# Normalize Harmonic Oscillator wave functions

The following are two harmonic oscillator wave functions:

S(x) = N exp(-max/2h)(2mwx/h)

S(x) = N exp(-max/2h)[8(mwx/h)^3 - 12mwx/h]

where N = .

Show that they are a) normalized, and b) orthogonal

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Note to student: Please find the answer in the attached file. All these problems involved the integration of one type of standard integral. It is worth to familiarize with this integral and Gamma function, if you are to do more of this type of problem. Thanks.

Solution:

If you note the given wave function, its exponential part has an x instead of x2. Also, a square root is missing towards the end.

Are these Normalized? We need to show that

Denote the Left hand side as I.

I =

I =

I =

I = 2

Let

d

Substituting in I,

I = 2

I =

is a standard integral =

I =

Use the given expression for N in the problem to find N for n = 1.

N(1) =

Substituting in I,

I =

This function is normalized.

Again, in the second function exponential part has an x instead of x2 and a square root is missing towards the end.

Let I =

Let

d

I =

I =

I =

There is a standard integral for these type of integrals.

is the Gamma function.

I =

I =

Find N for n = 3 using the given expression for N.

N(3) =

Substituting in I,

I =

is normalized as well.

(3)

To show that these two wave functions are orthogonal, we need to show that

I =

Let

d

I =

I =

I =

I =

I =

I = 0

Hence these two wave functions are orthogonal.

Â© BrainMass Inc. brainmass.com December 24, 2021, 7:39 pm ad1c9bdddf>https://brainmass.com/physics/wavefunction/normalize-harmonic-oscillator-wave-functions-207617