Lowering Operator Simple Harmonic Oscillator n=3

[njdevils45]

Homework Statement

Show that application of the lowering Operator A^- to the n=3 harmonic oscillator wavefunction leads to the result predicted by Equation (5.6.22).

Homework Equations

Equation (5.6.22): A^-Ψ_n = -iΨ_n-1√n

The Attempt at a Solution

I began by saying what the answer should end up becoming. I said that in the end I should get -iΨ₂√3 and that Ψ₂ was equal to the equation I have above. Then I took the equation for Ψ₃ and applied the lowering operator to it in an attempt to get what the prediction is. However, once I applied the lowering operator, I end up getting stuck after taking the derivatives and I get something that looks absolutely nothing like the prediction. I'm confident I took the derivatives correct though since they were never really complicated to begin with. I'm just wondering if maybe my setup is incorrect?

 

[TSny]

Your expression for the lowering operator $A^-$ is not quite right. If you check the dimensions of $\frac{d}{dx}$ and $\alpha^2 x$ you will find that these two expressions don't have the same dimensions if $\alpha$ is defined as $\frac{m \omega}{\hbar}$. There is a simple fix for this. If that's not the source of your difficulty, then it would help to see the details of your work.

 

[njdevils45]

I'll be honest I found these equations from online. None of them were given to me in the problem, it just expects me to either know the equations or look them up. I guess it's possible I'm using the wrong lowering operator. Is it possible that the α in the denominator should be in a square root and the α next to the "x" should not be raised to the 2nd power?

 

[TSny]

I guess it's possible I'm using the wrong lowering operator. Is it possible that the α in the denominator should be in a square root and the α next to the "x" should not be raised to the 2nd power?

Yes, that's right.

 

[njdevils45]

Ok great! In that case I'll try again thursday when I have time. Thank you!