Download First Problem Set for EPL202

EPL202: Problem Set 1
1. Consider a thermal neutron, that is, a neutron with speed v
corresponding to average thermal energy at the temperature T=300K. Is it
possible to observe a diffraction pattern when the beam of such neutrons
fall on a crystal? (b) In a large accelerator, an electron can be provided with
energy over 1 GeV=109 eV. What is the de Broglie wavelength corresponding
to such electrons.
2. Consider the following wavefunction for a quantum mechanical particle
(say electron) ( x)  Ae
(a)
(b)

( x  x0 ) 2
4l 2
.x
From the normalization condition find out the constant A ,
Why is this normalization condition necessary.
Find out the average position of this particle, namely x .
What is the physical meaning of x .
(c)
(d)
Find out the average momentum of this particle ,  p x  .The

momentum operator can be introduced as  i
x
Calculate the following quantities, namely
x  ( x   x) 2  & p x  ( p x   p x ) 2  and verify that

. Interpret this result physically.
2
(e)
Repeat all the calculations for a plane wave,   e ikx and list
all the difficulties you run.
3. Consider the following general form of the wave-fucntion which has
1
been introduced in the class  ( x) 
a(k )e ikx dk . Show that the

2
average momentum associated with such a particle is given
by  p x    dp x p x | a(k ) | 2 . Interpret this result.
p x  x 
1 0 0
4. Let us consider the following operator matrix 0 0  1 . Find out
0 1 0
the eigenvalues and the eigenfunctions of this operator. Is this
operator Hermitian?
5. Prove the following properties of a hermitian operator. (a) A hermitian
operators has real eigenvalues. (b) Eigenvectors of hermitian operator
with distinct eigenvalues are orthogonal.
6. Write down the operators used for the following quantities in quantum

2 
mechanics x, p x , p x , p, Lz , L2 . Check if they are hermitian or not.

, xˆ  ipˆ x
7. Check if the following operators are hermitian
x
8. A wave function is given by ( x)  Ae

( x  x0 ) 2
4l 2
. Find out its fourier
transform. Interpret this result in terms of uncertainty in position
and momentum. ( very similar to the first problem).
9. Write down the definition of the Dirac  function. Can the wave
function of an electron be given by a Dirac  function. Write down the
fourier transformation of  ( x)   ( x) . What does it mean?