1000 Solved Problems in Modern Physics

104 2 Quantum Mechanics – I

2.9 In atomic physics the atomic units are as follows:

(i) (a) The Bohr radius^2 /mee^2 is used as the unit of length. (b) The energy

is measured in multiples of the ionization energy of hydrogenmee^2 / 2 ^2

(c)^2 =1(d)e^2 = 2

(ii) In atomic units the Schrodinger equation

−

^2

2 me

∇^2 u−

e^2 u

r

=Eu

would read as

−∇^2 u−

2 u

r

=Eu

2.10 (a) Let the separation between the two nuclei each of chargeqbe 2d, then the
negative chargeQon the electrons is at a distancedfrom either nuclei
Total potential energy due to electrostatic interaction between three objects
is
qQ
d

+

qQ

d

+

q^2

2 d

≤ 0

Taking the equality sign and cancellingq

Q=−

q

4

(b) LetT 0 be the initial kinetic energy andp 0 the momentum of the ion andT

the kinetic energy andpthe momentum of the composite molecule andQ

the excitation energy.

T 0 =T+Q Energy conservation (1)

p 0 =p Momentum conservation (2)

∴(2mT 0 )^1 /^2 =(2. 2 mT)^1 /^2 (3)

The mass of the composite being 2mas the excitation energy is expected

to be negligible in comparison with the mass of the molecule. From (3) we

get

T=

T 0

2

(4)

Using (4) in (1), we findQ=T 20 =^10

− 19

2 =^5 ×^10

− (^20) J
2.11(a) Stationary orbits will be such that the circumference of a circular orbit is
equal to an integral number of deBroglie wavelength so that constructive
interference may take place i.e. 2πr=nλ
Butλ=h/p
∴L=rp=nh/ 2 π (Bohr’s quantization condition) (1)