1000 Solved Problems in Modern Physics

3.2 Problems 145

boundaries, write down general expressions for the wavefunctions in these

regions and the form the time-independent Schrodinger equation takes in

each region. What ratio of wavefunction amplitudes is needed to determine

the transmission coefficient?

(c) Write down the boundary conditions forψand dψ/dxatx=0 andx=L.

(d) A full algebraic solution for these boundary conditions is time consuming.

In the approximation for a tall or wide barrier, the transmission coefficient

Tis given by

T= 16

(

E

W

)(

1 −

E

W

)

e−^2 αL, whereα^2 = 2 m

(W−E

^2

)

DetermineTfor electrons of energyE=2 eV, striking a potential of

valueW=5 eV and widthL= 0 .3nm.

(e) Describe four examples where quantum mechanical tunneling is observed.

3.45 A particle of massmmoves in a 2-D potential well,V(x,y)=0for0<x<a

and 0<y<a, with walls atx= 0 ,aandy= 0 ,a. Obtain the energy eigen

functions and eigen values.

3.46 A particle of massmis trapped in a 3-Dinfinite potential well with sides of

length a each parallel to thex-, y-, z-axes. Obtain an expression for the number

of statesN(N1) with energy, say less thanE.

3.47 A particle of massmis trapped in a hollow sphere of radiusRwith impenetra-

ble walls. Obtain an expression for the force exerted on the walls of the sphere

by the particle in the ground state.

3.48 Starting from Schrodinger’s equation find the number of bound states for a par-

ticle of mass 2,200 electron mass in a square well potential of depth 70 MeV

and radius 1. 42 × 10 −^13 cm.

[University of Glasgow 1959]

3.49 A beam of particles of momentumk 1 are incident on a rectangular potential

well of depthV 0 and widtha. Show that the transmission amplitude is given by

τ=

4 k 1 k 2 e−ik^1 a

(k 2 +k 1 )^2 e−ik^2 a−(k 2 −k 1 )^2 eik^2 a

wherek 2 =

[

2 m(E−V 0 )

^2

] 1 / 2

Show thatττ∗=1 whenk 2 a=nπ. Further, show graphically the varia-

tion ofT, the transmission coefficient as a function ofE/V 0 , whereEis the

incident particle energy.

3.50 (a) What are virtual particles? What are space-like and Time-like four momen-
tum vectors for real and virtual particles?
(b) Derive Klein – Gorden equation and deduce Yukawa’s potential.