 # Homework #3

```Physics 220
Homework #3
Spring 2015
Due Wednesday 4/26/15
1.
A particle of mass m is moving in one dimension in a potential V ( x,t ) . The wave
function for the particle is Ψ ( x,t ) = Axe
and k are constants.
⎛ km ⎞ 2
⎛3 k ⎞
−⎜
x −i⎜
t
⎝ 2! ⎠⎟
⎝ 2 m ⎠⎟
e
for −∞ < x < ∞ , where A
a. Show that V is independent of t , and determine V ( x ) .
b. Normalize the wave function and determine the constant A .
c. Using the normalized wave function, calculate x , x 2 , p , and p 2 .
2.
Determine which of the following one-dimensional wave functions represent state of
definite momentum. For each wave function that does correspond to a state of
definite momentum, determine the momentum.
a. ψ ( x ) = eikx
b. ψ ( x ) = xeikx
c. ψ ( x ) = sin ( kx ) + i cos ( kx )
d. ψ ( x ) = eikx + e−ikx
3.
Griffith’s problem 1.4.
4.
Griffith’s problem 1.15.
5.
!
Suppose that a wave function Ψ ( r,t ) is normalized. Show that the wave function
!
eiθ Ψ ( r,t ) , where θ is an arbitrary real number, is also normalized.
6. Griffith’s problem 1.17 parts a – e.
7. Griffith’s problem 2.3
8. Suppose that ψ 1 and ψ 2 are two different solutions of the time-independent
Schrodinger wave equation with the same energy E .
a. Show that ψ 1 + ψ 2 is also a solution with energy E .
b. Show that cψ 1 is also a solution of the Schrodinger equation with energy E .
9. A particle of mass m is moving in one dimension near the speed of light so that the
p2
relation for the kinetic energy E =
is no longer valid. Instead, the total energy is
2m
given by E 2 = p 2 c 2 + m 2 c 4 . So, we can no longer use the Schrodinger equation.
Suppose that the wave function for the particle Ψ ( x,t ) is an eigenfunction of the
energy operator and an eigenfunction of the momentum operator, and also assume
there is no potential energy V . Derive a linear differential equation for Ψ ( x,t ) .
10. Determine the odd solutions to the finite square well. Determine the energy of the
single bound state with E < V0 . Normalize your solutions in each region to determine
the unknown coefficient A in each region. Plot your solution for ψ 2 (x) .
11. Determine the normalization coefficients for the second energy state of the even
solutions to the finite square well. That is, renormalize the solutions and determine
B in each region for E3 . Plot your solution for ψ 3 (x) , along with the solutions for
ψ 2 (x) from above and ψ 1 (x) from class.
12. Griffith’s problem 2.40
13. Griffith’s problem 2.47
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