Download Relativity Problem Set 9

Relativity Problem Set 9
Prof. J. Gerton
Due Tuesday November 8, 2011 at the beginning of class
Problem 1
(10 pts.) The quantum harmonic oscillator
A Quantum Harmonic Oscillator (QHO) in one dimension describes a quantum particle moving in the harmonic potential
V (x) =
mω 2 2
x.
2
The energy levels are quantized according to
1
En = ~ ω n +
,
2
(1)
(2)
where n = 0, 1, 2, ... is an integer quantum number.
(a) The wave function for the QHO in its ground state (n = 0) is a Gaussian, of the
form
2
ψ(x) = A e−B x .
(3)
Using the Schroedinger equation with the harmonic potential V (x) and the ground
state energy E0 , find the constant B.
(b) Find the constant A by imposing the normalization of the wave function.
Problem 2
(10 pts.) Expectation values for a harmonic oscillator
Consider the wave function ψ1 (x), describing the first excited state of a QHO with
frequency ω.
(a) Compute the expectation value of the momentum of a QHO in the first excited
state, that is, compute
Z +∞
(4)
hpi =
dx ψ1∗ (x) p̂ ψ1 (x),
−∞
where a star (*) indicates a complex conjugation. Notice that the ordering of the
factors inside the integral matters!
1
2
(b) Compute the expectation value of the position operator x̂ in the first excited
state.
(c) Compute the expectation value of the kinetic energy operator K̂ in the first
excited state.
Problem 3
(10 pts.) The Evanescent Wave
A beam of free (unbound) particles, all with mass m, is incident from the left (x =
−∞) on a step-up potential of the form
(
0, for x < 0,
V (x) =
(5)
V0 , for x ≥ 0.
We now consider the case where the total energy of each particle is smaller than the
potential height, E < V0 .
(a) Write down the wave function ψ(x) in the region x > 0.
(b) Recall that for a beam of free particles, ψ ∗ (x)ψ(x) gives the number of particles
per unit distance. Using this, discuss whether it would be possible to find a
particle in the region x > 0 if a measurement were made on the system.
(c) What is the probability that an incident particle will be reflected by the potential
barrier?
(d) Are your answers to (b) and (c) contradictory? Think hard and offer a tentative
explanation for this.
Problem 4
(10 pts.) The double well - Part 1
We consider the potential


+∞ for x < −a and for x > a;
V (x) = 0
for −a ≤ x < −b and for b < x ≤ a


V0
for −b ≤ x ≤ b;
(6)
Consider a particle of mass m and energy E < V0 moving in such potential.
(a) Draw the potential V (x).
(b) What is the wave function for x > a and x < −a?
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(c) Using the conservation of energy relation, find the momentum of the particle in
all regions of the well. Indicate with p the momentum in the regions [−a, −b]
and [a, b], while you indicate the momentum in the region [−b, b] as iκ (yes, the
momentum is imaginary in this region!).
(d) Can the particle be in the region [−b, b] in classical mechanics? What happens
here?
Problem 5
(10 pts.) The double well - Part 2
We refer to Problem 4 above.
(a) Write down the wave functions for the particle in the various regions of the double
well potential. Remember that in any region, the wave function is of the form
ψ(x) = C1 e−ik̃i x + C2 eik̃x ,
(7)
where k̃i is the momentum in the i−th region, given by the solution to (c) above,
and C1,i and C2,i are arbitrary constants for the region i. In particular, what is
the wave function in the region [−b, b]? Notice that in the definition of the wave
function you should have six coefficients total.
(b) Impose the symmetry of the wave function, that is, impose
ψ(−x) = ψ(x).
(8)
This will reduce the number of the coefficients from six to three. What is the
wave function in [−b, b] after the symmetrization?
(c) Impose the continuity of the wave function everywhere, and the continuity of its
derivative at x = −b (or at x = b, which is the same thanks to the symmetrization). From these equations, you should derive the expression
p = k tanh(κb) tan(p(b − a)),
(9)
where tanh(x) is the hyperbolic tangent and tan(x) is the usual tangent.
(d) Write the equation above in terms of E, V0 , m, a and b.
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