1000 Solved Problems in Modern Physics

170 3 Quantum Mechanics – II

∫a

0

ψn∗(x)ψn(x)dx= 1

A^2

∫α

0

sin^2

(nπx

a

)

dx= 1

(

A^2

2

)(

x−cos

(

2 nπx

a

))∣

∣

∣

a

0

=A^2 a= 1

Therefore,A=

(

2

a

) 1 / 2

(9)

The normalized wave function is

ψn(x)=

(

2

a

)^12

sin

(nπx

a

)

(10)

Using the value ofαfrom (7) in (3), the energy is

En=

n^2 h^2

8 ma^2

(11)

(c) probabilityp=

∫a

0 |ψ^3 (x)|

(^2) dx

=

∫ 23 a

a 3

(

2

a

)

sin^2

(

3 πx

a

)

dx=

1

3

(d)ψ(n) and probability densityP(x) distributions forn = 1 ,2and3are

sketched in Fig 3.6

Fig. 3.6

3.19 The Schrodinger equation for then–psystem in the CMS is

∇^2 ψ(r,θ,φ)+

(

2 μ

^2

)

[E−V(r)]ψ(r,θ,φ)=0(1)