15 votes
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Computational methods for finding the energy eigenvalues of the time-independent Schrodinger equation with arbitrary potential

Finding the eigenvalues for the Schrödinger equation is really similar to finding the eigenvalues for the wave equation. You start with your differential equation $$\left[-\frac{1}{2}\nabla'^2 + V(r)\...
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  • 8,111
13 votes
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Forcing an ODE solver to preserve the norm

The best approach is to use an ODE solver that is guaranteed to conserve the norm of the initial condition, i.e., for which $\|y_n\| = \|y_0\|$ for all $n\in\mathbb{N}$. Such solvers exist, and are ...
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10 votes

Non-hermitian discretizations in quantum mechanics

If your discrete equations represent your problem properly you should have the right eigenvalues, although numerically you can get a small imaginary part when solving the eigenvalue problem. As an ...
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8 votes
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Transparent boundary conditions for finite element simulation of TDSE

Kirill's answer is a good method but it is hampered that the fact that you need to tune it to a specific range in energy, so it is not appropriate if you have a broad-band wavepacket you need to ...
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8 votes
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Solving the time dependent Schrödinger equation with leapfrog integration in 1D

I read through the paper you linked and they give the stability condition for this method to be (eq. A6) $$ \frac{-2}{\Delta t} \le V \le \frac{2}{\Delta t} - \frac{2}{m \Delta r^2} $$ This has to be ...
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7 votes

Calculating partial trace of array in NumPy

I didn't follow exactly your notation so I can't say for sure what this would look like for your example, but two suggestions: if you really want your life made simple, check out ...
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  • 191
6 votes
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What is the current state of the art in solving higher dimensional parabolic PDEs (multi-electron Schrödinger equation)

The solutions for the equation are in $$\psi \in \mathbb{C}^{3M}\times\mathbb{R}^+ \enspace .$$ If the number of electrons is small enough you can just use any traditional method. Like a domain ...
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5 votes

Are there simple ways to numerically solve the time-dependent Schrödinger equation?

I can recommend using the finite-difference time-domain (FDTD) method. I even wrote a tutorial some time back that should answer most of your questions: J. R. Nagel, "A review and application of the ...
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5 votes

Transparent boundary conditions for finite element simulation of TDSE

In the literature these boundary conditions go by the name of absorbing boundary conditions (or nonreflecting, open, radiation, invisible, far-field), and this is a well-known topic. One clear ...
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5 votes

Python, numpy and complex functions (PDE's)

Caveat: I did not read beyond the statement So, what I did was to define functions for the potential and the second derivative, and use Euler's method. So there may be other issues with your code, ...
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5 votes

Calculating partial trace of array in NumPy

I don't have enough reputation to comment, but I wanted to give a basic complement of gIS's answer. Simply put, if we have the decomposition over $V=V_1\otimes V_2$, which we represent with, say, the ...
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5 votes
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Normalizing a density matrix at each iteration

As always with numerical problems, it's a good idea to simplify things as much as possible to isolate what's really going wrong. You can simplify the problem by considering the evolution of a pure ...
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5 votes
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Numerical solution to the infinite well problem

I think that the main problem might be with the solver you are using. The Hamiltonian (matrix) in this case is Hermitian, it is even symmetric since it is purely real. You could use ...
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4 votes
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Integrating radial Schrodinger equation with Lennard-Jones potential using Runge-Kutta with adaptive step size ends up with a step-size of zero

I know this problem well from my own research: it is given by the fact that the equation is very stiff. Thus, it is likely that you're doing nothing wrong (--although I haven't inspected your code). ...
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  • 2,792
4 votes

Is it possible to eliminate the inner sum to evaluate numerically?

Numerically the sum isn't terribly easy: it behaves as $$ \sum_{n\geq 2, k\geq 1} \frac{1}{n(n^2-1)} \frac{e^{-4/k}}{(k^2-n^{-2})^2}, $$ so it converges linearly. Most numerical methods that can be ...
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4 votes

Zero Eigenvalues in Lanczos Algorithm

The SciPy tutorial explicitly states Note that ARPACK is generally better at finding extremal eigenvalues: that is, eigenvalues with large magnitudes. In particular, using ...
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4 votes
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Discrepancies between numerical and analytical solution for particle in a finite potential well?

I think that your potential for the numerical case is wrong. The potential should be a big positive number, so the solution tends to zero outside the well when the value of the potential increases. ...
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4 votes
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Numerical solution of zero-potential time-dependent Schrödinger equation in 1D

Regarding the boundary conditions: Don't be fooled by Wikipedia. Yes, the scenario in the picture suggests an absorption at the boundaries, and yes, one could use absorbing boundary conditions in ...
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4 votes
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How can I evaluate more accurate energy eigenvalues from Schrodinger equation using shooting method?

Your problem was the lower limit of integration. It should have been $-x_e$ instead of 0, since $x_e$ is the equilibrium point for the potential and not the minimum distance. After correcting that, ...
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4 votes

Problems with the time-dependent Schrödinger equation solutions

First, you're solving the Schrödinger equation not in a box, but rather you apply periodic boundary conditions. That is, it's more like solving the TDSE on a ring. But this just as a comment and ...
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4 votes
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Computing eigenvalues of Schrodinger equation with spin

When modelling spin in the Schrödinger equation, one has several alternatives which need to be chosen in advance. I'll copy an excerpt from a work of mine to give an overview (it considers atomic ...
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  • 2,792
3 votes
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Numerical Solution to Schrödinger Equation--Multiple Wells

While I cannot help you with your specific implementation, I want to point out to an alternative method (as already indicated in a comment to phil's answer) : Marston's "Fourier Grid Hamiltonian" (FGH)...
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  • 772
3 votes

Zero Eigenvalues in Lanczos Algorithm

While the smallest eigenvalues of a sparse 40k x 40k matrix might be obtained in a reasonable amount of time using ARPACK and some handy tricks through the SciPy wrapper, as mentioned here, the ...
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3 votes
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Quantum Chemical Calculations is there a book for which method to use with what problem?

There is no universal answer to this. You will have to find publications that have similar set-ups and already did the benchmarking, or you will have to do it yourself. Once you understand how the ...
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3 votes

imaginary time propagation to find ground state wavefunction

It's a bit heavy, but not all of it must be read to get a useable method : Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, Bao and Du. I believe -- it'...
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  • 161
3 votes
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ground state from the Schroedinger equation with a central potential what happens to the origin

As mentioned by @Kirill, the differential equation for the hydrogen is slightly different. After the separation of variables, you end up with $$\left[ - \frac{\hbar^2}{2\mu} \left({1 \over r^2}{\...
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3 votes

Parallelization of element-wise matrix multiplication

There's no element wise multiplication operation in the BLAS library. Your best approach is probably to just implement the operation yourself using (e.g.) OpenMP threading. Before you do this, you ...
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3 votes

Are there simple ways to numerically solve the time-dependent Schrödinger equation?

A few answers and comments here conflate confusingly the TDSE with a wave equation; perhaps a semantics issue, to some extent. The TDSE is the quantized version of the classical non-relativistic ...
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3 votes

Modifying finite difference solution to Schrodinger eqn to account for fermion/boson effects

It is not that bosons are "encouraged" to occupy the same place, that is quite wrong. Here is an article on the spin-statistics theorem which describes the relationship between particle statistics and ...
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