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### Create Symmetric and Antisymmetric Wave Functions for a Three-or-More-Particle Systems

In quantum physics, you can put together the symmetric and antisymmetric wave functions of a system of three or more particles from single-particle wave functions. The symmetric wave function looks like [more…]

### How Particles Tunnel Through Potential Barriers That Have Greater Energy

When a particle doesn't have as much energy as the potential of a barrier, you can use the Schrödinger equation to find the probability that the particle will tunnel through the barrier's potential. You [more…]

### Determining the Energy Levels of a Particle in a Box Potential

In quantum physics, to be able to determine the energy levels of a particle in a box potential, you need an exact value for X(*x*) — not just one of the terms of the constants A and B. You have to use the [more…]

### How to Normalize the Wave Function in a Box Potential

In your quantum physics course, you may be asked to normalize the wave function in a box potential. Here's an example: consider the wave function [more…]

### When to Use Spherical Coordinates Instead of Rectangular Coordinates

In quantum physics, you sometimes need to use spherical coordinates instead of rectangular coordinates. For example, say you have a 3D box potential, and suppose that the potential well that the particle [more…]

### Applying the Schrödinger Equation in Three Dimensions

In quantum physics, you can apply the Schrödinger equation when you work on problems that have a central potential. These are problems where you're able to separate the wave function into a radial part [more…]

### Determining the Angular Part of a Wave Function

In quantum physics, you can determine the angular part of a wave function when you work on problems that have a central potential. With central potential problems, you're able to separate the wave function [more…]

### Determining the Radial Part of a Wave Function

In quantum physics, you can determine the radial part of a wave function when you work on problems that have a central potential. With central potential problems, you’re able to separate the wave function [more…]

### Applying the Radial Equation Outside the Square Well

In quantum physics, you can apply the radial equation outside a square well (where the radius is greater than *a*). In the region *r* > *a*, the particle is just like a free particle, so here's what the radial [more…]

### How to Simplify and Split the Schrödinger Equation for Hydrogen

In quantum physics, you may need to simplify and split the Schrödinger equation for hydrogen. Here's the usual quantum mechanical Schrödinger equation for the hydrogen atom: [more…]

### Finding the Total Energy Equation for Three-Dimensional Free Particle Problems

At some point, your quantum physics instructor may want you to find the total energy equation for three-dimensional free particle problems. The total energy of the free particle is the sum of the energy [more…]

### How to Add Time Dependence and Get a Physical Equation for Three-Dimensional Free Particle Problems

At some point, your quantum physics instructor may want you to add time dependence and get a physical equation for a three-dimensional free particle problem. You can add time dependence to the solution [more…]

### Working with Three-Dimensional Rectangular Potentials

This article takes a look at a 3D potential that forms a box, as you see in the following figure. You want to get the wave functions and the energy levels here. [more…]

### How to Work with a Cubic Potential

In quantum physics, when working with a box potential, you can make things simpler by assuming that the box is actually a cube. In other words, L = L* _{x}* [more…]

### Working with Three-Dimensional Harmonic Oscillators

In quantum physics, when you are working in one dimension, the general particle harmonic oscillator looks like the figure shown here, where the particle is under the influence of a restoring force — in [more…]

### Applying the Spherical Bessel and Neumann Functions to a Free Particle

In quantum physics, you can apply the spherical Bessel and Neumann functions to a free particle (a particle which is not constrained by any potential). The wave function in spherical coordinates takes [more…]

### Finding the Limits for Small and Large Rho of a Free Particle

In quantum physics, you can find the limits for small and large rho of a free particle. According to the spherical Bessel equation, the radial part of the wave function for a free particle looks like this [more…]

### Applying the Radial Equation Inside the Square Well

In quantum physics, you can apply the radial equation inside a square well (where the radius is greater than zero and less than *a*). For a spherical square well potential, here's what the radial equation [more…]

### Finding the Schrödinger Equation for the Hydrogen Atom

Using the Schrödinger equation tells you just about all you need to know about the hydrogen atom, and it's all based on a single assumption: that the wave function must go to zero as [more…]

### How to Find the Commutator of Operators

In quantum physics, the measure of how different it is to apply operator A and then B, versus B and then A, is called the operators’ *commutator*. Here’s how you define the commutator of operators A and [more…]

### How to Find the Heisenberg Uncertainty Relation from Scratch

If you’ve read through the last few sections, you’re now armed with all this new technology: Hermitian operators and commutators. How can you put it to work? You can come up with the Heisenberg uncertainty [more…]

### How to Work with Eigenvectors and Eingenvalues

In quantum physics, when working with kets, it is useful to know how to use eigenvectors and eigenvalues. Applying an operator to a ket can result in a new ket: [more…]

### How to Find the Inverse of a Large Matrix

Finding the inverse of a large matrix often isn’t easy, so quantum physics calculations are sometimes limited to working with unitary operators, U, where the operator’s inverse is equal to its adjoint, [more…]

### How to Derive the Schrödinger Equation

In quantum physics, the Schrödinger technique, which involves wave mechanics, uses wave functions, mostly in the position basis, to reduce questions in quantum physics to a differential equation. [more…]

### Measuring the Energy of Bound and Unbound Particles

In quantum physics, you can solve for the allowable energy states of a particle, whether it is bound, or trapped, in a potential well or is unbound, having the energy to escape. [more…]