Download Problem Set 10

Problem Set 10
Prof. J. Gerton
Due Tuesday November 22, 2011 at the beginning of class
Problem 1
(10 pts.) Raising/lowering operators
The ground state for a quantum harmonic oscillator (QHO) of frequency ω is
ψ0 (x) =
mω 1/4
π~
mω x2 .
Exp −
2~
(1)
This state satisfies the equation â− ψ0 (x) = 0, where the lowering operator â− is
r
mω
p̂
â− =
x̂ + i
,
(2)
2~
mω
and p̂ = −i~∂/∂x. Similarly, we define the raising operator as
r
mω
p̂
x̂ − i
.
â+ =
2~
mω
(3)
(a) Compute the wave function ψ1 (x) for the first excited state, using the equation
ψ1 (x) = â+ ψ0 (x),
(4)
(b) Check if the wave function ψ1 (x) thus obtained is normalized.
(c) Show that ψ1 (x) solves the Schroedinger equation
−
~2 ∂ 2 ψ1 (x) mω 2 x2
+
ψ1 (x) = E1 ψ1 (x),
2m ∂x2
2
(5)
where E1 is the energy of the first excited state of the QHO (look up in the notes
for its value).
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Problem 2
(10 pts.) Two-dimensional harmonic oscillator
Consider a two-dimensional QHO, with frequency ω0 and mass m.
(a) Find an expression for the total energy, Enx ,ny , in terms of the independent quantum numbers nx and ny . Hint: It might be helpful to draw an analogy with the
1D QHO discussed in the lecture and/or the 2D infinite square well considered
in HW8.
(b) Find the ground-state wavefunction, ψ02D (x, y), using the fact that the motion is
separable in the x and y directions. Again, an analogy with the 1D case might
be helpful.
(c) Write down the two-dimensional Schrödinger equation and show explicitly that
ψ02D (x, y) is a solution.
Problem 3
(10 pts.) Step-Down Potential
We consider a uniform beam of particles of mass m moving in the potential
(
V0 , for x < 0,
V (x) =
0, for x ≥ 0.
(6)
The energy of each particle is E > V0 . The particles are initially moving to the right,
coming from x = −∞.
(a) Write down the wave function for x < 0. Here, are there left- and right-moving
components of the wavefunction? Why?
(b) Write down the wave function for x > 0. Here, are there left- and right-moving
components of the wavefunction? Why?
(c) Write down the boundary conditions at x = 0 and solve for the amplitude coefficients of the reflected and transmitted waves, in terms of the amplitude coefficient
of the incident wave.
(d) What is the probability that an incoming particle will be reflected from the stepdown potential? Transmitted? Compare these answers to the classical result.
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Problem 4
(10 pts.) Potential well for a free particle
A particle of mass m and energy E moving in a region in which there is no potential
energy encounters a potential dip of width L and depth −V0 :
(
0, for x < 0 and x > L,
V (x) =
(7)
−V0 , for 0 < x < L.
Show that the reflection probability is given by
i
hp
2
2m(E + V0 ) L/~
sin
hp
i
R=
.
sin2
2m(E + V0 ) L/~ + 4(E/V0 ) [(E/V0 ) + 1]
Problem 5
(8)
(10 pts.) Tunnel effect
The transmission coefficient for particles of energy E crossing a region with potential
U0 > E and size L is
−1
U02
2
sinh (2α L)
,
(9)
T = 1+
4E(U0 − E)
where
r
α=
2m (U0 − E)
.
~2
(10)
(a) Show that in the limit αL 1 the transmission coefficient is approximately given
by
16E (U0 − E) −4α L
e
.
(11)
T =
U02
(b) A source is emitting a beam of monoenergetic electrons at a rate N = 106 s−1 ,
each electron having total energy E = 1 GeV. The electron beam is incident from
the left, and crosses a region of size L = 50 fm, where a potential U0 = 1.2 GeV
acts. What is the power of the beam (in W or GeV/s), before and after it has
passed through the potential?
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