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
Accepted
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 ...
- 826
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 ...
- 191
7
votes
Accepted
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 ...
- 2,391
6
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 ...
- 61
5
votes
Accepted
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 ...
- 10.1k
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
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 ...
- 12.2k
4
votes
Accepted
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
Accepted
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 ...
- 3,042
4
votes
Accepted
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 ...
- 3,042
4
votes
Accepted
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 ...
- 3,042
3
votes
Accepted
Solving Schrodinger equation numerically
I saw @BenCrowell comment and wanted to check how hard it was.
If you write the variational formulation of the Schrödinger equation you would get
$$\Pi[\psi] = \frac{1}{2} \langle \nabla \psi, \...
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3
votes
Matrix exponential of hermitian matrix with eigenvectors from generalized eigenvalue problem
I guess your problem is the following. Your working in a non-orthogonal basis $\phi_i$, and solved the time-independent Schrödinger equation
$$\mathbf H \mathbf C = E \mathbf S \mathbf C$$
Here $\...
- 3,042
3
votes
Accepted
Quantum Chemical Calculations is there a book for which method to use with what problem?
There is no universal answer to this. You will have to find publications that have similar set-ups and already did the benchmarking, or you will have to do it yourself. Once you understand how the ...
- 166
3
votes
imaginary time propagation to find ground state wavefunction
It's a bit heavy, but not all of it must be read to get a useable method :
Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, Bao and Du.
I believe -- it'...
- 161
3
votes
Accepted
ground state from the Schroedinger equation with a central potential what happens to the origin
As mentioned by @Kirill, the differential equation for the hydrogen is slightly different. After the separation of variables, you end up with
$$\left[ - \frac{\hbar^2}{2\mu} \left({1 \over r^2}{\...
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3
votes
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 ...
3
votes
Accepted
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)...
- 792
3
votes
Calculating partial trace of array in NumPy
If you use qutip, partial trace operations are already built-in. For example, here is how you can compute the partial trace of a random density matrix over three ...
- 131
3
votes
Accepted
Simple open-source Quantum chemistry or DFT code in C/C++
Quantum Chemistry codes can get very complicated very fast. Even if you limit yourself to DFT, there are many functionals to support. There's going to be a trade off here for you. You can get a simple,...
- 181
3
votes
Best books and notes for beginners in the parallel FORTRAN 90 programming
I'm assuming by 'supercomputer' you're talking about HPC and distributed memory systems, as opposed to shared memory workstations, in which case you have a couple of options when it comes to Fortran.
...
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3
votes
Accepted
MATLAB: Faber approximation of the matrix exponential to solve Liouville-von-Neumann equation
Not sure whether this provides a full answer, but at least it provides some thoughts and hints.
Concept
I think the expression $\exp(\mathcal{Lt})\rho$ needs clarification. This is not necessarily a ...
- 353
3
votes
Accepted
How to do Weierstrass-transform in MATLAB?
What you want is the convolution between two functions $f = |\Psi|^2$ and $g = g_{\sigma_x}(x)$, $h = (f * g)(x)$.
You can compute the Fourier transform of $h$, to get
$$\mathcal{F}\{h\} = \mathcal{...
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3
votes
What will be the impact of quantum computing on existing numerical techniques (e.g. CFD)?
There are two major reasons that there is no tag quantum-computing (at least yet) in Computational Science SE:
There is a specialized Quantum Computing StackExchange community.
I am not aware of ...
- 8,602
3
votes
Accepted
Recommendation for a fixed-step ODE solver?
I think DifferentialEquations.jl in Julia has a very comprehensive suite of ODE solvers, including the ones you mentioned (Adams-Bashfort and GBS) and many others. This Julia library is becoming more ...
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Only top scored, non community-wiki answers of a minimum length are eligible