For the following functions, name the most restricted space to which they
belong: nuclear space, Hilbert space, extended space, or none of the above.
- (2 pts) f(x) = sin(x);
- (2 pts) f(x) = sin(x)ex2 ;
- (2 pts) f(x) = 1
- (2 pts) f(x) = ex2 .