Download Quantum Mechanics: PHL555 Tutorial 2

Quantum Mechanics: and its applications
EPL202
Date: 15.02.10
Problem Set 4
1. Show that in r , ,  co-ordinate the representation of the angular
momentum operators are the following


L1  L x  i (sin 
 cos  cot 
)




L2  L y  i ( cos 
 cos  cot 
)



L3  L z  i

 1 

1
2 
L  (i ) 
(sin 
)


sin 2   2 
 sin  
2. (a)Show that the eigenstates of the operator L2 are given by spherical
(2l  1)(l  m)! m
Pl (cos  )e im .
harmonics Yl m ( ,  )  (1) m
4 (l  m)!
2
2
(b) Show that the following equation is satisfied by Pl m (x)
(1  x 2 )
d 2 Pl m
dPl m
m2

2
x

(
l
(
l

1
)

) Pl m  0
dx
dx 2
(1  x 2 )
© Show that the spherical harmonics are also eigenstates of the parity
operator.
3. The wavefunction of a particle subjected to a spherically symmetric

potential V (r ) is given by  (r )  ( x  y  3z ) f (r )
(a)
Is  an eigenfunction of L2 ? If so what is its corresponding
eigenvalue. If not what are the possible values we shall
obtain when we shall measure L2 .
(b)
What are the probabilities of finding out the particle in
various m states?
4. (a) A particle is in a spherically symmetric potential is known to be an
eigenstate of L2 & Lz with eigenvalues l (l  1) 2 and m . Prove that the
expectation values between states | l, m satisfy
 l (l  1) 2  m 2  2 
2
2
(a) Lx    L y   0,  Lx    L y   

2


1
5. Consider a system made up of two spin particles. Observer
2
A specializes in measuring the spin components of one of the particles
( s1x , s1z etc. )while the observer B measures the spin component of the
other particle. Suppose the system is known to be in the spin-singlet
state, that is stotal  0 . (a) What is the probability of for observer A to

obtain s1 z  when the observer B makes no measurement. ?Same
2

problem for s1 x  .
2
(b) Observer B determines the spin of particle 2 to be in the state

s 2 z  with certainty. What can be then said about the observer A’s
2
measurement if (i) A measures
s1z and (ii) if A measures s1x ? Justify
your answer.
1
atoms goes through a series of Stern Gerlach type
2
of measurements as follows

(a)
The first measurement accepts s z  atoms and rejects
2

s z   type of atoms.
2

(b)
The second measurement accepts s n  atoms and rejects
2

s n   type of atoms, where s n is the eigenvalue of the
2

operator S  nˆ with n̂ is an unit vector making an angle  in
6. A beam of spin
the x-z plane with respect to the z  axis.

(c)
The third measurement accepts s z  atoms and rejects
2

s z   type of atoms
2


What is the intensity of the final s z   beam when the s z  beam
2
2
surviving the first measurement is normalized to unity? How must we orient
the second measuring apparatus if we are to maximize the intensity of the

final s z   beam ?
2
7. Find out the bound state solutions for the particle in the following
spherically symmetric square well potential
0, r  a
V0 ,r  a
V

8 A system of two angular momentum of respective magnitude
1
1
j1  1 & j 2  , is described by the basis | j1  1, m1  | j 2  , m2  . The system
2
2
is in a state | JM  , where J is the total angular momentum and M is the z component of J. Consider in particular , the states (a)
3
3
1
| J  , M   & (b) | J  12 , M   . For each state calculate the probability of
2
2
2
measuring each pair of possible values (m1 , m2 ) & find the expectation values
of J 1z & J 2 z (c) Calculate the expectation values of J y in the state
|J 
1
1
,M  
2
2