Download Quantum Mechanics: EPL202 : Problem Set 1 Consider a beam of

Quantum Mechanics: EPL202 : Problem Set 1
1. Consider a beam of light passing through two parallel slits. When
either one of the slits is closed, the pattern observed on a screen
placed beyond the barrier is a typical diffraction pattern. When both
slits are open, then one obtains the two slit diffraction pattern ( an
interence pattern within a diffraction envelope. Explain why this
pattern is not the two single-slit diffraction patterns superposed. Can
this phenomena be explained in terms of classical particlelike photons?
Is it possible to demonstrate the particle aspects of light in this
experimental setup?
2. Consider the following wavefunction for a quantum mechanical particle

( x  x0 ) 2
(say electron) ( x)  Ae 4l .x
(a)
From the normalization condition find out the constant A ,
Why is this normalization condition necessary.
(b)
Find out the average position of this particle, namely x .
2
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

p x x  . 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.
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?
r


10. Consider a wavefunction in three dimension (r )  Ae a0 . Extract the
information content of this wave-fucntion by operating it position
operator , momentum operators (3), angular momentum operators (3),
Hamilton operator ( K.E+ some spherically symmetric potential
energy). Particularly identify the operator for which this is an
eigenfunction.