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### How to Find Square-Well Energy Levels

In quantum physics, if you know the boundary conditions of a square well, you can find theenergy levels of an electron.

The equation [more…]

### How to Find the Normalized Wave Function for a Particle in an Infinite Square Well

In quantum physics, if you are given the wave equation for a particle in an infinite square well, you may be asked to normalize the wave function. For example, start with the following wave equation: [more…]

### Find the Wave Function of a Particle in an Infinite Square Well over Time

In quantum physics, you can use the Schrödinger equation to see how the wave function for a particle in an infinite square well evolves with time*.* The Schrödinger equation looks like this: [more…]

### How to Find the Eigenfunctions of L^{2} in Spherical Coordinates

Your quantum physics instructor may ask you to find the eigenfunctions of L^{2} in spherical coordinates. To do this, you start with the eigenfunction of [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…]

### How to Determine the Allowed Energies of a Hydrogen Atom

When you apply the quantum mechanical Schrödinger equation for a hydrogen atom, the quantization condition for the wave function of *r* to remain finite as [more…]

### How an Increase in *r* Affects the Appearance of Hydrogen Wave Functions

In a hydrogen atom, the wave functions change as you change the orbital radius, *r*. So what do the hydrogen wave functions look like? Given that [more…]

### How to Calculate the Energy Degeneracy of a Hydrogen Atom in Terms of *n*, *l*, and *m*

Each quantum state of the hydrogen atom is specified with three quantum numbers: *n* (the principal quantum number), *l*(the angular momentum quantum number of the electron), and [more…]

### How to Calculate the Spin of an Electron in a Hydrogen Atom

When calculating the spin of an electron in a hydrogen atom, you need to allow for the spin of the electron, which provides additional quantum states. Given the following equation, where the wave function [more…]

### What Happens to a Wave Function When You Swap Two Particles

When you swap the order of operations for two particles in a multi-particle atom, this affects their wave function. Whether the wave function is symmetric under such operations gives you insight into whether [more…]

### How to Determine Harmonic Oscillator Eigenstates of a System

In quantum physics, when you have the eigenstates of a system, you can determine the allowable states of the system and the relative probability that the system will be in any of those states. [more…]

### How to Find the Energy of a Perturbed System Based on the Wavelength Parameter

In quantum physics, in order to find the energy of a perturbed system, E* _{n}*, you need to start by calculating the energy and wave function of an unperturbed system. You start with the energy: [more…]

### How to Find the Second-Order Corrections to Energy Levels and Wave Functions

In quantum physics, in order to find the second-order corrections to energy levels and wave functions of a perturbed system, E* _{n}*, you need to calculate E [more…]

### How to Find the Energy of a Charged Oscillator Using the Perturbation Theory

In quantum physics, when you have the exact eigenvalues for a charged oscillator in a perturbed system, you can find the energy of the system. Based on the perturbation theory, the corrected energy of [more…]

### How to Find the Eigenvalues and Eigenvectors for Degenerate Hamiltonians

Using quantum physics, you can determine the *f*eigenvalues and matching eigenvectors for systems in which the energies are degenerate. Take a look at this unperturbed Hamiltonian: [more…]

### How to Work with Particle Scattering and Cross-section Equations

Quantum physicists use large particle accelerators to discover more about particle scattering on the subatomic level. You can think of a scattering experiment in terms of [more…]

### How to Relate the Angles between Center-of-Mass and Lab Frames

Quantum physics experiments take place in the lab frame, but you do scattering calculations in the center-of-mass frame, so you have to know how to relate the angle between the two frames. [more…]

### How to Translate Cross-Sections between Center-of-Mass and Lab Frames

In quantum physics, once you relate the angles of the scattered particles in the lab frame and the center-of-mass frame, you can translate the differential cross section — the bull's eye when you're aiming [more…]

### How to Find the Resulting Angles in a Lab-Frame Collision between Particles of Equal Mass

Say you have two particles of equal mass colliding in a lab frame (where one particle starts at rest). You want to show that the two particles end up traveling at right angles with respect to each other [more…]

### How to Derive the Incident and Scattered Wave Functions of Spinless Particles

From a time-independent quantum physics point of view, you can derive the incident wave and scattered wave functions of two spinless nonrelativistic particles. To do so, you need to assume that the interaction [more…]

### How to Relate the Scattering Amplitude and Differential Cross Section of Spinless Particles

The scattering amplitude of spinless particles is crucial to understanding scattering from the quantum physics point of view. To see that, take a look at the current densities, J [more…]

### How to Find the First-Order Corrections to Energy Levels and Wave Functions

In quantum physics, in order to find the first-order corrections to energy levels and wave functions of a perturbed system, E* _{n}*, you need to calculate E [more…]

### How to Find the Expectation Value of a Dice Roll

Given that everything in quantum physics is done in terms of probabilities, making predictions becomes very important. And the biggest such prediction is the expectation value. The expectation value of [more…]