Commutation Relations

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.

Physical Basis of Quantum Mechanics

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

Schrödinger Equation

Brief Derivation of Schrödinger Equation Relation Between the Wave Function and Probability Density Stationary States Heisenberg Uncertainty Principle Some Consequences of the Uncertainty Principle

Operators, Eigenfunctions, and Symmetry

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

Motion in One Dimension

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

Discrete Eigenvalues and Bound States

Harmonic Oscillator Spectrum and Eigenstates Analytical Method for Solving the Simple Harmonic Oscillator Coherent States Charged Particles in an Electromagnetic Field WKB Approximation

Time Evolution and the Pictures of Quantum Mechanics

The Heisenberg Picture: Equations of Motion for Operators The Interaction Picture The Virial Theorem

Angular Momentum

Commutation Relations Angular Momentum as a Generator of Rotations in 3D Spherical Coordinates Eigenvalue Quantization Orbital Angular Momentum Eigenfunctions

Central Forces

General Formalism Free Particle in Spherical Coordinates Spherical Well Isotropic Harmonic Oscillator Hydrogen Atom WKB in Spherical Coordinates

The Path Integral Formulation of Quantum Mechanics

Feynman Path Integrals The Free-Particle Propagator Propagator for the Harmonic Oscillator

Continuous Eigenvalues and Collision Theory

Differential Cross Section and the Green's Function Formulation of Scattering Central Potential Scattering and Phase Shifts Coulomb Potential Scattering

Multidimensional problems entail the possibility of having rotation as a part of solution. Just like in classical mechanics where we can calculate the angular momentum using vector cross product, we have a very similar form of equation. However, just like any observable in quantum mechanics, this angular momentum is expressed by a Hermitian operator. Similar to classical mechanics we write the angular momentum operator $\mathbf {L} \!$ as:

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

Working in the spatial representation, we have $\mathbf {r}$ as our radial vector, while $\mathbf {p}$ is the momentum operator.

$\mathbf {p} =-i\hbar \nabla$

Using the cross product in Cartesian coordinate system, we get a component of ${\mathbf {L}}\!$ in each direction:

$L_{x}=yp_{z}-zp_{y}={\frac {\hbar }{i}}\left(y{\frac {\partial }{\partial z}}-z{\frac {\partial }{\partial y}}\right)\!$

Similarly, using cyclic permutation on the coordinates x, y, z, we get 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 (the Einstein summation convention of summing over repeated indices is understood here)

$L_{\mu }=\epsilon _{\mu \nu \lambda }r_{\nu }p_{\lambda }\!$

with

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

Or we simply say that the even permutation gives 1, odd permutation gives -1, otherwise, we get 0.

We can immediately verify the following commutation relations:

$[L_{\mu },r_{\nu }]=i\hbar \epsilon _{\mu \nu \lambda }r_{\lambda }$

$[L_{\mu },p_{\nu }]=i\hbar \epsilon _{\mu \nu \lambda }p_{\lambda }$

$[L_{\mu },L_{\nu }]=i\hbar \epsilon _{\mu \nu \lambda }L_{\lambda }$ (this relation tells us :$\mathbf {L} \times \mathbf {L} =i\hbar \mathbf {L}$)

and

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

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

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

For example,

${\begin{aligned}\left[L_{\mu },r_{\nu }\right]&=[\epsilon _{\mu \lambda \rho }r_{\lambda }p_{\rho },r_{\nu }]=\epsilon _{\mu \lambda \rho }[r_{\lambda }p_{\rho },r_{\nu }]=\epsilon _{\mu \lambda \rho }r_{\lambda }[p_{\rho },r_{\nu }]\\&=\epsilon _{\mu \lambda \rho }r_{\lambda }{\frac {\hbar }{i}}\delta _{\rho \nu }=\epsilon _{\mu \lambda \nu }r_{\lambda }{\frac {\hbar }{i}}\\&=i\hbar \epsilon _{\mu \nu \lambda }r_{\lambda }\end{aligned}}$

Also, note that for $L^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}=L_{\mu }L_{\mu }$,

${\begin{aligned}\left[L_{\mu },L^{2}\right]&=\left[L_{\mu },L_{\nu }L_{\nu }\right]\\&=L_{\nu }\left[L_{\mu },L_{\nu }\right]+\left[L_{\mu },L_{\nu }\right]L_{\nu }\\&=L_{\nu }i\hbar \epsilon _{\mu \nu \lambda }L_{\lambda }+i\hbar \epsilon _{\mu \nu \lambda }L_{\lambda }L_{\nu }\\&=i\hbar \epsilon _{\mu \nu \lambda }L_{\nu }L_{\lambda }-i\hbar \epsilon _{\mu \lambda \nu }L_{\lambda }L_{\nu }\\&=i\hbar \epsilon _{\mu \nu \lambda }L_{\nu }L_{\lambda }-i\hbar \epsilon _{\mu \nu \lambda }L_{\nu }L_{\lambda }\\&=0.\end{aligned}}$