
Check the allowed energies of the centered infinite square well are consistent with equation 2.30, and confirm that the wave function can be obtained from the equation 2.31, and sketch the first three solution.

Answer to Problem 2.36P
The allowed energies of the centered infinite square well are consistent with equation 2.30, and the wave function can be obtained from the equation 2.31. The sketch the first three solution is shown below.
Explanation of Solution
Write the solution of classical simple harmonic oscillator.
The time boundary conditions are
Apply this condition in equation1.
Add equation (II) and (III), and solve.
Where j = 1,2,3,…
Subtract equation (II) and (III), and solve.
If
Let
Then the equation (I) becomes,
Normalise the function in the equation (VI).
If
Let
Then the equation (I) becomes,
Normalise the function in the equation (VIII).
In these both cases, the energy will get as follows.
This is in agreement with equation 2.30 for a well of width 2a.
Use
So graph for the above function for first three solution is shown below.
It is same as Figure 2.2, except that some are upside down.
Conclusion:
Therefore, the allowed energies of the centered infinite square well are consistent with equation 2.30, and the wave function can be obtained from the equation 2.31. The sketch the first three solution is shown in Figure 1.
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