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Modern Physics
Found in: Page 279
Modern Physics

Modern Physics

Book edition 2nd Edition
Author(s) Randy Harris
Pages 633 pages
ISBN 9780805303087

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Short Answer

Consider a cubic 3D infinite well.

(a) How many different wave functions have the same energy as the one for which (nx,ny,nz)=(5,1,1)?

(b) Into how many different energy levels would this level split if the length of one side were increased by 5% ?

(c) Make a scale diagram, similar to Figure 3, illustrating the energy splitting of the previously degenerate wave functions.

(d) Is there any degeneracy left? If so, how might it be “destroyed”?

(a) There are two other wave functions having the same energy as the one for which nx,ny,nz=5,1,1 .

(b) This energy level would split into two new energy levels if the length of one side is increased by 5% .

(c) The scale diagram is as follows

(d) There will still be two wave functions having the same energy. This can be removed by changing any of the side lengths by a small amount.

See the step by step solution

Step by Step Solution

Step 1: Given data

There is a 3D infinite cubic well.

Step 2: Energy in an infinite well

The energy of a particle of mass m in a 3D infinite well of sides Lx, Lyand Lz is

E=π222mnx2Lx2+ny2Ly2+nz2Lz2 ..... (I)

Here, is the reduced Planck's constant and nx, ny and nz are the quantum numbers

Step 3:Determining the degeneracy of energy in a cubic well


For a cubic well, the energy in equation (I) reduces to

E=π222mL2nx2+ny2+nz2 ..... (II)

Thus the energy when any one of nx,ny,nz is 5 and the rest two are 1 are the same. Hence the wave functions corresponding to nx,ny,nz=5,1,1, nx,ny,nz=1,5,1 and nx,ny,nz=1,1,5 have the same energies. There are three wave functions having the same energy.

Step 4: Determining the splitting in energy level


Let the length along X axis, that isLx be increased by 5% . The new length is then $1.05L$ . The energy in equation (II) becomes


Thus the wave function corresponding to nx,ny,nz=5,1,1 will now have lower energy than the wave functions corresponding to nx,ny,nz=1,5,1 andnx,ny,nz=1,1,5 . The previous energy level will thus split into two separate energy levels.

Step 5: Plotting the energy levels


The plot of the initial and final energy levels are as follows

Step 6: Determining the residual degeneracy


There are still two wave functions corresponding to nx,ny,nz=1,5,1 and nx,ny,nz=1,1,5 that are in the same energy level. Thus a degeneracy of 2 is left. This can be removed if one of the lengths along x or y axis is slightly changed making the energy equation equal to equation (I).

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