Expectation value of Momentum

Smythe

I have a question which asks to determine the value of the expectation value of the momentum <p>. Where A=√3

I have attached my solution, and under it is the given solution.

You'll notice they are different; have I gone wrong somewhere?

Thanks

Morbert

Smythe wrote: »

I have a question which asks to determine the value of the expectation value of the momentum <p>. Where A=√3

I have attached my solution, and under it is the given solution.

You'll notice they are different; have I gone wrong somewhere?

Thanks

Do they give the wave function?

Your integration is correct.

http://www.wolframalpha.com/input/?i=integrate+x%5E2+from+0+to+1&lk=4&num=1

Smythe

Morbert wrote: »

Do they give the wave function?

Sorry, yes I should have included that. It states:

A particle limited to the x axis has a wavefunction ψ=Ax between x=0 and x=1 and zero elsewhere.

The answer you got is definitely correct. It looks like they worked out that the integral of x rather than x^2 by mistake.

Lbeard

Morbert wrote: »

Do they give the wave function?

Your integration is correct.

http://www.wolframalpha.com/input/?i=integrate+x%5E2+from+0+to+1&lk=4&num=1

Damn....I didn't know Wolfram could do that.

Morbert

Smythe wrote: »

Sorry, yes I should have included that. It states:

A particle limited to the x axis has a wavefunction ψ=Ax between x=0 and x=1 and zero elsewhere.

Hmm, so the wave function is not an eigenfunction of momentum. Very strange.

[latex]\int \psi^* \hat{p} \psi dx[/latex]
[latex]\Rightarrow -i\hbar \int \psi^* \frac{\partial \psi}{\partial x}dx[/latex]
[latex]\Rightarrow -i\hbar \int Ax A\frac{\partial x}{\partial x}dx[/latex]
[latex]\Rightarrow -i\hbar A^2 \int x dx[/latex]
[latex]\Rightarrow -3i\hbar \left[\frac{x^2}{2}\right]^1_0[/latex]
[latex]\Rightarrow -3i\hbar/2[/latex]

Ah I see what went wrong. The provided solution's description of how to get the answer is wrong, rather than the answer itself.

Smythe

Morbert wrote: »

Hmm, so the wave function is not an eigenfunction of momentum. Very strange.

[latex]\int \psi^* \hat{p} \psi dx[/latex]
[latex]\Rightarrow -i\hbar \int \psi^* \frac{\partial \psi}{\partial x}dx[/latex]
[latex]\Rightarrow -i\hbar \int Ax A\frac{\partial x}{\partial x}dx[/latex]
[latex]\Rightarrow -i\hbar A^2 \int x dx[/latex]
[latex]\Rightarrow -3i\hbar \left[\frac{x^2}{2}\right]^1_0[/latex]
[latex]\Rightarrow -3i\hbar/2[/latex]

That's great. Thanks very much for that.

So it appears that the original provided solution is still incorrect with respect to the fact that it is not negative, or alternatively it should have the 'i' in the denominator if left as a positive value.

Would that be correct?

Morbert

Smythe wrote: »

That's great. Thanks very much for that.

So it appears that the original provided solution is still incorrect with respect to the fact that it is not negative, or alternatively it should have the 'i' in the denominator if left as a positive value.

Would that be correct?

I believe so.