Question

Need some quantum mechanics help

Answer 1

\[\huge \Psi (x,t) = Ae^{i(kx-wt)}\] I need this to satisfy both the time dependent Schrodinger equation and the classical wave equation.

Answer 2

So the schroindger equation is \[-\frac{ h^2}{ 2m } \frac{ d^2 \psi (x,t) }{ dx^2 }+U(x)\psi = ih \frac{ d \psi(x,t) }{ dt }\] d's are differentials, and the classical wave equation is \[\frac{ h^2k^2 }{ 2m }+U_0=h \omega\]

Answer 3

I'm not exactly sure how to do this as I've not done multi variable calculus before but, I think I do know the basics.
l would think,
\[\large-\frac{ h^2}{ 2m } \frac{ d^2 Ae^{i(kx- \omega t)} }{ dx^2 }+U(x)\psi = ih \frac{ d Ae^{i(kx- \omega t)}) }{ dt }\]

Answer 4

\[\large-\frac{ h^2}{ 2m } \frac{ d (ik)Ae^{i(kx- \omega t)} }{ dx }+U(x)\psi = ih -i \omega Ae^{i(kx- \omega t)}\] does this look right so far?

Answer 5

\[\large - \frac{ h^2 }{ 2m } i^2k^2Ae^{i(kx- \omega t)}+U(x) \psi = ih - i \omega Ae^{i(kx- \omega t)}\]

Answer 6

Mhm, \[\huge Ae^{i(kx- \omega t)}\] would get cancelled out, one i will get cancelled out but it doesn't seem to be right.

Answer 7

Oh the schrodinger wave equation should be \[-\frac{ h^2}{ 2m } \frac{ d^2 \psi (x,t) }{ dx^2 }+U(x)\psi (x,t) = ih \frac{ d \psi(x,t) }{ dt }\] psi(x,t) in the middle as well.

Answer 8

Actually I got most of it, now I'm at this point: \[-\frac{ h^2 }{ 2m } i^2k^2+U(x)=-i^2hw\]

But now that I want to cancel the i^2 I think there is a little problem since the U(x) is there, so should I still divide? But it will be -U(x)/i^2 though.

Answer 9

i^2 is just -1

Answer 10

Oh right!

Answer 11

Thanks :)! That fixes things

Answer 12

It totally slipped my mind haha, thank you very much.

Answer 13

Are you sure the psi in the middle is also a function of x and t?

... i've only seen this eq once ... and avoided it since.

Answer 14

Yes, it is :) but it gets cancelled out.

Answer 15

Cancelled out as far as this question goes, but to answer your question, yup it's there!