1000 Solved Problems in Modern Physics

(Tina Meador) #1

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)=


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
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