Problem 1:

Let A and B be two observables of a system with a two dimensional state space, and suppose measurements are made of A, B, and A again in quick succession.  Show that the probability that the second measurement of A gives the same result as the first is independent of the initial state of the system.

Let {|a1>,|a2> } be an othonormal eigenbasis of A and let {|b1>,|ba> } be an othonormal eigenbasis of B.
A|ai> = ai|ai>,  B|bi> = bi|bi>.
|b1> = cos(θ)|a1> + sin(θ)e|a2>,  |b> = -sin(θ)|a1> + cos(θ)e|a2>
is the most general expansion of the |bi> in terms of the |ai>.
<b1|b2> = 0,  <b1|b1> = <b2|b2> = 1.
The initial state can be written as some linear combination of |a1> and a2>.
|Ψ> = c1|a1> + c2|a2>.


Problem 2:

Find <x> and ∆x for the nth stationary state of a free particle in one dimension restricted to the interval 0 < x < a.  Show that as n--> ∞ these become the classical values.

The eigenfunctions of the infinite well (U(x) = 0,  0 < x < L,  U(x) = ∞ everywhere else) are
ψn(x) = (2/a)1/2sin(nπx/a)  with eigenvalues  En = n2π2ħ2/(2ma2).
<x> = ∫0a dx Ψ*(x) x Ψ*(x)
<x> = (2/a∫0a dx  x sin2(nπx/L) = a/2.
Δx = <(<x2> - <x>2)>1/2.
<x2> = (2/a∫0a dx  x2 sin2(nπx/L) = a2/3 - a2/(2n2π2).
Δx2 = a2/3 - a2/4 - a2/(2n2π2) =  a2/12 - a2/(2n2π2).
Δx2 -->  a2/12  as n --> ∞.

Classically the particle bounces back and forth.  It spends the same amount of time at each position.
Therefore xavg = a/2 for a large number of particles with arbitrary initial positions.
x2rms = (1/a)∫0a dx (x - a/2)2 =(x3/3 - ax2/2 + a2x/4)|0a = a2/12.

Problem 3:

A system makes transitions between eigenstates of H0 under the action of the time dependent Hamiltonian H0 + Wcosωt, W << H0
Find an expression for the probability of transition from ||ψ1> to ||ψ2>, where ||ψ1> and ||ψ2> are eigenstates of H0 with eigenvalues E1 and E2
Show that this probability is small unless E2 - E1 ≈ ħω.
[This shows that a charged particle in an oscillating electric field with angular frequency w will exchange energy with the field only in multiples of E ≈ ħω.]


Problem 4:

Consider a one-dimensional system, with momentum operator p and position operator q.
(a) Show that [q,pn] = iħ n pn-1.
(b) Show that [q,F(p)] = iħ ∂F/∂p, if the function F(p) can be defined by a finite polynomial or convergent power series in the operator p.
(c) Show that [q,p2F(q)] = 2iħ pF(q) if F(q) is some function of q only.


Problem 5:

If baryon number were not conserved in nature, the wave functions for neutrons (n) and anti-neutrons (n) would not be stationary mass-energy eigenfunctions.
Rather, the wave functions of such particles would be oscillating, time-dependent superposition of neutron and anti-neutron components, given by


where n = 1,  n = 2.

Suppose that such oscillations occur and the Hamiltonian of the system, neglecting all degrees of freedom except for the one corresponding to the oscillations, is


where  E1 = m + U1  and  E2 = m + U2  are energies for n and n individually, and α is a real mixing amplitude.
In an external magnetic field B, the potential energies are U1 = μB and U2 = -μB where μμn = -μn. 
Suppose that at time t = 0 the initial state is that of a neutron.
(a)  Calculate the time-dependent probability for observing an anti-neutron.  Determine the period of oscillation.
(b)  Describe the effect of increasing the external magnetic field.


Let |n1> and |n2> denote the eigenstates of the operator that distinguishes the neutron from the anti-neutron. 
In this basis the matrix of H is

Let us rewrite H.

H = H1 + H2.  H1 = ½(E1 + E2)I,  H2 = ½(E1 - E2)K.
Define tanθ = 2α/(E1 - E2), then
Since every vector is an eigenvector of I, the eigenvectors of K are the eigenvectors of H.
The eigenvalues λ of K are found by solving det(K - λI) = 0.
λ2 = 1 +  tan2θ,  λ± = ±1/cosθ.
For the eigenvectors we have |ψ±> = c1|n1>+  c2|n2>.
For the eigenvector corresponding to λ+:  (1 - 1/cosθ) c1 + tanθ c2 = 0,
c2 = -c1(cosθ/sinθ)- 1/sinθ) = c1tan(θ/2).
|c1|2 + |c2|2 = 1  is satisfied if c2 = sin(θ/2), c1 = cos(θ/2),  |ψ+> = cos(θ/2)|n1> + sin(θ/2)|n2>.
Similarly, for the eigenvector corresponding to λ- we find c1  = -c2tan(θ/2).
|c1|2 + |c2|2 = 1  is satisfied if c1 = -sin(θ/2), c2 = cos(θ/2),  |ψ-> = -sin(θ/2)|n1> + cos(θ/2)|n2>.

+> and |ψ-> are the eigenvectors of H with eigenvalues E± = ½(E1 + E2) ± ½(E1 - E2)/cosθ.
We have |n1> = cos(θ/2)|ψ+> - sin(θ/2)|ψ->, |n2> = sin(θ/2)|ψ+> + cos(θ/2)|ψ->.
At t = 0 we have |ψ(0)> = |n1>.
|ψ(t)> = cos(θ/2)exp(-iE+t/ħ)|ψ+> - sin(θ/2)exp(-iE-t/ħ)|ψ->.
The probability of observing an antineutron at time t is Pn(t) = |<n2|ψ(t)>|2.
<n2|ψ(t)> = cos(θ/2) sin(θ/2)[exp(-iE+t/ħ) - exp(-iE-t/ħ)].
Pn(t) =  ¼sin2θ[2 - exp(i(E+ - E-)t/ħ) + exp(i(E+ - E-)t/ħ)] = ½sin2θ [1 - cos((E+ - E-)t/ħ)].
Period of oscillation:  2π/T = (E+ - E-)/ħ,  T = h/(E+ - E-).
Let B = Bk, μB = μzB. 
Then E1 = m + μzB and E2 = m - μzB, ½(E1 + E2) = m,  ½(E1 - E2) = μzB.
tanθ = α/μzB,  1/cos2θ = 1 + (α/μzB)2,  sin2θ = α2/(μz2B2 + α2).
E± = m ± (μz2B2 + α2)1/2, E+ - E- = 2(μz2B2 + α2)1/2.
Pn(t) = ½[α2/(μz2B2 + α2)] [1 - cos(2(μz2B2 + α2)1/2t/ħ)].

(b)  The larger B, the shorter the period of the oscillations.  However, The larger B, the smaller the amplitude of the oscillations.