Solutions due on or before 13:00 on Friday 19 th
December.
Your solutions should be handed to me in person.
You are expected to work alone on these problems.
Each problem carries equal marks and this assignment will count as 20% of the grade for this course.
Question 1
A particle moving in one-dimension is in a stationary state whose wavefunction is
 


0 x



A
0

 1
 cos a x

 -
x a


-a a x
  a
Where A and a are real constants
(a) Sketch this wavefunction.
(b) Is this a physically acceptable wavefunction. Justify your answer
(c) Evaluate the constant A so that the wavefunction is normalised
(d) Evaluate the uncertainties
 x and
 p and verify that
 x
 p > h/4

(e) Find the classically allowed region.
Question 2
At time t = 0 a particle is represented by the wave function

 


Ax


A
0
 b a
 x
 b
 a if if a
0

 x x
 otherwise
 b a
(a) Normalise the wavefunction (that is, find A in terms of a and b).
(b) Sketch

(x,0) as a function of x.
(c) Where is the particle most likely to be found at t=0.
(d) What is the probability of finding the particle to the left of a, i.e. at x < a? Check your result in the limiting cases of b = a and b = 2a.
(e) What is the expectation value of x and x
2
?
Question 3
A one-dimensional potential barrier has the shape shown in the figure below. Find an expression for
L
E
V
1
V
2
V = 0 x the reflection and transmission coefficient for a particle of mass m coming from the left wthan an energy such that V
1
< E < V
2
.

Question 4
(a) Determine the 'classical' probability distribution for a harmonic oscillator. That is, what is the probability of finding a classical particle at position x if it is undergoing simple harmonic motion.
How does probability this compare with the probability distribution of a quantum oscillator ?
(b) Confirm by substitution that the wavefunction for the first excited state of a one-dimensional harmonic oscillator,

1

N
1
H
1
  exp
 y
2
2
, where x =
 y, H
1
(y) = 2y and
 

2 mk

1 4
, is a solution to the Schrodinger equation for the harmonic oscillator and show that its energy is
3
2

 
3
2 h

.
(c)The oscillations of the atoms about their equilibrium positions in the molecule HI can be modelled as a harmonic oscillator. Calculate the reduced mass of this system. The force constant for the bond between the hydrogen and iodine atoms has a value k = 313.8 N.m-1. Use this information to evaluate the separation of the vibrational energy levels and predict the wavelength of the light needed to induce a transition between adjacent energy levels.
5) An Experimenter asks for funding to build an experiment to observe visually through an optical microscope the quantum behaviour of a small harmonic oscillator. According to his proposal, the oscillator consists of an object 10
-6
m in diameter and mass of 10
-15
kg. It vibrates on the end of a thin fibre with a maximum amplitude of 10
-5
m and frequency 1000 Hz. You are the referee for this proposal and you must decide the answer to the following questions:
(a) What is the approximate vibrational quantum number for the system in the state described?
(b) What would be its energy in electron-volts if it were in its lowest vibrational state? Compare this energy with the average thermal energy kT (1/40 eV) of air molecules at room temperature.
(c) What would be the classical amplitude of vibration if it were in its lowest vibrational state?
Compare this with the wavelength of visible light (about 500 nm) by which it is observed.
(d) Would you, as referee of this proposal, recommend award of the requested funding to carry out the research? Justify your answer.