PC3130/3201 QUANTUM MECHANICS
TUTORIAL 5
March 2000
1.
Using the perturbation method, calculate the lowest energy up to first order of
a particle of mass m in a one-dimensional potential well given by
V (x) = t
x
a
-a < x < a
=
x < -a; x > a
where t is a small constant. The first state function 1(x) and energy E1 of a
particle of mass m inside an infinite square well are given by
1 
1
x
 22
and E1 
,
cos
a
2a
8ma 2
and


2
0
2.
 cos 2  d 
2
16

1
4
An electron in the Coulomb potential (40)-1 qe/r has the normalised
groundstate wave function (a3)-1/2 exp(-r/a). Assuming the nuclear charge is
uniformly distributed on the surface of a sphere of radius r0, find the first order
correction to the groundstate energy of the hydrogen atom, treating deviation
from Coulomb potential energy as a perturbation.
Repeat the calculation by assuming the nucleus to be a uniformly charged solid
sphere of radius r0
0
Note:
r x
a = (40  2 / (qe2m) = 0.5 A
(40)-1 qe2 / a = 27.2 eV,
~
r0 = 10-13 cm
3.
A charged particle is bound in a harmonic potential V = kx2/2 . The system is
placed in an external electric field  that is constant in space and time.
Calculate the shift of the energy of the ground state to order  2 .
4.
The first excited state of the hydrogen atom has fourfold degeneracy, and the
eigenfunctions in atomic units are: -
1
2
r 1 1/ 2
)
2 4
 200  2( ) 3/ 2 e  r / 2 (1  )(
 210 
1 1 3/ 2  r / 2 3 1/ 2
( ) e r ( ) cos
4
3 2
 211  (  )
1 1 3/ 2 r / 2 3 1/ 2
( ) e r ( ) sin e i
8
3 2
The hydrogen atom is placed in a constant electric field  along the x3~
direction. The perturbation potential can be taken as
H = qe  . x = qe  x3
~
~
Show
(a)
  nm x3  nm  = 0 for m  m


by using x3 ,  3  0
(b)
  nm x3  nm  = 0
by considering the parity of the state  nm (r,,)
(c)
zero.
the possible values for the first order energy correction are 3qe and
Note:


0
r n1e  r dr  (n)