1000 Solved Problems in Modern Physics

198 3 Quantum Mechanics – II

ever the conditionk 2 a=nπis satisfied, the width of the maxima becomes

broader as the electron’s incident energy increases. The occurrence of nearly

complete transparency to incident electrons in the atoms of noble gases is

known as the Ramsuer-Townsend effect.

The conditionk 2 a =nπimplies thata =nλ/ 2 ,n = 1 , 2 ..., that is

whenever the barrier contains an integral number of half wavelengths leading

to complete transparency. Interference phenomenon of this type is analogous

to the transmission of light in optical layers.

3.50 (a) particles which can not be observed are called virtual particles
A 4-vector momemtum isp=(p,iE) so that
(4−momentum)^2 =(3 momentum
Space
)^2 −(energy)^2
time
The componentsp 1 , 2 , 3 are said to be spacelike and the energy compo-
nentE, timelike. Ifqdenotes the 4-momentum transfer in a reaction i.e.
q=p−p′, wherep, p′are initial and 4-momenta, then
q^2 >0 is spacelike as in the scattering process
q^2 <0 is timelike as for (mass)^2 of free particle
q^2 =0 is lightlike
(b) The relativistic relationship between total energy, momentum and mass for
the field quantum is
E^2 −p^2 c^2 −m^2 c^4 =0(1)
We can now use the quantum mechanical operators

E→i

∂

∂t

andp→−i∇

To transform (1) into an operator equation

−^2

∂^2

∂t^2

+^2 ∇^2 −m^2 c^4 = 0

representing the force between nucleons by a potentialφ(r,t) which may

be regarded as a field variable, we can write

[

∇^2 −

1

c^2

∂^2

∂t^2

−

m^2 c^2

^2

]

φ= 0

as the wave equation describes the propagation of spinless particles in

free space.

The time independent part of the equation is

[

∇^2 −

m^2 c^2

^2

]

φ=0(2)

Form=0, this equation is the same as that obeyed in electromagnetism,

for a point charge at the origin, the appropriate solution being

φ(r)=

(

e

4 πε 0

)

1

r

(3)

whereε 0 is the permittivity.