Qal QuantP1S

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Solution:

  • [22MΔ2+V]W(x,y,z)=EW(x,y,z)

In our case, using separation of variables, we will get 3 differential equations for x, y and z. W(x,y,z)=w(x)w(y)w(z)

d2w(x)dx2m2w(x)=0(1)
The same will be for y and z.

Solution of equation (1) is following
w(x)=Asin(mx)+Bcos(mx)
w(y)=Csin(ky)+Dcos(ky)
w(z)=Esin(qz)+Fcos(qz)


  • Applying B. C. at x=y=z=0 wave function should be zero, that means B=D=F=0. We have

w(x)=Asin(mx)
w(y)=Csin(ky)
w(z)=Esin(qz)

Also, w(a)=0 which gives Asin(ma)=0,m=πnxa. For y component Csin(kb)=0,k=πnyb and for z Esin(qc)=0,q=πnzc

A, C and E are normalization constants

1A2=sin2(πnx/a)dx=a2, limits are from 0 to a.

The eigenfunction for each component will be

w(x)=2asin(πnxx/a)
w(y)=2bsin(πnyy/b)
w(z)=2csin(πnzz/c)

The eigenenergies

Enx=π22nx22Ma2, Eny=π22ny22Mb2, Enz=π22nz22Mc2
Total energy is sum of these energies.


  • E=π22n22M2a2, where n2=n2x+n2y+n2z, n=1,2,3...



2.)Solution:


a.) H=(w1vvw2)