Showing the expectation values of a system are real quantities

[FisiksIdiot]

Homework Statement

A one-dimension system is in a state described by the normalisable wave function Ψ(x,t) i.e. Ψ → 0 for x → ±∞.

(a) Show that the expectation value of the position ⟨x⟩ is a real quantity. [1]

(b) Show that the expectation value of the momentum in the x-direction ⟨p⟩ is a real quantity, too. Hint: using integration by parts and normalisability show that ⟨pˆx⟩ = ⟨p⟩∗. [4]

Homework Equations

1=N∫ψ*ψ.dx

<x>=∫ψ* xψ.dx over all space

<p>=∫ψ* -ih dψ/dx.dx over all space

The Attempt at a Solution

The difficult aspect of this for me is determining what the correct wave function is. Using the information given I assumed that the correct wave function was e^-ax2/2 e^iEt/h (where a/2 is an arbitrary constant) as it fits the above requirements (I could be wrong.)

However, upon normalising and calculating <x> and <p>, the values obtained will of course will be 0 and therefore real as my assumption was a symmetric wave function. This is all well and good, however the question explicitly states to use integration by parts to solve for <p>.

<p>=N∫(e^-ax2/2 e^iEt/h) -ih d/dx(e^-ax2/2 e^-iEt/h).dx over all space

which gave:

<p>=N iha∫xe^-ax2.dx over all space

which cannot be integrated by parts as far as I understand-perhaps it can?. Have I got the wrong end of the stick somewhere in my thinking?

Thanks in advance.

 

[Bryson]

You wave function is incorrect. Do not plug it in until you have done the math. You do not even need a "test" wavefunction for part a.

 

[FisiksIdiot]

Thanks for the prompt reply, but I don't quite follow. Could you clarify what you mean by 'not needing a wave function'? I feel I am approaching the problem from the wrong angle.

 

[Bryson]

Hmm, perhaps forget what I mentioned previously. You simply need to reconsider your wave function. Perhaps try something of the form $\psi = e^{\pm i(kx-E t/ \hbar)}$. . .