Suppose I have a 1-D harmonic oscilator with angular velocity ##\omega## and eigenstates ##|j>## and let the state at ##t=0## be given by ##|\Psi(0)>##. We write ##\Psi(t) = \hat{U}(t)\Psi(0)##. Write ##\hat{U}(t)## as sum over energy eigenstates.
I've previously shown that ##\hat{H} = \sum_j...
Right. That makes a lot of sense (and explains why they're telling my to vary ##V_0/t##.. Changing this little thing gets a much nicer curve, I'll still need to verify if it's the correct curve, but I call this progress :) Thanks!
Edit: Turns out I'm ~2 orders of magnitude off now.. Progress...
I'm working on an assignment where I'm required to numerically find the energy of a delta-potential's bound state. To do this, we've converted the time-independent schrödinger equation to an eigenvalue problem with E the eigen value, ψ the eigen vector and H a matrix as follows:
with ##t =...
I agree. The derivation gets a little hairy though, and already got 5 pages of those, so I decided to keep it out for now. I can always add it, but that's a lot of work which is time I'll have to pay for somewhere else. Usually I derive everything I use, even little things such as the inertial...
Yeah, I found that too. But I like this because it shows a different angle of the situation, which is what I was attempting to do. Doing mechanics without differential equations is like doing special relativity without lorentz transformations, it's just not happening. But that doesn't mean you...
I have no objection against the calculus. I'm just trying to explore the science: to find out if there are other avenues that would still work. That includes a different way of measuring the torsion constant if such a way exists (provided that you don't directly measure it but use the...
Yeah, I agree.
The source you've given is still using differential equations of motion though. Is that really something you're fundamentally stuck to?
Edit: No, I'm unable to measure the period in a vacuum chamber, the anti-vibration desks just wouldn't be able to be moved.
"Where air resistance does become relevant is in finding the value for $\kappa$ (torsion spring coefficient)." ^^
Well, mainly because I just want to explore the science behind it. I can solve the differential equation, or I can try to find a different way which should also work ^^
I'm only...
Homework Statement
I'm currently working on the "cavendish" experiment and wish to use/develop a method separate from the casus we've been provided. Now I've nicely calculated and derived everything I need to know, including all the corrections that have to be made for the mass of the rod, the...