Computation of Wavefunctions of the Harmonic Oscillator
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** Please see the attached file for the full problem description **
Use Gaussian multiplication on the Hermite polynomials in the attached document. These give the un-normalized wave functions for the levels of the harmonic oscillator.
The 3rd and 4th Hermite polynomials are, respectively:
H_2(x) = 4x^2 - 2
H_3(x) = 8x^3 - 12x
When multiplies by the Gaussian, e^-ax^2 /2 (please see the attached file), these give the un-normalized wave functions for the v = 2 and 3 levels of the quantum harmonic oscillator.
a. Show that the resulting wave functions are orthogonal to each other.
b. Find the normalization constants for the two wave functions.
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Solution Summary
In this solution, we use the orthogonality of the Hermite polynomials to compute the un-normalized wavefunctions psi_2 and psi_3 of the harmonic oscillator.
Solution Preview
** Please see the attachment for the complete solution **
We are given the following Hermite polynomials:
(please see the attached file)
a. We wish to show that the un-normalized wave-functions
(please see the attached file)
and
(please see the attached file)
are orthogonal over the interval (please see the attached file) where (please see the attached file) is a positive constant.
We need to show the following:
(please see the attached file)
We have:
(please see the attached file)
Now we make the substitution:
(please see the attached ...
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