Position expectation value in harmonic oscillator

[Trajito]

Hello,

I want to find <x_ft_f|x(t)|x_it_i> in harmonic oscillator.

I tried to insert the complete set of energy eigenstates to the right and the left side of x(t), but it yields somewhat more complicated stuff.

Thank you

 

[BruceW]

Shouldn't the bra and ket contain the same quantum state when you're trying to find the expectation value? Also, I thought the harmonic oscillator eigenstates were stationary states (not depending on time)?
If you think about it, the expectation value of position is simply the position around which the particle is oscillating, since the system is symmetric. I.E. if the system takes x=0 as the position the particle oscillates around, then the expectation value of position is simply 0.

 

[matonski]

Shouldn't the bra and ket contain the same quantum state when you're trying to find the expectation value?

The more general term is "matrix element" when the bra and ket refer to different states.

Also, I thought the harmonic oscillator eigenstates were stationary states (not depending on time)?

You can find the expectation value of an operator in any state, not just an eigenstate.

If you think about it, the expectation value of position is simply the position around which the particle is oscillating, since the system is symmetric. I.E. if the system takes x=0 as the position the particle oscillates around, then the expectation value of position is simply 0.

The expectation value depends on the state. It is only zero in some states.

 

[Matterwave]

Hello,

I want to find <x_ft_f|x(t)|x_it_i> in harmonic oscillator.

I tried to insert the complete set of energy eigenstates to the right and the left side of x(t), but it yields somewhat more complicated stuff.

Thank you

What states are the initial and final states in that matrix element?

 

[BruceW]

O I get it now, his problem is a general quantum state, not an energy eigenstate of the system.
I wouldn't try to insert the complete set of energy eigenstates, instead just put in the actual wavefunction, and try to do the integration over all space.

 

[Trajito]

There is no "actual" wave function. Only that we know is particle is found at the position x_i at the time t_i, and is found at the position x_f at the time t_f. The question is what is the expectation value of x at a time t which is in the interval [t_i, t_f].

Maybe if I write my trial solution, it can be more clear.

<x>_t = <x_ft_f|x(t)|x_it_i> = <x_ft_f|n><n|x(t)|m><m|x_it_i> where |n> and |m> are the energy eigenstates. Then,

<x>_t = $$\sum_{n,m}\Psi_{n}^{*}(x_{f},t_{f})\Psi_{m}(x_{i},t_{i})$$<n|x(t)|m>

<n|x(t)|m> = $$\frac{1}{\sqrt{2}}e^{-i(E_{m}-E_{n})t}$$ <n|a+a⁺|m> where we take $$\hbar$$ and $$\omega$$ to be 1.

<n|a+a⁺|m> = $$\sqrt{m}\delta_{n,m-1}+\sqrt{n}\delta_{n,m+1}$$

Here, substitution and making use of Hermite polynomials yields complex valued expectation values.

 

[BruceW]

I don't think
$$<n|x(t)|m> = \frac{1}{\sqrt{2}}e^{-i(E_{m}-E_{n})t}<n|a+a^{+}|m>$$
is right, I reckon it should just equal $<n|a+a^{+}|m>$ (times some constant, maybe).
Then you would get a real expectation value. But it looks like the expectation value would be infinite, since it is a sum of all the energy eigenstates. I think this may be due to the difficulties in normalising a quantum state of exact position.

I'm not an expert, but maybe it would work better if you say that the state has a very small spread in position at certain points in time. (Rather than exact position, since I think that is what is making this problem difficult).

 

[Trajito]

We must have a term involving t, otherwise the result wouldn't depend on t, which makes the expectation value of x the same at any time, which is impossible since at t=t_i it should be equal to x_i and at t=t_f it should be equal to x_f.

 

[SpectraCat]

There is no "actual" wave function. Only that we know is particle is found at the position x_i at the time t_i, and is found at the position x_f at the time t_f. The question is what is the expectation value of x at a time t which is in the interval [t_i, t_f].

Maybe if I write my trial solution, it can be more clear.

<x>_t = <x_ft_f|x(t)|x_it_i> = <x_ft_f|n><n|x(t)|m><m|x_it_i> where |n> and |m> are the energy eigenstates. Then,

<x>_t = $$\sum_{n,m}\Psi_{n}^{*}(x_{f},t_{f})\Psi_{m}(x_{i},t_{i})$$<n|x(t)|m>

<n|x(t)|m> = $$\frac{1}{\sqrt{2}}e^{-i(E_{m}-E_{n})t}$$ <n|a+a⁺|m> where we take $$\hbar$$ and $$\omega$$ to be 1.

<n|a+a⁺|m> = $$\sqrt{m}\delta_{n,m-1}+\sqrt{n}\delta_{n,m+1}$$

Here, substitution and making use of Hermite polynomials yields complex valued expectation values.

Hmmm .. that doesn't look like an expectation value to me. Expectation values always have the same wavefunction on both sides as far as I know. To be honest, I am not sure I really understand the formulation of the question ... you are representing the particle positions at t_i and t_f as delta functions at x_i and x_f, respectively, right? Now you want an expression for the expectation value of the position during the interval between t_i and t_f? That seems like a problem for Feynman path integrals ... you need a properly weighted sum over all possible paths the particle could have taken to get from x_i to x_f. I suppose it might then be possible to calculate the expectation value of position in the given time interval.

However, I still think the question is ill-posed because you have measured the particle's position twice. I can use a propagator to describe the time-evolution of the wavefunction starting from a delta function at t_i until just before the second measurement. Maybe it is sort of a trick questions and that is what is required? Because you can certainly get an expectation value that way. The fact that you know the position of the particle at t_f seems kinda irrelevant to me ...

 

[BruceW]

SpectraCat has got the right idea - just use the time dependent Schrödinger equation to calculate the state of the system in between the times $$t_i$$.
The only other thing to decide is what exact form should the Dirac delta function take?

 

[A. Neumaier]

There is no "actual" wave function. Only that we know is particle is found at the position x_i at the time t_i, and is found at the position x_f at the time t_f. The question is what is the expectation value of x at a time t which is in the interval [t_i, t_f].

This is not given by your attempted formula.

For the harmonic oscillator you can use Ehrenfest's theorem to get the exact dynamics of the expectation.
But you can impose a boundary state only at a _single_ time.

But maybe you wanted to do the following?
Suppose you prepared the oscillator in a particular state at time t_i. Then you can determine the state at any time by solving the time-dependent Schroedinger equation in a ladder basis and compute the probability of measuring a position at time t_f by squaring the matrix element you wrote down.