Questions tagged [quantum-mechanics]

Numerical methods and problems involving the solution of the Schrodinger equation and related subatomic models.

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Numerical Solution of the Schrödinger equation for hydrogen

I'm trying to solve the Schrödinger equation for the hydrogen atom in the following form numerically: $$\left[-\frac{\hbar^2}{2m}\frac{d^2}{dr^2}+V(r)+\frac{\hbar^2l(l+1)}{2mr^2}\right]R(r)=ER(r).$$ ...
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Numerov method for 1D potential barrier problem: numerical errors

I wanted to solve a 1D potential barrier problem. It seemed easy enough to do: set a travelling wave coming from the left, part of it gets reflected by the barrier, and part of it gets gets ...
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112 views

Computing eigenvalues of Schrodinger equation with spin

I want to solve a 2-dimensional particle in box problem with two electrons in the quantum well.I would like to take into account spin of electrons and Coulomb interactions to compute singlet and ...
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149 views

How to compute the Eigenvalue and Eigenstates of Quantum well with Effective mass using finite difference method in Python?

I want to compute the eigenvalues and eigenstates of a quantum well with different effective masses of electron in the barrier and in the quantum well. As can be seen [1]: https://github.com/mholtrop/...
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Numerically solving schrodinger equation [duplicate]

Consider the potential given above: $$U(r) = \frac{U_0}{\exp[(r - r_0)/\epsilon] + 1}\, .$$ How to solve the Schrodinger equation with this potential numerically and find the eigenvalues?
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35 views

Weird behavior in for solving TISE in harmonic oscillator potential using the shooting method

I was solving the time independent Schrödinger equation using the shooting method for harmonic oscillator potential. This is the code that I wrote for that with the results (code is written in julia): ...
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1answer
90 views

Numerical solution to the infinite well problem

I've used the following code to implement it ...
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47 views

Ising model in Python (Magnetization Scaling)

I am trying to implement the Ising Model in Python for Gibbs Distribution: $$\pi(x) = \frac{1}{Z(\beta)} e^{-\beta H(x)}$$ \begin{align*} p(x,y)&=r(x,y) \cdot \min \left( \frac{\pi(y)}{\pi(x)},1 \...
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61 views

Problems with the time-dependent Schrödinger equation solutions

I have this Mathematica problem that solves numerically the time-dependent Schrödinger equation in a box: ...
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246 views

Numerov method for solving Schrödinger equation

I have just begun learning computer science to apply it to Physics and I am trying to write a code for solving Schrödinger's equation of the harmonic oscillator (setting $V=\frac{x^2}{2}$) in one ...
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84 views

Finding the extrema of a transition probability function for a quantum walker on a graph

The goal Implement some Python code to find the extrema points of a function that is strongly oscillating. The background Let $G$ be a connected graph with $n$ points with Laplacian matrix $L(G)$. We ...
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95 views

Python routine to calculate shape resonances of H2

I am currently doing a project in which my aim is to write a program that can be used to calculate single and multi-channel shape resonances. So I'm looking at bound states and quasi-bound states. ...
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1answer
149 views

A problem with Poisson equation

I'm computing the Hartree potentials of atoms by solving the Poisson equation and I use hydrogen atom as a test case. The Poisson equation for hydrogen atom in atomic units is given by $$\nabla^2 V_H =...
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135 views

Why is Time evolving block decimation so efficient?

I have a short question about Time evolving block decimation (TEBD). During a lecture I was told that this method is very efficient in evolving 1D quantum spin systems with only nearest neighbor ...
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1answer
736 views

How can I evaluate more accurate energy eigenvalues from Schrodinger equation using shooting method?

I am trying to use the "shooting method" for solving Schrodinger's equation for a reasonably arbitrary potential in 1D. But the eigenvalues so evaluated in the case of potentials that do not have hard ...
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2answers
235 views

2d Schrodinger Equation via matrix diagonalization in C

I am trying to solve the time-independent Schrodinger equation in two dimensions via discrete matrix diagonalization. I want the energy eigenvalues and the corresponding eigenfunctions for a given ...
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1answer
104 views

If not MATLAB, what software/programming language should I use to simulate/animate wave functions in various potentials + more? (example given)

I want to integrate programming into my learning in math and science in a very specific way. I want to create visualizations and simulations of concepts I am learning. When I learn a numerical method ...
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1answer
184 views

Operator splitting to solve time dependent Schrödinger equation

I encountered the split operator method to solve the time dependent Schrödinger equation during a lecture. I understand the method on a theoretical basis (I think at least), but I'm struggling to ...
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2answers
398 views

Numerical solution of zero-potential time-dependent Schrödinger equation in 1D

I want to solve numerically the one-dimensional time-dependent Schrödinger equation $$i \psi_t(x,t)=-\frac{\hbar}{2m} \psi''(x,t)$$ My issue is that I don't have the physical background to understand ...
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1answer
236 views

Recommendation for a fixed-step ODE solver?

