1000 Solved Problems in Modern Physics

3.2 Problems 137

The Born approximation

Here the entire potential energy of interaction between the colliding particles is
regarded as a perturbation. The approximation works well when the kinetic energy
of the colliding particles is large in comparision with the interaction energy. It there-
fore supplements the method of partial waves.

σ(θ)=|f(θ)|^2 (3.27)

where

f(θ)=−K−^1

∫∞

0

rsinKr V(r)dr (3.28)

and

K= 2 ksin

θ

2

,k=p. (3.29)

3.2 Problems..................................................

3.2.1 Wave Function ....................................

3.1 An electron is trapped in an infinitely deep potential well of widthL= 106 fm.

Calculate the wavelength of photon emitted from the transitionE 4 →E 3. (See

Problem 3.18).

3.2 Givenψ(x)=

(π

α

)−^14

exp

(

−α

(^2) x 2
2

)

, calculate Varx

3.3 Ifψ(x)=x 2 N+a 2 , calculate the normalization constantN.

3.4 Find the flux of particles represented by the wave function

ψ(x)=Aeikx+Be−ikx

3.5 For Klein – Gordon equation obtain expressions for probability density and

current. Explain the significance of the result.

3.6 (a) Find the normalized wave functions for a particle of massmand energyE

trapped in a square well of width 2aand depthV 0 >E.

(b) Sketch the first two wave functions in all the three regions. In what respect

do they differ from those for the infinite well depth.

3.7 The Thomas-Reich-Kuhn sum rule connects the complete set of eigen func-

tions and energies of a particle of massm. Show that

(

2 μ

^2

)∑

k

(Ek−Es)|xsk|^2 = 1