Spherical Coordinates

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 now write down the Cartesian components of the angular momentum operator in spherical coordinates. We will make use of this result later in determining the eigenfunctions of the angular momentum squared and of one of its components.

The Cartesian coordinates $x,\!</math<math>y,\!$ and $z\!$ can be written in terms of the spherical coordinates $r,\!$ $\theta ,\!$ and $\phi \!$ as follows:

$x=r\sin \theta \cos \phi ,\!$ $y=r\sin \theta \sin \phi ,\!$ $z=r\cos \theta \!$

Denote the state $\langle \mathbf {r} \!\,|=\langle \mathbf {r} \!\,\theta \phi |$

If you choose $\alpha \!$ along the z-axis then the only coordinate that will change is phi such that $\phi \rightarrow \phi +\alpha$. Now the state is written as:

$\langle \mathbf {r} \!\,\theta \phi |\left(1+{\frac {i}{\hbar }}\alpha L_{z}\right)=\langle \mathbf {r} \!\,\theta \phi +\alpha |$

Working to first order in alpha the right hand side becomes:

$\langle \mathbf {r} \!\,\theta \phi |+\alpha {\frac {\partial }{\partial \phi }}\langle \mathbf {r} \!\,\theta \phi |$

Therefore $L_{z}={\frac {\hbar }{i}}{\frac {\partial }{\partial \phi }}$

Now choose $\alpha \!$ along the x-axis then the cartesian coordinates are changed such that: $x\rightarrow x\!$ , $y\rightarrow y-\alpha z\!$, and $z\rightarrow z+\alpha y\!$,

from these transformations it can be determined that $\delta \theta =-\alpha \sin \phi \!$ since $\delta z=\alpha y\!$ and since x does not change it can be determined that $\delta \phi =-\alpha \cot \theta \cos \phi \!$.

This means that the original state is now written as: $\langle \mathbf {r} \!\,\theta \phi |\left(1+{\frac {i}{\hbar }}\alpha L_{x}\right)=\langle \mathbf {r} \!\,\,\theta -\alpha \sin \phi \,\phi -\alpha \cot \theta \cos \phi |$

Expanding the right hand side of the above equation as before to the first order of alpha the whole equation becomes: $\langle \mathbf {r} \!\,\theta \phi |L_{x}={\frac {\hbar }{i}}\left(-\sin \phi {\frac {\partial }{\partial \theta }}\,-\cot \theta \cos \phi {\frac {\partial }{\partial \phi }}\!\right)\langle \mathbf {r} \!\,\theta \phi |$

Therfore $L_{x}={\frac {\hbar }{i}}\left(-\sin \phi {\frac {\partial }{\partial \theta }}\,-\cot \theta \cos \phi {\frac {\partial }{\partial \phi }}\!\right)$

Using the same techinque, choose $\alpha \!$ along the y-axis and the coordinates will change in a similar fashion such that it can be shown that $L_{y}={\frac {\hbar }{i}}\left(\cos \phi {\frac {\partial }{\partial \theta }}-\cot \theta \sin \phi {\frac {\partial }{\partial \phi }}\!\right)$

Problem

(Richard L. Liboff, Introductory Quantum Mechanics, 2nd Edition, pp. 377-379)

Show, using the above results, that the operator,

${\hat {R}}_{\Delta \phi }=\exp \left({\frac {i}{\hbar }}\Delta \phi {\hat {L}}_{z}\right),$

when applied to a function $f(\phi )\!$ of the azimuthal angle $\phi ,\!$ rotates the angle $\phi \!$ to $\phi +\Delta \phi .$ That is, show that

${\hat {R}}_{\Delta \phi }f(\phi )=f(\phi +\Delta \phi ).$

Solution