Timeline for Animated Wave propagation using split operator method
Current License: CC BY-SA 3.0
24 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 12 at 23:47 | comment | added | Jens | @ayr For this differential equation, you can't specify spatial boundary conditions (that's what you'd do in the time-independent Schrödinger equation which is of elliptical type). Instead, the entire solution is determined by the initial condition - no spatial boundaries at all. | |
| Mar 12 at 15:04 | comment | added | ayr | Please explain how the boundary conditions are set here? Are Dirichlet and Neumann conditions used? | |
| Aug 1, 2015 at 19:44 | comment | added | user0322 | I try this one but feels missing some concepts: Table[Plot3D[{-I NIntegrate[Conjugate[\Psi 1[x, y, t] + \Psi 2[x, y,t]]*(xD[\Psi 1[x, y, t] + \Psi 2[x, y,t], {y, 1}]-y D[\Psi 1[x, y, t] + \Psi 2[x, y, t], \{x,1\}])\}, \{x, -5, 5\}, \{y, -5, 5\}], \{t, 0, tMax, tMax/20\}] | |
| Aug 1, 2015 at 15:10 | comment | added | user0322 | I would like to plot the expectation value of the angular momentum <Lz> as a function of t but seems a problem with the integration. Any comments would be greatly appreciated. | |
| Jul 6, 2015 at 19:07 | comment | added | user0322 | I really appreciated for your time and for the solution. This is what I wanted and now sure of calculating the expectation value of the angular momentum. | |
| Jul 6, 2015 at 2:42 | history | edited | Jens | CC BY-SA 3.0 | Norm preservation |
| Jul 6, 2015 at 2:37 | history | edited | Jens | CC BY-SA 3.0 | Norm preservation |
| Jul 5, 2015 at 19:45 | comment | added | user0322 | I tryed this [Beta]1[t_] := NIntegrate[Abs[[CapitalPsi]1[x, y, t]]^2 + Abs[[CapitalPsi]2[x, y, t]]^2, {x, -5, 5}, {y, -5, 5}]; but not constant. I guess I made a mistake. ListPlot[Table[{t, [Beta]1[t]}, {t, 0, tMax}], PlotStyle -> Thickness[0.005], PlotRange -> All] | |
| Jul 5, 2015 at 19:12 | history | edited | Jens | CC BY-SA 3.0 | Omitted a term that was zero anyway, but should be there for completeness |
| Jul 5, 2015 at 19:09 | comment | added | Jens | I'm pretty sure that you can get this conservation law only if you integrate over all space, not locally at every point individually. It's the continuity equation. | |
| Jul 5, 2015 at 16:42 | comment | added | user0322 | I would like to ask one comments regarding the the total number of particles as a function of time over the phase space. I tyed using Plot[Abs[[CapitalPsi]1[1, 1, t]]^2 + Abs[[CapitalPsi]2[1, 1, t]]^2, {t, 0, tMax}, PlotRange -> All] for fixed values of x & y but the number of particles is not constant. And would like over the possible ranges in x & y, like {x, -5,5},{y,-5,5}.Any comments would be greatly appreciated. | |
| May 14, 2015 at 13:49 | comment | added | user0322 | ,everything is perfect now. Thank you for the detailed explanation! | |
| May 5, 2015 at 19:56 | comment | added | Jens | As I said, it's an arbitrary choice, you can change it to whatever you need. There's nothing more to it. | |
| May 5, 2015 at 19:32 | comment | added | user0322 | I completely agree with you. My question is that when taking the Pauli matrices the extra term 1/2? | |
| May 5, 2015 at 18:48 | comment | added | Jens | You can put in any parameters you like. It depends on the physical constants of your problem, such as Fermi velocity etc. But choosing the numerical values of the constants in the equation is not a Mathematica issue. | |
| May 5, 2015 at 18:09 | comment | added | user0322 | the term1/2 in the Pauli matrices makes no difference on the plot? | |
| May 4, 2015 at 19:27 | comment | added | user0322 | When I insert it a little bit changed the shape of the plot. | |
| May 4, 2015 at 19:25 | comment | added | Jens | Yes, you're right - that was intentional. Such prefactors aren't important in linear differential equations, but of course they do affect your normalization integral. Since the question asked for a plot, I left out the prefactor because it doesn't affect the displayed shape. | |
| May 4, 2015 at 19:22 | comment | added | user0322 | thank you so much! In the initial state you forgot the term: b/(2 Sqrt[[Pi]]) | |
| May 4, 2015 at 19:17 | comment | added | Jens | You can't make a plot of that because there are no more variables to vary when you integrate. Also, the integration has to be within the NDSolve domain. | |
| May 4, 2015 at 18:52 | comment | added | user0322 | I tryed this but couldn't give the plot Plot3D[Integrate[ Abs[[CapitalPsi]1[x, y, 1]]^2 + Abs[[CapitalPsi]2[x, y, 1]]^2, {x, -10, 10}, {y, -10, 10}], ColorFunction -> "Rainbow"] | |
| May 4, 2015 at 18:26 | comment | added | Jens | The 2 is added and subtracted only to offset the 3D plots vertically, so you can see them better. You can change the plot to Plot3D[ Abs[\[CapitalPsi]1[x,y,t]]^2+Abs[\[CapitalPsi]2[x,y,t]]^2, ... and that will produce the modulus squared. For more information on the styling, also have a look at the options of the Plot3D command. | |
| May 4, 2015 at 14:57 | comment | added | user0322 | ,it is a classic solution. This is what I wanted. But I have some unclear points like 'Re[Ψ1[x, y, t]] - 2, 2 + Re[Ψ2[x, y, t]]'. I don't understand the term 2. And I would like to plot the integral of the squared absolute value of the wavefunction over space (The probablity density function |ψ1[x, y, t]|^2+|ψ2[x, y, t]|^2 at any time t=5). Later on I will calculate the expectation value of the angular momentum. | |
| May 4, 2015 at 5:08 | history | answered | Jens | CC BY-SA 3.0 |