1000 Solved Problems in Modern Physics

182 3 Quantum Mechanics – II

<E>=<ψ|H|ψ>

=<(C 1 ψ 1 +C 2 ψ 2 )|H|(C 1 ψ 1 +C 2 ψ 2 )>

=<C 1 ψ 1 +C 2 ψ 2 |C 1 Hψ 1 +C 2 Hψ 2 |>

=<C 1 ψ 1 +C 2 ψ 2 |C 1 E 1 ψ 1 +C 2 E 2 ψ 2 |>

=C^21 E 1 +C 22 E 2

=

C^21 ω

2

+

C 223 ω

2

=

1

2

ω(C 12 + 3 C 22 )

whereω=

(

k

m

) 1 / 2

3.33 The ground state is

ψ=

(

2

a

) 1 / 2

sin(πx/a)

The wave function corresponding to momentumpis

ψi=( 2 π)−^1 /^2

∑

k

Ckeikx

The probability that the particle has momentum betweenpandp+dpis given

by the value of

|Ck|^2 , whereCkis the overlap integral

Ck=(2π)−

(^12)

(

2

a

) 1 / 2 ∫a

0

eikxsin

(πx

a

)

dx

Itegrating by parts twice,

Ck=(πa)

12 (

eika+ 1

)(

π^2 −k^2 a^2

)− 1

The required probability is

|Ck|^2 =πa

(

eika+ 1

)(

e−ika+ 1

)(

π^2 −k^2 a^2

)− 2

= 4 πa cos^2

(

ka

2

)

(

π^2 −k^2 a^2

)− 2

3.34 The transmission coefficient is given by

T=e−G (1)

G=

2



∫b

a

[2m(U(r)−E)^1 /^2 dr (2)

Put

U(r)=

zZe^2

r

(3)

for the Coulomb potential energy between the alpha particle and the residual

nucleus at distance of separationr.