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### 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}### 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…]