Expectation value and momentum for an infinite square well

Matt Q.
Messages
5
Reaction score
0

Homework Statement

√[/B]
A particle in an infinite square well has the initial wave function:

Ψ(x, 0) = A x ( a - x )

a) Normalize Ψ(x, 0)

b) Compute <x>, <p>, and <H> at t = 0. (Note: you cannot get <p> by differentiating <x> because you only know <x> at one instance of time)

Homework Equations

The Attempt at a Solution


For part a, I figured out A = \sqrt{30 / a^5}

I'm sort of confused for part b. For <x>, I set up the integral like this:

\int_{0}^{a}x Ψ(x, 0)^2 dx

And got \frac{a^6 A^2}{60}, but I'm not sure if I got it right.

For <p> and <H>, I don't know how to set up these integrals. How would I set up these integrals? I don't need you to solve it for me, I just wanted to know how to set them up.

Thank you for reading and helping.
 
Physics news on Phys.org
How does the momentum operator look like in position space, i.e. ##p_x\psi(x) =\ldots##?
 
Matt Q. said:
For <x>, I set up the integral like this:

\int_{0}^{a}x Ψ(x, 0)^2 dx

And got \frac{a^6 A^2}{60}, but I'm not sure if I got it right.
If you plug in your result for ##A##, you get ##\langle x \rangle = \frac a2##. Given the symmetry of ##\Psi(x,0)##, does the answer seem reasonable?
 
blue_leaf77 said:
How does the momentum operator look like in position space, i.e. ##p_x\psi(x) =\ldots##?

I figured it out and got 0 for <p>. That seems reasonable right?

vela said:
If you plug in your result for ##A##, you get ##\langle x \rangle = \frac a2##. Given the symmetry of ##\Psi(x,0)##, does the answer seem reasonable?

Ah I didn't think of that. It makes sense.
 
I also figured out <H> which turns out to be ##\frac{5 h^2}{m a^2}## which seems pretty reasonable too right?
 
Matt Q. said:
I figured it out and got 0 for <p>. That seems reasonable right?
Yes.
 
Hi, I had an exam and I completely messed up a problem. Especially one part which was necessary for the rest of the problem. Basically, I have a wormhole metric: $$(ds)^2 = -(dt)^2 + (dr)^2 + (r^2 + b^2)( (d\theta)^2 + sin^2 \theta (d\phi)^2 )$$ Where ##b=1## with an orbit only in the equatorial plane. We also know from the question that the orbit must satisfy this relationship: $$\varepsilon = \frac{1}{2} (\frac{dr}{d\tau})^2 + V_{eff}(r)$$ Ultimately, I was tasked to find the initial...
The value of H equals ## 10^{3}## in natural units, According to : https://en.wikipedia.org/wiki/Natural_units, ## t \sim 10^{-21} sec = 10^{21} Hz ##, and since ## \text{GeV} \sim 10^{24} \text{Hz } ##, ## GeV \sim 10^{24} \times 10^{-21} = 10^3 ## in natural units. So is this conversion correct? Also in the above formula, can I convert H to that natural units , since it’s a constant, while keeping k in Hz ?
Back
Top