Orbital Angular Momentum Eigenfunctions

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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

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

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