1000 Solved Problems in Modern Physics

142 3 Quantum Mechanics – II

Use the constants to represent the amplitude of the reflected and trans-

mitted particle streams respectively and take

k^21 =

2 mE

^2

andk^22 =

2 m(Vb−E)

^2

(b) At the boundaries to the potential barrier,ψand dψ/dxmust be continu-

ous. Equate the solutions that you have atx=0 andx=aand manipulate

these equations to derive the following expression for the transmission

amplitude

τ=

4 ik 1 k 2 e−ik^1 a

[(ik 1 +k 2 )^2 e−k^2 a]−[(ik 1 −k 2 )^2 ek^2 a]

3.31 In Problem 3.30,

(a) Show that the fraction of transmitted particles is given byFtrans=τ∗τ,

which when calculated evaluates to

Ftrans=

[

1 +

Vb^2 sinh^2 (k 2 a)

4 E(Vb−E)

]− 1

(b) How wouldFtransvary if E>Vb.

3.32 A particle is trapped in a one dimensional potential given by=kx^2 /2. At a

timet =0 the state of the particle is described by the wave functionψ=

C 1 ψ 1 +C 2 ψ 2 , whereψis the eigen function belonging to the eigen valueEn.

What is the expected value of the energy?

3.33 A particle is trapped in an infinitely deep square well of widtha. Sud-

denly the walls are separated by infinite distance so that the particle becomes

free. What is the probability that the particle has momentum betweenpand

p+dp?

3.34 The alpha decay is explained as a quantum mechanical tunneling. Assuming
that the alpha particle energy is much smaller than the potential barrier the
alpha particle has to penetrate, the transmission coefficient is given by

T≈exp

{(

−

2



)∫b

a

[2m(U(r)−E)]^1 /^2 dr

}

The integration limits a and b are determined as solutions to the equation

U(r)=E, whereU(r) is the non-constant Coulomb’s potential energy. Cal-

culate the alpha transmission coefficient and the decay constantλ.

3.35 The one-dimensional square well shown in Fig. 3.2 rises to infinity atx= 0

and has rangeaand depthV 1. Derive the condition for a spinless particle of

massmto have (a) barely one bound state (b) two and only two bound states

in the well. Sketch the wave function of these two states inside and outside the

well and give their analytic expressions.