Orbital Angular Momentum Eigenfunctions

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

Worked Problem about angular momentum.

We will now find the orbital angular momentum eigenfunctions $|l,m\rangle$ in terms of position. Recall from the previous section that

${\hat {L}}_{z}|l,m\rangle =m\hbar |l,m\rangle .$

If we act on the left with a position eigenvector $\langle r,\theta ,\phi |,$ then this becomes

$\langle r,\theta ,\phi |{\hat {L}}_{z}|l,m\rangle =-i\hbar {\frac {\partial }{\partial \phi }}\langle r,\theta ,\phi |l,m\rangle =m\hbar \langle r,\theta ,\phi |l,m\rangle ,$

or, introducing $\psi _{l,m}(r,\theta ,\phi )=\langle r,\theta ,\phi |l,m\rangle ,$

${\frac {\partial \psi _{l,m}}{\partial \phi }}=im\psi _{l,m}.$

We may now separate out the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} dependence from the $r\!$ and $\theta \!$ dependences; i.e.,

$\psi _{l,m}(r,\theta ,\phi )=g(r,\theta )\Phi (\phi ).\!$

Solving for the $\phi \!$ dependence, we obtain

$\psi _{l,m}(r,\theta ,\phi )=g(r,\theta )e^{im\phi }.\!$

We may now determine the $\theta \!$ dependence by using the fact that

${\hat {\mathbf {L} }}^{2}|l,m\rangle =l(l+1)\hbar ^{2}|l,m\rangle .$

Writing this in the position basis as before, we obtain

$L^{2}=-\hbar ^{2}\left({\frac {1}{\sin \theta ^{2}}}{\frac {d^{2}}{d\phi ^{2}}}+{\frac {1}{\sin \theta }}{\frac {d}{d\theta }}\left(\sin \theta {\frac {d}{d\theta }}\right)\right)$

Where the eigenvalues of $L^{2}\!$ are:

$L^{2}|l,m\rangle =\hbar ^{2}l(l+1)|l,m\rangle$

We proceed by using the property of $L_{+}\!$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_-\!} , defined by

$L_{\pm }={\frac {\hbar }{i}}e^{\pm i\phi }\left(\pm i{\frac {d}{d\theta }}-\cot \theta {\frac {d}{d\phi }}\right)$

to find the following equation

$\langle r,\theta ,\phi |L_{+}|l,l\rangle =-i\hbar e^{i\phi }\left(i{\frac {\partial }{\partial \theta }}-\cot \theta {\frac {\partial }{\partial \phi }}\right)\langle r,\theta ,\phi |l,l\rangle =0$

Using the above equations, we get

$\left({\frac {\partial }{\partial \theta }}-l\cot \theta \right)\langle r,\theta ,\phi |l,l\rangle =0$

And the solution is

$\langle r,\theta ,\phi |l,l\rangle =f(r)e^{il\phi }(\sin \theta )^{l}$

where $f(r)\!$ is an arbitrary function of $r\!$. We can find the angular part of the solution by using $L_{-}\!$. It turns out to be

$P_{l}^{m}(x)={\frac {(-1)^{m}}{2^{l}l!}}(1-x^{2})^{m/2}{\frac {d^{l+m}}{dx^{l+m}}}(x^{2}-1)^{l}$

And we know that $Y_{l}^{m}(\theta ,\phi )\!$ are the spherical harmonics defined by

$Y_{l}^{m}(\theta ,\phi )=(-1)^{l}{\sqrt {{\frac {2l+1}{4\pi }}{\frac {(l-m)!}{(l+m)!}}}}P_{l}^{m}(\cos \theta )e^{im\phi }$

where the function with cosine argument is the associated Legendre polynomials defined by:

$P_{l}^{m}(x)=(-1)^{m}(1-x^{2})^{m/2}{\frac {d^{m}}{dx^{m}}}P_{l}(x)$

with

$P_{l}(x)={\frac {1}{2^{l}l!}}{\frac {d^{l}}{dx^{l}}}(x^{2}-1)^{l}$

And so we then can write:

$P_{l}^{m}(x)={\frac {(-1)^{m}}{2^{l}l!}}(1-x^{2})^{m/2}{\frac {d^{l+m}}{dx^{l+m}}}(x^{2}-1)^{l}$

Central forces are derived from a potential that depends only on the distance r of the moving particle from a fixed point, usually the coordinate origin. Since such forces produce no torque, the orbital angular momentum is conserved.

We can rewrite the angular momentum as

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

As has been shown, angular momentum acts as the generator of rotation.

An exercise with angular momentum.