Download Relativity Problem Set 8

Relativity Problem Set 8
Prof. J. Gerton
Due Tuesday November 1, 2011 at the beginning of class
Problem 1
(10 pts.) Solving a PDE
Using the separation of variables technique, find an explicit form of the function
ψ(x, t) satisfying the equation
∂
∂
+
ψ(x, t) = e−x ψ(x, t).
(1)
∂t ∂x
Problem 2
(10 pts.) Free particle
A free particle is prepared to have the wave function
( h
2 i
N 1 − xa
for −a ≤ x < a,
ψ(x) =
0
elsewhere,
(2)
where a is a known parameter with units of length and N is a normalization constant.
Given a wave function ψ(x), the expectation value hf (x)i of a function f (x) is
defined as
Z +∞
hf (x)i =
ψ ∗ (x) f (x) ψ(x) dx,
(3)
−∞
where a star (*) means complex conjugation.
(a) Find N .
(b) Using Eq. (3), compute hxi, hx2 i and find the standard deviation
p
σx = hx2 i − (hxi)2 ,
(4)
of the wave packet in Eq. (2).
(c) Using the uncertainty principle, estimate the momentum spread and the kinetic
energy of the particle.
1
2
Problem 3
(10 pts.) The infinite well
We consider the infinite potential well in one dimension:
(
+∞ for x < 0 and x > L,
V (x) =
0
for 0 < x < L.
(5)
In this well, a particle of mass m has the wave function
ψn (x, t) = A sin
n π x −iEn t/~
e
,
L
(6)
where A is a normalization constant, n is an integer called the quantum number and
En is the energy spectrum (see class notes).
(a) Find the constant A, which is independent of n.
(b) Using the uncertainty principle, find the minimum momentum pmin .
(c) Compute the probability to find the particle between 0 and L/4 for the ground
state n = 1 and for the first excited state n = 2. The probability to find the
particle in a certain region is given by Eq. (3) with f (x) = 1 and with the region
of integration limited to the portion considered.
Problem 4
(10 pts.) Two-dimensional infinite well
We consider a two-dimensional square well, with side L and on the (x-y) plane.
(a) What is the minimum momentum of a particle in this well? What is its minimum
energy?
(b) The generic wave function for the well is
ψ(x, y) = A sin
ny π y
nx π x
sin
,
L
L
(7)
where nx and ny are integers. Find the constant A.
(c) Write down a general expression for the energy of the system with specific quantum numbers nx , ny .
(d) Write down the ground state wave function. What is the energy of the particle
in the ground state?
course name
PS #
3
Problem 5
(10 pts.) Computations for a particle in a well
An electron is trapped in an infinite well. The ground state has energy E1 = 1.26 eV.
(a) What is the width of the well?
(b) How much energy must be added to the particle to reach the excited state n = 3?
(c) If the electron sits in the state n = 2, how much energy should it gain to jump
to n = 4?
course name
PS #