Would any one be kind enough to walk me through finding a general solution of this partial differential equation? I've tried pretty hard, and I'm kind of getting the separation of variables, but really I'm lost. This is not a homework assignment (it was a week and a half ago- I've already turned it in), so I'm not asking anyone to do my homework for me. It's just that I'm sure I will see this on the exam, and one walk through would really help me to understand it. All the tutorials I've seen skip a million steps and don't really explain it. So, thanks to whoever helps!

Here is the Schrödinger equation:

(-ћ^2/2m) [ (∂^2 (Ψ (x, y))/∂x^2 ) + (∂^2 (Ψ (x, y))/∂y^2) ] = EΨ(x, y)

As you can see, there is a second partial derivative of Ψ (x, y) with respect to x, and one with respect to y. When I tried to solve this, I moved the (-ћ^2/2m) to the other side of the equation and let E/(-ћ^2/2m) = - k^2, then added k^2 Ψ(x, y) to both sides, giving:


∂^2 (Ψ (x, y))/∂x^2 ) + (∂^2 (Ψ (x, y))/∂y^2 + k^2 Ψ(x, y) = 0

Just to clean it up, I'll just write Ψ instead of Ψ(x, y). At this point we are all on the same page about what variables the function contains.

∂^2 Ψ /∂x^2 + ∂^2 Ψ/∂y^2 + k^2 Ψ = 0



If that helps any. Anyway, if anyone has the time, I would like to see this worked out using separation of variables to a general solution (not just a particular solution- please show me the whole thing, if you will). Also, if someone wanted to be very nice, please also work this out by first converting to polar coordinates.

I know I'm asking a lot, but it's pretty crucial that I understand how this process works. To whomever is extra nice and wants to show me how to do this, thank you very much!

I DO know how to solve a PDE that does not have the additional function on the other side of the equation, however. I can do a standard wave equation. If anyone can tell me what to do with the EΨ(x, y) and the (-ћ^2/2m) I think I would be able to solve it. Thanks.

Hi Grasshopper, nice to see you again!

There are two possible cases, depending on whether E is positive or negative.

In one case, you have a wave with sine and cosine, which can be propagating or standing. This is the case when the particle is free, at least on the portion of space with sin and cos solutions.

In the other case, the solution is just an exp function, characteristic of a fading wave, where the particle doesn't propagate because the particle is trapped somewhere else. The particle has some probability of presence in this portion of space because it's a wave - call it tunnel effect if you wish.

Both cases are united if you treat the solution with complex numbers. Sine and cosine are then combinations of exponentials.

People traditionally write sin(kx*x) and sin(ky*y) - or cos or exp or cosh or sinh - and k^2 = kx^2 + ky^2 where k is the wavenumber with components along X and Y. Then h*k/2pi is the momentum, which is linked to E and m.

In a real (but modelled...) physical problem, one would use his intuition to choose the right function (sin, exp, propagate...) on each portion of space, and probably adjust coefficients so that different portions make proper transitions.

Good luck!

E can't be negative in this case because there is no potential term. Only positive energy solutions are physical.

As for writing the Laplacian in circular coordinates, it's a fairly tedious and uninspiring thing to do. You can look it up on wolfram mathworld - just google. Otherwise it's spending half an hour or so being careful with derivatives and coordinate transformations. Remember that x = r cos theta and y = r sin theta.

lol, good to see you too!

mr_homm has pretty much taken care of the turning it to polar coordinates part. The only thing I need is to make sure that I really do know how to use the separation of variable technique.

At this point, I'm still missing something. But hey, you learn as you go...



anyway, I'll look up the Laplacian, but really I need the practice with the half-hour of careful derivatives.

There's really nothing all that magical about separation of variables. In your example you have a function of both x and y, Ψ(x, y) and by separating variables you simply replace that by a product of functions of x on it's own and y on it's own, viz; Ψ(x, y) = X(x) Y(y) Doing this reduces the problem because the partial derivatives only act on a single variable. for example

∂^2 (Ψ (x, y))/∂x^2 = ∂^2 (X(x) Y(y) ))/∂x^2 = Y(y) ∂^2 X(x)/∂x^2 = Y(y) d^2 X(x) / dx^2

you can then rearrange the equation such that it is in the form;

functions and derivatives of x = functions and derivatives of y

If this is the case both sides must equal a constant and you have reduced the problem of a PDE in 2 variables to 2 ODE's. For the Cartesian coordinates you have here the solutions will be complex exponentials, but doing this after you've gone to spherical coordinates will give you a different and more complicated solution. I've never done this problem, but a reasonable guess at solutions is a Bessel function and a Legendre polynomial.

Well, my problem is that I don't really know how to solve an ODE of this form:

d^2X/dx^2 + d^2Y/dy^2 = -k^2 XY


As I understand it, the trick is to assume that each is equal to the same constant. So, letting that constant be p^2, you could have:

d^2X/dx^2 -Xp^2 = 0

and

d^2Y/dy^2 -Y(p^2 k^2) = 0

right?

If that is the case, then I could just use the auxiliary function to solve these, right?

For the first one:

m^2 - p^2 = 0

==> m1 = p, m2 = -p, ==> X = c1e^px + c2e^-px

correct?

Then there would be something similar with Y, except it would include the imaginary number in the exponential (because m^2 + k^2 = 0 has no real root).

And of course, c1e^px is nonsense because the wave would exponentially increase outside of the boundary, so I'd multiply the second part of the X solution by the Y solution, right?

And from there, just sum up every linear combination of that?