My problem involves the solution of a second-order ODE with a fixed-step (input and output). Specifically, this ODE is the radial part of Dirac and Schrödinger equation for a spherical symmetric ...
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95 views

Numerical integration of SDE: choice of $dt$ and algorithm

I am working on the following Stochastic Differential Equation (SDE) in the Quantum Mechanics context: $$dX_{t} = a X_{t} dt + b X_{t} dW$$ where $X_{t}$ is my stochastic varible, $dt$ is my ...
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1answer
245 views

Numerical solution to the Landau-Zener problem

I tried to use a midpoint method and numerically solve the Schrödinger equation for the original Landau-Zener (LZ) problem: a $2\times 2$ Hamiltonian $$\left(\begin{array}{c} \alpha t\\ \delta \end{...
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1answer
249 views

Numerov method for Schrodinger equation

While learning about numerical methods for solving the Schrödinger equation I came across Numerov's method. I want to get the solution for the harmonic oscillator by alreading giving the eigenvalues. ...
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1answer
582 views

What will be the impact of quantum computing on existing numerical techniques (e.g. CFD)?

Quantum computing seems to be a very active and promising development area in computer science. However, I am curious as to what impact (if any) quantum computing will have on existing classical ...
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193 views

How to do Weierstrass-transform in MATLAB?

I have a diagonalization problem. I have the eigenstates correctly, and I want to do a Gaussian-smearing (Weierstrass-transform) on them. So I have the wave functions ($\Psi$), and the continuous ...
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273 views

Finding second excited state of Schrödinger equation with secant Runge Kutta method

In our assignment, we are required to find the energies of the ground state and the first two excited states of the Schrödinger equation in a harmonic potential: $$V = \frac{50 x^2}{(10^{-11})^2}\, .$...
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82 views

How to numerically calculate the transition dipole integral in periodic systems?

Now I have wave functions $\psi_a$ and $\psi_b$ of two states in Gaussian CUBE format. I'd like to evaluate the transition dipole moment integral $\pmb\mu$ between these two states. As my simulation ...
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276 views

MATLAB: Faber approximation of the matrix exponential to solve Liouville-von-Neumann equation

EDIT: I moved the full code to my github page so the post can be read more easily. I am writing a script to take the Faber approximation approach outlined in Hassan Fahs paper (free access) and apply ...
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1answer
459 views

Incorporating a potential barrier in a wave-packet simulation (Fourier Transform method)

I'm trying to simulate the scattering of a wave-packet at a potential barrier in Python. I'm using a Fourier Transform method (not sure if its the same as the Split-Step method), where I apply Fourier ...
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47 views

Hydrogen-like wavefunction as starting guess for atomic solver?

I've been looking into radial solvers for quantum wave equations (Schroedinger and Dirac). In both cases, the suggestion seems to be to go with the "shooting method", with integration schemes of ...
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1answer
3k views

Double potential well with Python

I'm trying to understand the Schrödinger equation and solving it a bit better, and I'm running into some doubts while coding, even though I am adapting the code to this situation. Also I tried asking ...
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1answer
128 views

Discrepancies between numerical and analytical solution for particle in a finite potential well?

Analytical Inside the box, the wavefunction is: \begin{equation} \frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} = E \psi(x) \iff \frac{d^2 \psi(x)}{dx^2} = k^2 \psi(x) \end{equation} where $k = \...
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1answer
605 views

Calculate partial trace of an outer product in Python?

