QM Harm. Osc. expectation values

[Tom Pietress]

QM Harmonic Oscillator, expectation values

Hello. I am working on a problem involving the 1-dimensional quantum harmonic oscillator with energy eigenstates |n>. The idea of the exercise is to use ladder operators to obtain the results. I feel I am getting a reasonably good hang of this, but my answer to one of the parts isn't comforting me. It goes like this:

Obtain the matrix elements $\left\langle m\right|\hat{P}^{2}\left|0\right\rangle$ for all m$$\geq$$1

The reason I'm unsure is that this gives me a negative answer - I might be wrong here but that doesn't seem right to me. What I've done (note: I see now that I use m both for mass and the matrix element index; but I believe they are easy to distinguish):

$$\left\langle m\right|\hat{P}^{2}\left|0\right\rangle =\left(i\sqrt{\frac{\hbar m\omega }{2}}\right)^{2}\left\langle m\right|(\hat{a}^{\text{+}}-\hat{a})^{2}\left|0\right\rangle =-{\frac{\hbar m\omega }{2}}\left\langle m\right|\left({\hat{a}}^{\text{+}}\right)^{2}-{\hat{a}}^{\text{+}}\hat{a}-\hat{a}{\hat{a}}^{\text{+}}+{\hat{a}}^{2}\left|0\right\rangle$$

The notation I use is â for the lowering operator and â+ for the raising operator.

My thoughts here are that the second and fourth term vanishes because they will apply the lowering operator on |0>. The first and third remains (although I will later also throw away the third):

$$-{\frac{\hbar m\omega }{2}}\left\langle m\right|\left[\sqrt{2}\left|2\right\rangle -1\left|0\right\rangle \right]=-{\frac{\hbar m\omega }{\sqrt{2}}}\left\langle m\right|2\left.\right\rangle$$

(Not sure if the kind of bracketing I did there is all right, but I believe it does the job.) The reason why 1|0> vanishes is that m never is 0 and thus <m|0> is always 0. If I have understood correctly, the integral on the right will be 0 for every m other than 2 and 1 for m=2. This I gather from the Kronecker delta. Thus the matrix elements will have the (negative!) value $$-{\frac{\hbar m\omega }{\sqrt{2}}}\$$ when m=2 and 0 for all other m.

I would appreciate any comments on what I've done here, since I'm only beginning to come to terms with Dirac notation and quantum mechanics as a whole.

 

[Jhero]

Thank you for your time and help!

Hello, thank you for your post. Your approach to the problem is correct, but there is a small mistake in your final answer. The Kronecker delta in the matrix element will only be 1 when m=2, not 1. Therefore, the correct answer should be -{\frac{\hbar m\omega }{\sqrt{2}}}\ when m=2 and 0 for all other m. This means that the matrix element for m=2 is indeed negative, but this is not a cause for concern. In quantum mechanics, negative values for expectation values are not uncommon and do not indicate any issues with the calculation. I hope this helps clarify your answer. Keep up the good work with your understanding of quantum mechanics!