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### How to Find the Hermitian Adjoint

In quantum physics, you’ll often work with Hermitian adjoints. The *Hermitian adjoint* — also called the *adjoint* or *Hermitian conjugate* — of an operator A is denoted [more…]

### How to Arrange Eingenvectors

In quantum physics, the eigenvectors of a Hermitian operator define a complete set of orthonormal vectors — that is, a complete basis for the state space. When viewed in this “eigenbasis,” which is built [more…]

### How to Find the Eigenvectors and Eigenvalues of an Operator

In quantum physics, if you’re given an operator in matrix form, you can find its eigenvectors and eigenvalues. For example, say you need to solve the following equation: [more…]

### How to Find a Wave-Function Equation in an Infinite Square Well

Infinite square well, in which the walls go to infinity, is a favorite problem in quantum physics. To solve for the wave function of a particle trapped in an infinite square well, you can simply solve [more…]

### How to Find Angular Momentum Eigenvalues

When you have the eigenvalues of angular momentum states in quantum mechanics, you can solve the Hamiltonian and get the allowed energy levels of an object with angular momentum. The eigenvalues of the [more…]

### Derive the Formula for the Rotational Energy of a Diatomic Molecule

Here’s an example that involves finding the rotational energy spectrum of a diatomic molecule. The figure shows the setup: A rotating diatomic molecule is composed of two atoms with masses [more…]

### Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators

In quantum physics, you can find the eigenvalues of the raising and lowering angular momentum operators, which raise and lower a state’s *z* component of angular momentum. [more…]

### How to Change Rectangular Coordinates to Spherical Coordinates

In quantum physics, to find the actual eigenfunctions (not just the eigenstates) of angular momentum operators like L^{2}and L* _{z}*, you turn from rectangular coordinates, [more…]

### Find the Eigenfunctions of L_{z} in Spherical Coordinates

_{z}

At some point, your quantum physics instructor may ask you to find the eigenfunctions of L* _{z}* in spherical coordinates. In spherical coordinates, the L

_{z}operator looks like this: [more…]

### Find the Missing Spot with the Stern-Gerlach Experiment

The Stern-Gerlach experiment unexpectedly revealed the existence of spin back in 1922. Physicists Otto Stern and Walther Gerlach sent a beam of silver atoms through the poles of a magnet — whose magnetic [more…]

### Fermions and Bosons

In analogy with orbital angular momentum, you can assume that *m* (the *z*-axis component of spin) can take the values –*s*, –*s* + 1, ..., *s* – 1, and *s*, where [more…]

### How Spin Operators Resemble Angular Momentum Operators

Because spin is a type of built-in angular momentum, spin operators have a lot in common with orbital angular momentum operators. As your quantum physics instructor will tell you, there are analogous spin [more…]

### Spin One-Half Matrices

In quantum physics, when you look at the spin eigenstates and operators for particles of spin 1/2 in terms of matrices, there are only two possible states, spin up and spin down. [more…]

### Pauli Matrices

In quantum physics, when you work with spin eigenstates and operators for particles of spin 1/2 in terms of matrices, you may see the operators S* _{x}*, S

*, and S [more…]*

_{y}### 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…]

### How Pair Production and Pair Annihilation Define Light Particles

By observing both pair production and pair annihilation, 20th-century physicists were able to prove that light has the characteristics of a particle. This process of discovery began in 1928, when the physicist [more…]

### How de Broglie Showed that All Particles Display Wave-Like Properties

In 1923, the physicist Louis de Broglie suggested that not only did waves exhibit particle-like aspects, but that the reverse was also true — all material particles should display wave-like properties. [more…]

### How to Use the Heisenberg Uncertainty Principle in Relation to Position and Momentum

Quantum physicists understand that matter exhibits wave-like properties, which means that matter, like waves, aren't localized in space. This fact inspired Werner Heisenberg, in 1927, to come up with his [more…]

### How Quantum Physics Converts Momentum and Position into Probabilities

Quantum physics, unlike classical physics, is completely nondeterministic. You can never know the *precise* position and momentum of a particle at any one time. You can give only probabilities for these [more…]

### How to Assemble Relative Probabilities into a Vector

In quantum physics, probabilities take the place of absolute measurements. Say you've been experimenting with rolling a pair of dice and are trying to figure the relative probability that the dice will [more…]

### How to Use Ket Notation in Quantum Physics

In quantum physics, ket notation makes the math easier than it is in matrix form because you can take advantage of a few mathematical relationships. For example, here’s the so-called Schwarz inequality [more…]

### How to Find the Wave Function of the Ground State of a Quantum Oscillator

In quantum physics, you can find the wave function of the ground state of a quantum oscillator, such as the one shown in the figure, which takes the shape of a gaussian curve. [more…]

### How to Find Commutators of Angular Momentum, L

In quantum physics, you can find commutators of angular momentum, L. First examine L* _{x}*, L

*, and L*

_{y}*by taking a look at how they commute; if they commute [more…]*

_{z}### How to Create Angular Momentum Eigenstates

You can create the actual eigenstates, | *l*, m >, of angular momentum states in quantum mechanics. When you have the eigenstates, you also have the eigenvalues, and when you have the eigenvalues, you can [more…]

### How to Use Creation and Annihilation Operators to Solve Harmonic Oscillator Problems

Creation and annihilation may sound like big make-or-break-the-universe kinds of ideas, but they play a starring role in the quantum world when you're working with harmonic oscillators. You use the creation [more…]