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### How the Compton Effect of Light Explains Wavelength Shift

Although Max Planck and Albert Einstein postulated that light could behave as both a wave and a particle, it was Arthur Compton who finally proved that this was possible. His experiment involved scattering [more…]

### How to Use Kets, the Hermitian Conjugate, and Bra-ket Notation

What do Dirac notation and the Hermitian conjugate have in common? They help physicists to describe really, really big vectors. In most quantum physics problems, the vectors can be infinitely large — for [more…]

### How to Use Operators for Quantities in Quantum Physics

In quantum physics, you can use operators to extend the capabilities of bras and kets. Although they have intimidating-sounding names like Hamiltonian, unity, gradient, linear momentum, and Laplacian, [more…]

### How to Use Linear Operators in Quantum Physics

In quantum physics, you need to know how to use linear operators. An operator A is said to be *linear* if it meets the following condition: [more…]

### How to Shift the Symmetric Square Well Around the Origin

Infinite square wells, in which the walls go to infinity, are a favorite in quantum physics problems. In some instances, you may want to shift things so that the square well is symmetric around the origin [more…]

### How Particles Pass Through Potential Barriers That Have Less Energy

When you are working with potential barrier of height V_{0} and width *a* where E > V_{0}, this means that the particle has enough energy to pass through the potential barrier and end up in the [more…]

### How to Find the Energy Eigenstate of a Harmonic Oscillator in Position Space

In quantum physics, you can use operators to determine the energy eigenstate of a harmonic oscillator in position space. The charm of using the operators [more…]

### How to Find Any Excited State of a Harmonic Oscillator

If you can determine the wave function for the ground state of a quantum mechanical harmonic oscillator, then you can find any excited state of that harmonic oscillator. [more…]

### How to Find the Energy Level of a Harmonic Oscillator: An Example

Your quantum physics instructor may ask you to find the energy level of a harmonic oscillator. The best way to learn how is through an example. Say that you have a proton undergoing harmonic oscillation [more…]

### How Quantum Physicists Describe a Black Body

One of the major ideas of quantum physics is *quantization*— measuring quantities in discrete, not continuous, units. The idea of quantized energies arose with one of the earliest challenges to classical [more…]

### How Physicists Solved the Photoelectric Effect of Light

The photoelectric effect was one of many experimental results that made up a crisis for classical physics around the turn of the 20th century. It was also one of Einstein’s first successes, and it provides [more…]

### How to Solve the Schrödinger Equation for Free Particles

There are plenty of free particles — particles outside any square well —in the universe, and quantum physics has something to say about them. The discussion starts with the Schrödinger equation: [more…]

### How to Estimate a Particle's Location by Applying Schrödinger's Equation to a Wave Packet

If you have a number of solutions to the Schrödinger equation, any linear combination of those solutions is also a solution. So that’s the key to getting a physical particle: You add various wave functions [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, R* _{nl}*(

*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…]

### How to Find the Total Energy of a Multi-Particle System

The Hamiltonian represents the total energy of all the particles in a multi-particle system. You can describe that system in quantum physics terms. The following figure shows a multi-particle system where [more…]

### How to Apply the Hamiltonian to a Neutral, Multi-Electron Atom

A multi-electron atom is the most common multi-particle system that quantum physics considers. You can apply a Hamiltonian wave function to a neutral, multi-electron atom, as shown in the following figure [more…]

### How to Classify Symmetric and Antisymmetric Wave Functions

You can determine what happens to the wave function when you swap particles in a multi-particle atom. Whether the wave function is symmetric or antisymmetric under such operations gives you insight into [more…]

### How to Decouple Different Particles into Linearly Independent Equations

In quantum physics, you can decouple systems of particles that you can distinguish — that is, systems of identifiably different particles — into linearly independent equations. To illustrate this, suppose [more…]

### How to Distinguish Particles in a Multi-Particle System

Quantum mechanically, identical particles in a multi-particle system don’t retain their individuality in terms of any measurable, observable quantity. You lose the individuality of identical particles [more…]

### Create Symmetric and Antisymmetric Wave Functions for Any System of N Particles

In quantum physics, many of the wave functions that are solutions to physical setups like the square well aren’t inherently symmetric or antisymmetric; they’re simply asymmetric. In other words, they have [more…]

### Create Symmetric and Antisymmetric Wave Functions for a Two-Particle System

If your quantum physics instructor asks you to create symmetric and antisymmetric wave functions for a two-particle system, you can start with the single-particle wave functions: [more…]