Question

Separation of variables, Schrodinger's Equation.
For the first part of a physics homework question, I am given an equation and am told to separate the variables in a certain way. I am pretty sure this is part of ordinary differential equations, which I have never learned. (Not a pre- or co-requisite of my course.) Here is what I am asked - any help would be appreciated! Thank you!

The Schrodinger equation for two dimensions is \[(\frac{\delta ^2}{\delta x^2} \frac{\delta ^2}{\delta y^2}) \psi (x,y) = - \frac{2m(E-U)}{\hbar ^2} \psi (x,y)\]
U is constant.

I am supposed to "separate variables by trying a solution of the form \[\psi (x,y) = f(x) g(y)\] then dividing by \[f(x) g(y)\]." It also says to call the separation constants \[C_x \]and\[ C_y\]

Answer 1

I'm sorry I'm a bit rusty on def eq, maybe Paul's could help? http://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx He has a dif eq page

Answer 2

Thank you for looking at it! :) I'll check it out!

Answer 3

Np, and you can always look at khan academy They might have something on Schrodinger's

Answer 4

Thank you for your help! Let's see how fast I can learn this stuff!

Answer 5

haha quantum mechanics?

Answer 6

http://www.khanacademy.org/math/differential-equations/first-order-differential-equations/differential-equations-intro/v/what-is-a-differential-equation
I didn't watch it but I think this is your problem :)

Answer 7

Yep! Or at least this ODE stuff! I mean, can't be that hard.. Quantum Mechanics... The course is on wave-particle theory... I think they should include more math as a prerequisite! Calc 3 is the only math corequisite. I guess it would make the degree take a lot longer to get, though, if ODE was required.. Back to work, now!....

Answer 8

Thank you for all your help!

Answer 9

And have a good night! :)

Answer 10

woww... ok well now that I'm curious if I get it I'll post it. Good luck, and good night!

Answer 11

Haha, okay! Please don't tax yourself for my sake!

Answer 12

nah I have PDE over the summer so it should be a good review. ODE is in the fall too so I should practice :)

Answer 13

I see! God luck in both! I'll be taking ODE in the fall, but reading about it over the summer!

Answer 14

I apologize if this problem does end up involving physics!

Answer 15

meh it might but it's still fun for a nerd like me

Answer 16

ok need more info, do you know the values for m, E, and h bar?

Answer 17

Constant... h bar is 1.055*10^-34 [some units]. I imagine that they should stay in the equations, too, to be exact.

I'm sorry I didn't say that they're all constant! That is a physics part... My bad! :\
The book didn't specify in the question, because it's taken for granted in a physics context.

Answer 18

ahh then it's not bad I'm going to simplify that fraction to P ok?

Answer 19

Watching Khan Academy's video you sent me, the "solution" thing makes so much more sense! I need to use ODE stuff to find \[\psi(x,y)\]and it should then, somehow, be expressed as a function of x multiplied by a function of y.

The "P" thing is fine with me!

Answer 20

well it's actually a PDE, due to the partials give me a few more minutes and I'll have it(hopefully)

Answer 21

Thank you! It doesn't hurt if you don't get it - so no pressure! You've already helped so much!

Answer 22

http://www.math.umn.edu/~olver/pd_/sv.pdf p 123 should help

Answer 23

I think it's in Greek though. Just kidding.... Thank you! I'll read more than the first sentence now...

Answer 24

p 122 at the bottom

Answer 25

It is almost exactly your problem :)

Answer 26

Thank you! Very much! I'll continue to look at it. I added it to my favorites, too. If you ever need help with something I might know, tag me if you want a hand. I owe you. I don't get on Open Study much because I have been very busy with my own work, but at least my e-mail will have a notification. I don't know how much help I could, be, but I can try.

Answer 27

Np and your welcome, I hate physics so hopefully I can avoid it, but at least now I know who to call for help ;) If you need any Calc 3 or upper math help you can still ask me, I should be at grad student level this time next year (fingers crossed) and random questions like this are good practice. Good luck with finals!

Answer 28

Cool, good luck! Yeah, if they stick you in a physics course for fun, I'll do my best to help! :)

Answer 29

b) For an infinite well,\[U=\left(\begin{matrix} 0 & 0<x<L, 0<y<L \\ \infty & otherwise \end{matrix} \right)\]
What should f(x) and g(y_ be outside this well?
What functions would be acceptable standing-wave solutions for f(x) and g(y) inside the well?
Are \[C_x\]and\[C_y\] positive, negative, or zero?
Imposing appropriate conditions, find the allowed values of Cx and Cy.

Answer 30

c) How many independent quantum numbers are there?

d) Find the allowed energies E.

e) Are there energies for which there is not a unique corresponding wave function?

Answer 31

thanks!!

Answer 32

Not a problem! Enjoy! If possible! :P

Answer 33

haha i am sure i will

Answer 34

That's scary.
Kidding!

Answer 35

I think the solution was:\[\frac{\delta ^2 f(x)}{f(x) \delta x^2}+\frac{\delta ^2 g(y)}{g(y) \delta y^2}=-\frac{2m(E-U)}{\hbar ^2}\]
I Think that was it...
And \[\frac{\delta ^2 f(x)}{f(x) \delta x^2}=C_x\]\[\frac{\delta ^2 g(y)}{g(y) \delta y^2}=C_y\]since the two terms are independent and add up to be a constant.

Answer 36

I'll ask a few people I know who are really good at this cause it's bothering me lol

Answer 37

Maybe it was physics-related! I know that separation of variables is an ODE procedure and is very helpful in physics, though!

Answer 38

lol did you at least get the answer?

Answer 39

I think that my post from 3 days ago is the answer. I based it on some other math in the textbook that I didn't understand. It was a similar process, in the book.

Answer 40

hmm ok well good luck with finals!

Answer 41

and you can bet I'll be hitting you up for physics help in the future

Answer 42

Also on second look and seeing that the end result is a constant, i agree with your answer

Answer 43

Sounds good! I like Open Study, so I should be on every once in a while! And I'll have more free time next semester. I'll be on in the summer, too.

Yeah! The two terms on the left must be constant because the terms are independent of each other and they add up to a constant! I learned that. And it makes sense, where independent here means that a change in the left term will not affect the value of the right term, which must be added to the left term to arrive at a constant.

Answer 44

I'll be here too and yea i kept forgetting that that big fraction was a constant lol

Answer 45

Yeah, tougher when it's less native to you!! Good luck in PDE! I'll be reading a book from Amazon to learn ODE so it's not as bad in the fall.