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Schrödinger's equation, potential function

Started by uutee Dec 22, 2003 at 1:18 PM 3 replies 1.8k views

Original Post

Dec 22, 2003 at 1:18 PM

Hi, I''m working on some PDE-based visual effects (strictly speaking not game development, mostly technical demos, but no homework stuff in any case =) Let psi denote a 2d or 3d field of complex numbers. Let N denote the nabla operator, ^ powering and * any kind of multiplication. Now we have: i * (doo psi / doo t) = -k * N^2 * psi + V*psi (i is the imaginary unit, t is the time, k is a constant and V is a potential function) Now I need to ask, is this the General Form of time-dependent Schrödinger''s equation? It has the wave component (N^2*psi) and a potential function (V*psi), but is this enough? I''ve implemented a simulator without the potential function and it produces pretty cool looking wavy effects, but I have no idea what kind of functions people use for the potential function. Does the potential function only depend on the spatial position and time, or does it depend on psi too? Of course as an effect coder I''m free to make such choices, but I''d like to know Expert''s Views on how this stuff goes in real theory =) Thanks in advance, - Mikko Kauppila

uutee

Dec 22, 2003 at 2:54 PM

Thanks for the quick reply AP =)

>>You seem to have left out a factor h-bar somewhere
>>but otherwise it looks ok.

But the h-bar really is just a factor? As far as I remember it only depends on particle momentum... so I can probably implicitly include the h-bar in my k-constant (physical exactness isn''t necessary, this is just largely about visuals)?

>>The potential function is only a function of position and >>time. Classic potential functions include particle in
>>a box and harmonic oscillator.

Ah, millions of thanks. I just wonder what these functions look like, analytically? I''ll try do some google searching too.

Lots of thanks!
- Mikko

Timkin

Dec 22, 2003 at 5:23 PM

Schroedinger''s Equation is just a special case of the more general differential Chapman Kolmogorov equation. If you ignore discrete state changes in psi (your density function) then you arrive at the Fokker-Planck equation, which is the over-arching equation for all drift-diffusion processes (of which, again, Schroedinger''s is a special case).

Your potential function can be anything you like... but not all potential functions are realisable.

Just a quick question for you... did you solve for psi on a grid, versus time, or did you solve the k''th order moments of psi (e.g., k=1,2,3)? How long does it take you to perform an iteration? Did you solve for arbitrary psi at t=0?

Cheers,

Timkin

uutee

Dec 22, 2003 at 8:22 PM

>>Just a quick question for you... did you solve for psi
>>on a grid, versus time, or did you solve the k''th
>>order moments of psi (e.g., k=1,2,3)? How long does it
>>take you to perform an iteration? Did you solve for
>>arbitrary psi at t=0?

Initially I set psi to be some "nice looking" field (discretized as a grid of complex numbers). Then in each iteration I evaluate psi forwards in time using Euler-Cromer integration. It''s quite stable, and definitely realtime on a 2d field.

A guy on IRC explained me the harmonic oscillator potential function... it did give some further randomness to the to the look of the simulation, but looked a bit like old sineplasmas, so I turned away from potential functions =) So yeah, now it''s just a wave solver.

- Mikko

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