Expectation values for a harmonic oscillator

[KaiserBrandon]

Homework Statement

I need to find <x>, <x²>, <p>, and <p²> for a particle in the first state of a harmonic oscillator.

Homework Equations

The harmonic oscillator in the first state is described by $$\psi$$(x)=A$$\alpha$$^1/2*x*e^{-$$\alpha$$*x2/2}. I'm using the definition <Q>=($$\int$$$$\psi$$1*Q*$$\psi$$)dx where $$\psi$$1 is the complex conjugate of $$\psi$$, and Q is the specific operator.

The Attempt at a Solution

I solved for <x>, and found it was zero. <p> I'll solve for in a similar fashion. However, for <x²> and <p²>, I am unsure of what operators I use. For the <x> operator, it is simply x, so for <x²>, would I use x² as an operator?
(note that I the only superscripts here are the ones above e, I don't know why latex is putting all of my symbols so high

 

[kuruman]

For the <x> operator, it is simply x, so for <x²>, would I use x² as an operator?

Yes, that's what you should use.

 

[dx]

(Put the whole equation in the tex barackets instead of individual symbols)

The operator for <x²> is simply multiplication by x², so <x²> = ∫ψ*(x)x²ψ(x)dx, and <p²> is

$$-\int \psi^{*}(x) \hbar^2\frac{\partial^2}{\partial x^2}\psi(x) dx$$

 

[KaiserBrandon]

ok, thank you so much guys. It's a good thing I have physics forum to at least make my new ventures into the realm of quantum mechanics a bit easier :)

 

[paranormal]

)I would like to clarify a few points about the question before providing a response. Firstly, it is not clear what is meant by "the first state of a harmonic oscillator." A harmonic oscillator can have multiple states, each with its own wavefunction and expectation values. It would be helpful to specify which state is being referred to in this question.

Secondly, the wavefunction provided in the problem does not seem to be normalized, which would make it difficult to use for calculating expectation values. It would be important to normalize the wavefunction before using it to calculate expectation values.

Now, for the response, let's assume that the first state of the harmonic oscillator refers to the ground state, which has a wavefunction of \psi(x)=Ae^(-\alpha*x^2/2). In this case, the expectation values can be calculated as follows:

<x> = \int_{-\infty}^{\infty} x|\psi(x)|^2 dx = 0 (since the integrand is an odd function)

<x^2> = \int_{-\infty}^{\infty} x^2|\psi(x)|^2 dx = \frac{1}{2\alpha} (since the integrand is a Gaussian function)

<p> = \int_{-\infty}^{\infty} \psi^*(x)(-i\hbar)\frac{\partial}{\partial x}\psi(x) dx = 0 (since the integrand is an odd function)

<p^2> = \int_{-\infty}^{\infty} \psi^*(x)(-i\hbar)\frac{\partial}{\partial x}\left[(-i\hbar)\frac{\partial}{\partial x}\psi(x)\right] dx = \hbar^2\alpha (since the integrand is a Gaussian function)

In summary, for the ground state of a harmonic oscillator, the expectation values for position and momentum are zero, while the expectation values for position squared and momentum squared are \frac{1}{2\alpha} and \hbar^2\alpha, respectively.