I have a python implementation of calculating the partial trace over select dimensions. ...
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1answer
807 views

Solving Schrodinger equation numerically

Here is the Schrodinger equation that is to be solved: A 1D hard wall potential in $[0, 1]$. The potential within the potential well is given by a linear combination of Gaussian dips $$v(x) = - \...
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1answer
147 views

Magnetization Vector from XY Model for an AntiFerromagnetic System

I am working on an XY model and I'm trying to calculate the magnetization and direction for an anti-ferromagnetic (AF) system. So I have a collection of spins in the $XY$ plane represented as vectors ...
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380 views

Avoiding divergent solutions with `odeint`? shooting method

I am trying to solve an equation in Python. Basically what I want to do is to solve the equation: $$ \frac{1}{x^2}\frac{d}{dx}\left(Gam \frac{dL}{dx}\right)+L\left(\frac{a^2x^2}{Gam}-m^2\right)=0 $$ ...
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1answer
343 views

Matrix exponential of hermitian matrix with eigenvectors from generalized eigenvalue problem

I want to calculate the following expression $$ \exp(-i\Delta t\mathbf{H}) $$ where $\mathbf{H}\in\mathbb{C}^{n\times n}$ is a hermitian matrix. Since I have a highly optimized eigensolver in the code ...
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1answer
153 views

Two variables integration matlab

I'm trying to solve physical problem in quantum mechanics of helium atoms, the solution require numerical integration over 2 variables. However when i'm trying to run the next code ...
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1answer
256 views

Non-hermitian discretizations in quantum mechanics

Consider the Schroedinger equation $$\left(-\frac12\frac{\partial^2}{\partial x^2} + V(x) \right) \psi(x) = E \psi(x)$$ The usual way to solve it is to introduce a discretization of $\psi(x)$. This ...
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717 views

Numerical solution to Time-dependent Schrodinger equation with time dependent hamiltonian

Currently I am facing the problem to solve numerically the following equation for a double well harmonic potential: $iℏ\frac{\partial}{\partial t}ψ(x,t)=−\frac{ℏ}{2m}\frac{\partial ^2}{∂x^2}ψ(x,t)+V(...
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1answer
161 views

Jump-Diffusion process: practical solver beyond Euler method?

A jump-diffusion process is a stochastic process where both continuous noise (in my case complex Wiener noise $dZ,dZ^*$ such that $dZ^2=dZ^{*2}=0,|dZ|^2=dt$) and discrete Jumps (in my case Poissonian $...
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1answer
123 views

Normalizing a density matrix at each iteration

I need to numerically evolve a density matrix using this formula(Actually I have more terms but right nows I am starting with this and facing problems): $$\dot\rho(t) = -i[H(t), \rho(t)]$$ $H(t)$ is ...
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Is there a software package for Quantum Monte Carlo estimation of the exchange correlation functional?

Quantum Monte Carlo (QMC) calculations historically have been used to parameterise the exchange-correlation functionals for Density Functional Theory (DFT). For example, this article explains a way to ...
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2answers
4k views

Solving the 1D Particle-in-a-Box using C++

I've just finished learning the physics behind the problem and would like to write a program in C++ than can solve the problem. I'm actually stuck at the start. I've quite a bit of research, the ...
3
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1answer
830 views

Numerical propagation of a density matrix using Liouville von Neumann equation

I want to look at time evolution of the density matrices of some, very simple, spin systems, but I am having trouble with my approach. I want to use a simple for-...
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1answer
168 views

Matrix exponential by eigenvectors - implementation issues

I posted a similar question yesterday but I deleted it since I think that I had to reformulate it after some insights. I want to calculate $$ \exp(-i\Delta t\,\mathcal{H}) = V\,\mathrm{diag}(\{\exp(-...
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3answers
6k views

Calculating partial trace of array in NumPy

A simulation I'm doing requires me to calculate the partial trace of a large density matrix. I am trying to calculate it using tools from numpy, but my code seems to be having some problems. For ...
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0answers
211 views

Second Quantization in Matlab

This question may be more suited for physics.stackexchange, but I saw this post was recommended for StackOverflow or Computational Science, so I'm asking my question here. I am trying to write a ...
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2answers
751 views

Schrödinger equation with time dependent Hamiltonian

I need to solve the Schrödinger equation with a time dependent Hamiltonian $$i\hbar \frac{\partial}{\partial t} \Psi = \left[-\frac{\hbar^2}{2m}\nabla^2 +\frac{1}{2} k(t)(x^2+y^2) + V(r)\right]\Psi $...
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184 views

lightweight implementation of semiempirical quantum chemistry (e.g. MNDO,AM-1,PM3)

I'm searching for semi-empirical quantum chemistry solver which would be easy to integrate into my own software. I found a few implementations which can be in principle used e.g. MOPAC, ORCA, SQM some ...