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

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 $\phi \!$ dependence from the $r\!$ and $\theta \!$ dependences; i.e.,

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

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

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

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

${\hat {L}}_{+}|l,l\rangle =0.$

In the position basis, the raising and lowering operators are given by

${\hat {L}}_{\pm }={\frac {\hbar }{i}}e^{\pm i\phi }\left(\pm i{\frac {\partial }{\partial \theta }}-\cot \theta {\frac {\partial }{\partial \phi }}\right).$

We thus obtain

$\left({\frac {\partial }{\partial \theta }}-l\cot \theta \right)g_{l}(r,\theta )=0.$

Solving the above equation, we find that the full wave function is

$\psi _{l,l}(r,\theta ,\phi )=f_{l}(r)e^{il\phi }(\sin \theta )^{l},\!$

where $f_{l}(r)\!$ is an arbitrary function of $r.\!$ Note that this function may (and, as we will see in the next chapter, does) depend on $l.\!$ We may now find the wave functions $\psi _{l,m}(r,\theta ,\phi )\!$ by repeated application of ${\hat {L}}_{-}.$ It turns out to be

$\psi _{l,m}(r,\theta ,\phi )=f_{l}(r)e^{im\phi }P_{l}^{m}(\cos \theta ),$

where

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

is an associated Legendre function. One may also write this as

$\psi _{l,m}(r,\theta ,\phi )=f_{l}(r)Y_{l}^{m}(\theta ,\phi ),\!$

where

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

are the spherical harmonics.

In the next chapter, we will be considering particles in central potentials, which are potentials that depend only on the distance $r\!$ of the moving particle from a fixed point, usually the coordinate origin. Since the resulting forces produce no torque, the orbital angular momentum is conserved. In quantum mechanical terms, this means that the angular momentum operator commutes with the Hamiltonian. Therefore, the results developed throughout this chapter will be very useful in discussing such potentials.

Problems

(1) A system is initally in the state,

$\psi (\theta ,\phi )={\frac {1}{\sqrt {5}}}Y_{1}^{-1}(\theta ,\phi )+{\sqrt {\frac {3}{5}}}Y_{1}^{0}(\theta ,\phi )+{\frac {1}{\sqrt {5}}}Y_{1}^{1}(\theta ,\phi ).$

Find the possible results of a measurement of ${\hat {L}}_{z}$ and the probabilities of finding each value.

Solution

(2) Classically, the Earth revolves around the sun counter-clockwise in the $xy\!$ plane with the sun at the origin. Quantum mechanically, what is the minimum angle that the angular momentum vector of the earth can make with the $z\!$ axis? Ignore the intrinsic spin of the Earth. The orbital angular momentum of the Earth is $\ 4.83\cdot 10^{31}{\text{J}}\cdot {\text{s}}.$ Compare the minimum angle with that of a quantum particle with $l=4.\!$

Solution

(3) A plane rotator (i.e., a particle confined to move on a unit circle) is in a state with a wavefunction $\psi (\phi )=A\sin ^{2}{\phi },\!$ where $\phi \!$ is the azimuthal angle.

(a) Determine the normalization constant, $A.\!$

(b) Find the probability of measuring different values of the $z\!$ component of the angular momentum ${\hat {L}}_{z}.$

(c) Find the expectation values of ${\hat {L}}_{z}$ and ${\hat {L}}_{z}^{2}.$

Solution