Expected value of x for quantum oscillator - integration help

[phosgene]

Homework Statement

I have a wavefunction $Cxe^{-ax^2}$ and I have to find the expected value of x.

Homework Equations

$∫_{-∞}^{∞} x^3 e^{-Ax^2} dx = 1/A^2 for A>0$

The Attempt at a Solution

I get an integral like this:

$<x>=|C|^2 ∫_{-∞}^{∞} x^3 e^{-Ax^2} dx$

After trying integration by parts and failing (miserably), I took the coward's way out and went to an integration table. I found this:

$∫_{-∞}^{∞} x^3 e^{-Ax^2} dx = 1/A^2 for A>0$

My question is this: my $a=ωm/ \hbar$ is always positive, right? I googled stuff on negative frequency, but I don't think that it applies in this situation (quantum oscillator).

 

[phosgene]

Actually, I think I got it using integration by parts, but I'd still like to know if my 'shortcut' is still valid.

 

[Office_Shredder]

If A<0, it's not that you need a different formula, it's that your integral doesn't converge at all.

Although for this specific integral I think you either wrote it down wrong or did the problem wrong because you're integrating an odd function and should just get 0 regardless of what a is (again, as long as it's positive so the integral converges)

 

[phosgene]

I realized that now using Wolfram Alpha...that was my first thought, but then I thought that it would be too easy. I need to stop convincing myself that an easy solution is wrong solution...

 

[paranormal]

I would first like to commend you for attempting to solve the problem on your own before resorting to looking at an integration table. This shows good problem-solving skills and a willingness to learn.

To answer your first question, yes, in the context of a quantum oscillator, the value of a=ωm/\hbar is always positive. This is because the frequency ω is related to the energy of the oscillator, which cannot be negative in quantum mechanics.

Regarding your attempt at integration, it is important to note that integration by parts may not always be the most efficient method for solving a particular integral. In this case, using the integration table was a valid approach and it gave you the correct result. However, it is always good to have a basic understanding of integration techniques and to try them out before resorting to looking at tables or using software.

In summary, your approach was correct and your result is valid. But as a scientist, it is important to have a deeper understanding of the underlying principles and techniques involved in solving a problem. Keep up the good work!