How Do You Calculate Expectation Values in Quantum Mechanics?

[Frank Einstein]

Homework Statement

A system's state of spin 1/2 is represented at t=0 by C*exp[-a²(p-p₀)²]*{{1,0},{0,1}} where the density matrix is represented in the base of eighenvalues of S_z and the spatial vector is represented in the continuum base of statesP_x, P_y, P_z.
Find <X>, <Px> and <ΔX>, <ΔPx>

Homework Equations

P_{A, ρ}=Tr(ρE_A)=ΣWi<ψi|A|ψi>
(ΔA)²=<A²>-<A>²

The Attempt at a Solution

To calculate <X> I change ψ(p) to ψ(x)=1/(π^0.25*√aħ)*exp[-0.5*(x/aħ)-ix(p₀/ħ)].
Now I have to find ΣWi<ψi|X|ψi>. Here it is where I start getting lost; I think I have to integrate the spatial part of ψ multiplied by x between +- infinite.
Can someone please tell me if I'm right?

Thank you very much.

 

[anathema]

Yes, you are correct. To calculate <X>, you need to integrate the spatial part of ψ multiplied by x over all space. This will give you the expectation value of the position operator. Similarly, to calculate <Px>, you need to integrate the spatial part of ψ multiplied by p over all space. To calculate <ΔX>, you need to take the square root of <X^2>-<X>^2, and similarly for <ΔPx>, you need to take the square root of <Px^2>-<Px>^2. Remember to use the appropriate normalization factor for the spatial part of ψ.