Physics 517 Homework Set #6 Autumn 2015 Due in class 11/18/16

1. The free particle propagator in three dimensions is given by K0(r, t; r′, t′) =(

m2πih̄(t−t′)

) 32

exp[ im2h̄(r−r′)2t−t′ ].

a. Show by explicit differentiation that for t > t′ K0 satisfies the free particle Schoredinger equation:(ih̄∂

∂t+h̄2∇2

2m

)K0(r, t; r′, t′) = 0.

b. Now suppose a potential V (r, t) acts on the particle, and that at initial time, t0 the wave function is givenby ψ0(r, t0). Verify that the wave function of the particle at a later time t is given by the solution to theintegral equation:

ψ(r, t) =

∫d3r′K0(r, t; r′, t0)ψ0(r′, t0) +

1

ih̄

∫ t

t0

dt′∫d3r′K0(r, t; r′, t′)V (r′, t′)ψ(r′, t′). (1)

Hint: show that the ψ(r, t) as given by the above equation is consistent with the boundary conditions andsolves the Schroedinger equation.

2. Consider a particle of mass m which at t = 0, is prepared in the state ψ(x, 0) = 1√V

if x ≤ R, but vanishes

for x > R. Thus the density is constant over a sphere of radius R and volume V = 4πR3/3. For t > 0, theparticle moves freely in three dimensions .

(a) Show that for x� R, and all t that

ψ(x, t) =

(2

πV

) 12(aR

ix2

) 32 [

sinx

a− x

acos

x

a

]ei x

2m/2h̄t,

where a = h̄t/(mR)

(b) You can do this part without doing (a). State the significance of the parameter a. Determine the behaviorof the probability distribution in the limits x� a and x� a.

3. The one-dimensional propagator for a particle in a uniform gravitational field is given by:

K(x′, T ;x0, 0) =

√m

2πih̄Texp

[im

2h̄

((x′ − x0)2

T− g(x′ + x0)T − g2T 3

12

)], (2)

where g accounts for gravitational acceleration and the motion is along the direction of the gravitational field.Suppose at t = 0 the initial wave function is

ψ(x0, 0) =1

(2πa2)1/4exp(− x2

0

4a2),

where a is a given constant length.

(a) Find ψ(x′, T ) and show that it describes a wave packet that undergoes uniform acceleration g. Discuss thespreading of the wave packet.

(b) Determine the three-dimensional version of Eq. (2), with the same gravitational potential.

4. Consider a Hamiltonian that depends on a continuous real parameter λ so that the time-independentSchroedinger equation is given by

H(λ)|ψ(λ)〉 = E(λ)|ψ(λ)〉. (3)

(a) Show that ∂E∂λ = 〈ψ|∂H∂λ |ψ〉.This result is known as the Feynman-Hellman theorem.

(b) Find a one-dimensional potential V (x) that gives bound states with energy eigenvalue separations (split-tings) that are independent of particle mass. Hint: One way is to use the Feynman-Hellman and virial theorems.

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5. Consider the classical harmonic oscillator of frequency ω for a particle of mass m. Let x(t) = A cosωt +B sinωt. . Choose A and B such that x(0) = x1 and x(T ) = x2, where the initial time is 0 and the final timeis T .

(a) Show that the action for the trajectory connecting x1 and x2 is Scl = mω2 sinωT

[(x2

1 + x22) cosωT − 2x1x2

].

(b) At t = 0, the wave function is given by ψ(x0, 0) = 1(2πa2)1/4

exp(− x20

4a2 ), where a is a given constant length,

with a� b ≡√h̄/(mω). For t > 0 the particle moves under harmonic oscillator forces, find the wave function

ψ(x, t)). From this determine roughly the time needed for the spatial extent of the wave function to be of theorder of b.

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