1000 Solved Problems in Modern Physics

(Romina) #1

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.
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