Matrix Mechanics position and momentum operator

[jcharles513]

Homework Statement

Stationary states are related by 1/m*p_ij = E_i-E_j/(i*hbar) *x_ij where p_ij = ∫ψ_i(x)*p*ψ_j (x) and similarly for x_ij

Homework Equations

Classical correspondence for equation of motion is d<x>/t = <p>/m
Schrodinger equation
Ehrenfest Theorem
Fourier Transform

The Attempt at a Solution

I know that ψ(x,t) = ψ_i(x)*e^-iE_it/hbar
And I would know how to do this sort of thing for a linear combination or a specific example i.e. infinite square well. I just don't know how to go about generalizing it. If you could get me started, I can go from there.

 

[mark726]

Hello,

Thank you for your post. It seems like you are asking for assistance in understanding how to generalize the relationship between stationary states. Let's break down the equation given in the forum post and see if we can make sense of it.

First, let's start with the classical correspondence for the equation of motion, d<x>/dt = <p>/m. This equation relates the average position <x> and average momentum <p> of a particle in classical mechanics. In quantum mechanics, these quantities are represented by operators, so we can rewrite this equation as:

d<x>/dt = <p>/m → [d<x>/dt] = <ψ|p|ψ>/m

Next, we can use the Schrodinger equation to relate the momentum operator to the energy operator:

p = -iħ(d/dx) → <ψ|p|ψ> = -iħ<ψ|(d/dx)|ψ>

Now, let's consider the Ehrenfest theorem, which states that the expectation value of an operator obeys the classical equation of motion. Using this, we can rewrite our equation as:

[d<x>/dt] = -iħ<ψ|(d/dx)|ψ>/m → [d<x>/dt] = -iħ/m * d/dt<ψ|x|ψ>

Finally, we can use the Fourier transform to relate the position and momentum operators in the position basis to the position and momentum operators in the momentum basis. This will give us our final equation:

[d<x>/dt] = -iħ/m * d/dt<ψ|x|ψ> → [d<x>/dt] = -iħ/m * d/dt<ψ|p|ψ>

This equation is very similar to the one given in the forum post, and we can see that they are related by a factor of 1/m*pij. This factor comes from the fact that the expectation value of the momentum operator in the momentum basis is given by pij = ∫ψi(p)*p*ψj(p)dp, where ψi(p) and ψj(p) are the momentum space wavefunctions for the states i and j, respectively.

I hope this helps to clarify the relationship between stationary states and the classical correspondence for the equation of motion. Let me know if you have any further questions.