Download Quantum Mechanics: PHL555 Tutorial 2

Quantum Mechanics: EPL202
Date:04.04.10
Problem Set 5
1. Consider the Hydrogen atom problem.
(a)
First write the Lagrangian in terms of of co-ordinate and
velocities of the electron and the proton
(b)
Now using center of mass and relative co-ordinate write
down the Hamiltonian of the problem in these two coordinates and their conjugate momentum
(c)
Show that the above problem gets effectively reduced to an
one body problem in which an electron is moving around a
massive immobile nucleus? Is this description possible if
mass of the electron and proton are of the same order.
(d)
Show that the relative co-ordinate Hamiltonian has
simultaneous eigenstates with L2 , Lz
(e)
(f)
Now using the above information show that the energy
eigenfunctions can be written as a product of a Radial part
and and angular part.
Show that the radial eigenvalue equation can be finally
d 2 u E ,l
1  l (l  1)
 (  
)u E ,l  0
2
4 y
dy
y2
reduced to the form

2 E

e2 , y  2
r
2
2 | E |
2
(g)
Find out the solution of the above equations as r  0, r  
(h)
Using this asymptotic form show that the radial solutions
are given by associated Laguerre polynomials
Now using the above results show that the boundary
conditions imposes the quantization condition which reads
for k number of terms in the polynomials as
k  l  1  n, k  0,1,2,3, Discuss the implication of the above
(i)
result.
2. Consider the one-dimensional harmonic oscillator problem assuming a
3
1
(n 2  n  ) .
perturbation H 1  x 4 . Show that E n1 
2
2
2(m /  )
3. Consider a Hamiltonian of the form
2 2
2
1
H 
( 2  2 )  k ( x12  x22 ) & H 1  x1 x2 . Treating H 1 as a
2m x1 x2
2
perturbation calculate the first order shift in the energy level of the
first excited state. Can you solve this problem exactly?
1
4. In the previous problem consider H 1  kx1 x 2 ( x12  x 22 ) as the
2
perturbation. Show that the ground state is not shifted, but the two
fold degenerate first excited state split.
5. For a hydrogen atom placed in a weak uniform magnetic field along the
, the Hamiltonian will be of the form

p2
e   B B
H  H 0  H1 , H 0 
 V (r ) & H 1 
BL 
Lz , where the second
2m
2m

part of the Hamiltonian represents the interaction with the magnetic
field. We have neglected the effects due to spin angular momentum of
the electron . Treat H 1 as a perturbation and show s(l  0) states are
not split , where as p(l  1) states are split into three states separated
by the energy interval  B B . This is known as normal Zeeman Effect
6. In the following problem we will consider the effect of proton
magnetic moment and hence the hyperfine structure. Since electron
and proton are spin ½ particle , thus there are four possible spin
states , namely | e , p , | e , p , | e , p , | e , p  . Show that the
1
operator P  [1   e   p ] , is the spin exchange operator, i.e. when
2
P operates on any state the spin directions are exchanged
P | e  p  | e  p  , etc. The  e &  p represent the Pauli spin
matrices for electron and proton respectively and they work only on
the electron and proton part of the wavefunction respectively.
7. This problem is about perturbation theory for degenerate states.
Consider the first excited state of the Hydrogent atom n=2. which is

4 fold degenrate. Apply an electric field of the form   Eˆz . Show
that the degeneracy is partially lifted. Find out the explicitly the
resulting energy levels with their energy and the eigenfunctions.
8.