1000 Solved Problems in Modern Physics

3.3 Solutions 189

None of the numbersnx,ny,ornzcan be zero, otherwiseψ(x,y,z) itself

will vanish.

For an infinitely deep potential wellEn= h

2

8 ma^2

[

n^2 x+n^2 y+n^2 z

]

. The com-
binationnx=ny=nz=0 is ruled out because the wave function will be
zero. Table 3.4 gives various energy levels along with the value ofg, the degen-
eracy. The values ofnx,nyand nzare such thatn^2 x+n^2 y+n^2 z= 8 ma^2 En/h^2 =
constant for a given energyEn. The energies of the excited states are expressed
in terms of the ground state energyE 0 =h^2 / 8 ma^2

Table 3.4

nx ny nz gEn

0 0 1 3-fold E 0 =h^2 / 8 ma^2

010

100

0 1 1 3-fold 2 E 0

101

110

1 1 1 Non-degenerate 3 E 0

0 0 2 3-fold 4 E 0

020

200

0 1 2 6-fold 5 E 0

102

120

021

201

210

1 1 2 3-fold 6 E 0

121

211

3.41 (a) Case (i)U 0 <E,Regionx 0
PuttingV(x)=0, Schrodinger’s equation is reduced to
d^2 ψ
dx^2

+

(

2 mE

^2

)

ψ=0(1)

which has the solution

ψ 1 =Aexp(ik 1 x)+Bexp(−ik 1 x)(2)

wherek^21 =

2 mE

^2

(3)

ψ 1 represents the incident wave moving from left to right (first term in (2))

plus the reflected wave (second term in (2)) moving from right to left

Regionx0:

d^2 ψ

dx^2

+

[

2 m(E−U 0 )

^2

]

ψ=0(4)

which has the physical solution