1000 Solved Problems in Modern Physics

152 3 Quantum Mechanics – II

3.2.7 Approximate Methods .........................

3.97 Consider hydrogen atom with proton of finite size sphere with uniform

charge distribution and radiusR. The potential is

V(r) =−

3 e^2

2 R^3

(R^2 −r^2 /3) forr<R

=−e^2 /r forr>R

Calculate correction to first order forn=1 andn=2 withl=0 states

[Adapted from University of Durham 1963]

3.98 A particle of massmand chargeqoscillating with frequencyωis subjected

to a uniform electric fieldEparallel to the direction of oscillation. Determine

the stationary energy levels.

3.99 Consider the Hermitian HamiltonianH= H 0 +H′, whereH′is a small

perturbation. Assume that exact solutionsH 0 |ψ>=E 0 |ψ>are known, two

of them, and that they are orthogonal and degenerate in energy. Work out to

first order inH′, the energies of the perturbed levels in terms of the matrix

elements ofH′.

3.100 The helium atom has nuclear charge+ 2 esurrounded by two electrons. The
Hamiltonian is

H=

(

−

^2

2 m

)

(∇ 12 +∇ 22 )− 2 e^2

(

1

r 1

+

1

r 2

)

+

e^2

r 12

where r 1 and r 2 are the position vectors of the two electrons with nucleus as

the origin, andr 12 =|r 1 −r 2 |is the distance between the two electrons. The

expectation value for the first two terms are evaluated in a straight forward

manner, the third term which is the interaction energy of the two electrons is

evaluated by taking the trial function as the product of two hydrogenic wave

functions for the ground state. The result is

<H>=

e^2 Z^2

a 0

−

4 e^2 Z

a 0

+

5 e^2 Z

8 a 0

=

(

e^2

a 0

)(

Z^2 −

27 Z

8

)

Thus, the energy obtained by the trial function is

E(Z)=

(

−

e^2

2 a 0

)(

27 Z

4

− 2 Z^2

)

Determine the ionization energy of the helium atom.

3.101 Consider the first-order change in the energy levels of a hydrogen atom due
to an external electric field of strengthEdirected along thez-axis. This phe-
nomenon is known as Stark effect.
(a) Show that the ground state (n=1) of hydrogen atom has no first-order
effect.
(b) Show that two of the four degenerate levels forn=2 are unaffected and
the other two are split up by an energy difference of 3eEa 0.