Commutation Relations

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Quantum Mechanics A
Schrödinger Equation
The most fundamental equation of quantum mechanics; given a Hamiltonian \mathcal{H}, it describes how a state |\Psi\rangle evolves in time.
Basic Concepts and Theory of Motion
UV Catastrophe (Black-Body Radiation)
Photoelectric Effect
Stability of Matter
Double Slit Experiment
Stern-Gerlach Experiment
The Principle of Complementarity
The Correspondence Principle
The Philosophy of Quantum Theory
Brief Derivation of Schrödinger Equation
Relation Between the Wave Function and Probability Density
Stationary States
Heisenberg Uncertainty Principle
Some Consequences of the Uncertainty Principle
Linear Vector Spaces and Operators
Commutation Relations and Simultaneous Eigenvalues
The Schrödinger Equation in Dirac Notation
Transformations of Operators and Symmetry
Time Evolution of Expectation Values and Ehrenfest's Theorem
One-Dimensional Bound States
Oscillation Theorem
The Dirac Delta Function Potential
Scattering States, Transmission and Reflection
Motion in a Periodic Potential
Summary of One-Dimensional Systems
Harmonic Oscillator Spectrum and Eigenstates
Analytical Method for Solving the Simple Harmonic Oscillator
Coherent States
Charged Particles in an Electromagnetic Field
WKB Approximation
The Heisenberg Picture: Equations of Motion for Operators
The Interaction Picture
The Virial Theorem
Commutation Relations
Angular Momentum as a Generator of Rotations in 3D
Spherical Coordinates
Eigenvalue Quantization
Orbital Angular Momentum Eigenfunctions
General Formalism
Free Particle in Spherical Coordinates
Spherical Well
Isotropic Harmonic Oscillator
Hydrogen Atom
WKB in Spherical Coordinates
Feynman Path Integrals
The Free-Particle Propagator
Propagator for the Harmonic Oscillator
Differential Cross Section and the Green's Function Formulation of Scattering
Central Potential Scattering and Phase Shifts
Coulomb Potential Scattering

In many multidimensional problems, we often deal with rotational motion of particles, and thus we are interested in treating angular momentum in the framework of quantum mechanics. The (orbital) angular momentum operator in quantum mechanics is given by the cross product of the position of the particle with its momentum:

\hat{\mathbf{L}}=\hat{\mathbf{r}}\times\hat{\mathbf{p}}

Working in the position representation, this becomes

\hat{\mathbf{L}}=\mathbf{r}\times\frac{\hbar}{i}\nabla.

Evaluating the cross product in the Cartesian coordinate system, we get a component of \mathbf{L}\! in each direction; for example,

\hat{L}_x=\hat{y}\hat{p}_z-\hat{z}\hat{p}_y=\frac{\hbar}{i}\left(y\frac{\partial}{\partial z}-z\frac{\partial}{\partial y}\right),

and similarly the other two components of the angular momentum operator. All of these can be written in a more compact form using the Levi-Civita symbol as

\hat{L}_{\mu}=\epsilon_{\mu\nu\lambda}\hat{r}_\nu\hat{p}_\lambda,

where

\epsilon_{\mu\nu\lambda} = 
\begin{cases}
+1, & \mbox{if } (\mu,\nu,\lambda) \mbox{ is } (1,2,3), (3,1,2) \mbox{ or } (2,3,1), \\
-1, & \mbox{if } (\mu,\nu,\lambda) \mbox{ is } (3,2,1), (1,3,2) \mbox{ or } (2,1,3), \\
0, & \mbox{otherwise: }\mu=\nu \mbox{ or } \nu=\lambda \mbox{ or } \lambda=\mu
\end{cases}

and we use the Einstein summation convention, in which sums over repeated indices are omitted. The above definition of the Levi-Civita symbol gives the "sign" of a permutation of 123 (it is 1 for even permutations, or -1 for odd permutations).

We can immediately verify the following commutation relations:

[\hat{L}_\mu,\hat{x}_\nu]=i\hbar\epsilon_{\mu\nu\lambda}\hat{x}_\lambda

[\hat{L}_\mu,\hat{p}_\nu]=i\hbar\epsilon_{\mu\nu\lambda}\hat{p}_\lambda

[\hat{L}_\mu,\hat{L}_\nu]=i\hbar\epsilon_{\mu\nu\lambda}\hat{L}_\lambda

The last relation may also be written as

\mathbf{L}\times\mathbf{L}=i\hbar\mathbf{L}.

Furthermore,

[\hat{\mathbf{n}}\cdot\hat{\mathbf{L}},\hat{\mathbf{r}}]=i\hbar(\hat{\mathbf{r}}\times\hat{\mathbf{n}})
[\hat{\mathbf{n}}\cdot\hat{\mathbf{L}},\hat{\mathbf{p}}]=i\hbar(\hat{\mathbf{p}}\times\hat{\mathbf{n}})
[\hat{\mathbf{n}}\cdot\hat{\mathbf{L}},\hat{\mathbf{L}}]=i\hbar(\hat{\mathbf{L}}\times\hat{\mathbf{n}})

For example,


\begin{align}
\left[\hat{L}_\mu,\hat{x}_\nu\right] &= [\epsilon_{\mu\lambda\rho}\hat{x}_\lambda \hat{p}_\rho,\hat{x}_\nu] = \epsilon_{\mu\lambda\rho}[\hat{x}_\lambda \hat{p}_\rho,\hat{x}_\nu] = \epsilon_{\mu\lambda\rho}\hat{x}_\lambda[\hat{p}_\rho,\hat{x}_\nu] \\
&= \epsilon_{\mu\lambda\rho}\hat{x}_\lambda\frac{\hbar}{i}\delta_{\rho\nu} = \epsilon_{\mu\lambda\nu}\hat{x}_\lambda\frac{\hbar}{i} \\
&= i\hbar\epsilon_{\mu\nu\lambda}\hat{x}_\lambda.
\end{align}

Also, note that for \hat{mathbf{L}}^2=\hat{L}_x^2+\hat{L}_y^2+\hat{L}_z^2=\hat{L}_{\mu}\hat{L}_{\mu},


\begin{align}
\left[\hat{L}_{\mu},\hat{L}^2\right] &= \left[\hat{L}_{\mu},\hat{L}_{\nu}\hat{L}_{\nu}\right] \\
&= \hat{L}_{\nu}\left[\hat{L}_{\mu},\hat{L}_{\nu}\right]+\left[\hat{L}_{\mu},\hat{L}_{\nu}\right]\hat{L}_{\nu} \\
&= \hat{L}_{\nu} i\hbar \epsilon_{\mu\nu\lambda} \hat{L}_{\lambda} + i\hbar \epsilon_{\mu\nu\lambda} \hat{L}_{\lambda} \hat{L}_{\nu} \\
&= i\hbar \epsilon_{\mu\nu\lambda} \hat{L}_{\nu}\hat{L}_{\lambda} - i\hbar \epsilon_{\mu\lambda\nu}\hat{L}_{\lambda}\hat{L}_{\nu} \\
&= i\hbar \epsilon_{\mu\nu\lambda} \hat{L}_{\nu}\hat{L}_{\lambda} - i\hbar \epsilon_{\mu\nu\lambda}\hat{L}_{\nu}\hat{L}_{\lambda} \\
&= 0. 
\end{align}

Therefore, the magnitude of the angular momentum squared commutes with any one component of the angular momentum, and thus both may be specified exactly in a given measurement.

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