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### 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…]

### 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…]

### 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 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…]