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Computational Physics Quantum Mechanics

Numerical Quantum Mechanics: The Time-dependent Schrödinger Equation (III)

Back in 2016, I wrote two blog posts (I and II) concerning how to solve the time-dependent Schrödinger equation (TDSE) numerically. Although I gave recipes for computing the spatial derivative that occurs in the TDSE, and for integrating the TDSE itself, I omitted a rather important aspect of the computation: boundary conditions (BC’s). In this […]

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Computational Physics Quantum Mechanics

Quantum Mechanics: Ground States for 2 Charged Particles in the 1D Infinite Square Well

In this blog post I want to have a look at the Coulomb interaction, the governing equation of electrostatics, in the context of quantum mechanics. In my undergraduate QM courses, when discussing multi-particle systems, the first thing textbooks tended to do was to turn off the Coulomb interaction to render the problems analytically tractable. The hydrogen […]

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Computational Physics Quantum Mechanics

Numerical Quantum Mechanics: The Time-dependent Schrödinger Equation (II)

In my previous blog post, I had an introductory look at the problem of integrating the time-dependent Schrödinger equation (TDSE) numerically, plotting the wave function at each time step, and creating an animation of the result. In this post I will describe how to solve the TDSE in more detail. Recall the TDSE for a spinless particle in a […]

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Computational Physics Quantum Mechanics

Numerical Quantum Mechanics: The Time-dependent Schrödinger Equation (I)

In quantum mechanics, particles are represented by ‘wave functions’, wavy entities that are extended in space. The square modulus of the wave at any location gives the probability density of finding the particle at that location. The behavior of the waves is given by the time-dependent Schrödinger equation – a wave equation that describes how the waves evolve in time. […]