Spherical Coordinates

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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 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,\! 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\!

Let us start with the x\! component of the angular momentum, \hat{L}_x. In Cartesian coordinates, this is

\hat{L}_x=-i\hbar\left (y\frac{\partial}{\partial z}-z\frac{\partial}{\partial y}\right ).

If we make use of the chain rule, then we obtain

\hat{L}_{x} = -i\hbar\left( -\sin\phi \frac{\partial}{\partial \theta} \, -\cot\theta\cos\phi \frac{\partial}{\partial\phi} \! \right).

Similarly, the y\! and z\! components may be found to be

\hat{L}_{y} = -i\hbar\left(\cos\phi \frac{\partial}{\partial\theta} - \cot\theta\sin\phi \frac{\partial}{\partial\phi} \! \right)

and

\hat{L}_{z} = -i\hbar\frac{\partial}{\partial \phi}.

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

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