Multiple Dimensions & Spherical Coordinates

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Translate the Schrödinger Equation to Three Dimensions

In quantum physics, you can break the three-dimensional Schrödinger equation into three one-dimensional Schrödinger equations to make it easier to solve 3D problems. In one dimension, the time-dependent [more…]

Solving the Wave Function of Small r and Large r Using the Schrödinger Equation

Your quantum physics instructor may ask you to solve for the wave function for a made-up particle of mass m in a hydrogen atom. To do this, you can begin by using a modified Schrödinger equation that solves [more…]

Combine the Solutions for Small r and Large r in the Schrödinger Equation

When you apply the quantum mechanical Schrödinger equation for a hydrogen atom, you need to put together the solutions for small r and large r. The Schrödinger equation gives you a solution to the radial [more…]

How to Keep a Function of r Finite as r Goes to Infinity

In quantum physics, when finding the solution for a radial equation for a hydrogen atom, you need to keep the function of r finite as r approaches infinity to prevent the solution from becoming unphysical [more…]

Calculate the Wave Function of a Hydrogen Atom Using the Schrödinger Equation

If your quantum physics instructor asks you to find the wave function of a hydrogen atom, you can start with the radial Schrödinger equation, Rnl(r), which tells you that [more…]

Calculate the Distance of an Electron from the Proton of a Hydrogen Atom

When you want to find where an electron is at any given time in a hydrogen atom, what you’re actually doing is finding how far the electron is from the proton. You can find the expectation value of [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 = Lx [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…]

Finding the x, y, and z Equations for Three-Dimensional Free Particle Problems

At some point, your quantum physics instructor may want you to find the x, y, and z equations for three-dimensional free particle problems. Take a look at the [more…]

Solving the Wave Function of R Using the Schrödinger Equation

If your quantum physics instructor asks you to solve for the wave function of the center of mass of the electron/proton system in a hydrogen atom, you can do so using a modified Schrödinger equation: [more…]

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