1000 Solved Problems in Modern Physics

156 3 Quantum Mechanics – II

3.123 Using the Born approximation, the amplitude of scattering by a spherically
symmetric potentialV(r) with a momentum transferqis given by

A=

∫∞

0

[

sin

(qr



)

qr



]

V(r)4πr^2 dr

Show that in the case of a Yukawa-type potential, this leads to an amplitude

proportional to (q^2 +m^2 c^2 )−^1.

3.3 Solutions..................................................

3.3.1 Wave Function ....................................

3.1 En=

n^2 h^2

8 mL^2

=

π^2 n^2 ^2 c^2

2 mc^2 L^2

=

π^2 ×(197.3MeV−fm)^2 n^2

2 x 0 .511(MeV)×(10^6 fm)^2

= 0. 038 n^2 eV

E 1 = 0 .038 eV,E 2 = 0 .152 eV,E 3 = 0 .342 eV,E 4 = 0 .608 eV

ΔE 43 =E 4 −E 3 = 0. 608 − 0. 342 = 0 .266 eV

λ=

1 , 241

0. 266

= 4 ,665 nm

3.2ψ(x)=(π/α)−^1 /^4 exp

(

−

α^2

2

x^2

)

Varx=<x^2 >−<x>^2

The expectation value

<x>=

∫∞

−∞

ψ∗xψdx= 0

becauseψand alsoψ∗are even functions whilexis an odd function. There-

fore the integrand is an odd function

<x^2 >=

(π

α

)− 1 / 2 ∫∞

−∞

x^2 exp(−α^2 x^2 )dx

Putα^2 x^2 =y;dx=^1 / 2 α

√

y

<x^2 >=

(

πα^5

)− 1 / 2 ∫∞

0

y^1 /^2 e−ydy

But

∫∞

0 y

1 / (^2) e−ydy=Γ(3/2)=√π/ 2
Varx=<x^2 >=(4α^5 )−^1 /^2