Free Particle in 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

A free particle is a specific case when $V_{0}=0\!$ of the motion in a uniform potential $V(r)=V_{0},$ so it is more useful to consider a particle moving in a uniform potential. The Schrödinger equation for the radial part of the wave function is

$\left(-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dr^{2}}}+{\frac {\hbar ^{2}}{2m}}{\frac {l(l+1)}{r^{2}}}+V_{0}\right)u_{l}=Eu_{l}.$

Let $k^{2}={\frac {2m}{\hbar ^{2}}}|E-V|.$ Rearranging the equation gives us

$\left(-{\frac {d^{2}}{dr^{2}}}+{\frac {l(l+1)}{r^{2}}}-k^{2}\right)u_{l}=0.$

If we now let $\rho =kr,\!$ then the equation reduces to the dimensionless form,

$\left(-{\frac {d^{2}}{d\rho ^{2}}}+{\frac {l(l+1)}{\rho ^{2}}}\right)u_{l}(\rho )=u_{l}(\rho )=d_{l}d_{l}^{+}u_{l}(\rho ),$

where $d_{l}\!$ and $d_{l}^{\dagger }\!$ are the raising and lowering operators,

$d_{l}={\frac {d}{d\rho }}+{\frac {l+1}{\rho }}$

and

$d_{l}^{\dagger }=-{\frac {d}{d\rho }}+{\frac {l+1}{\rho }}.$

Because $d_{l}^{\dagger }d_{l}=d_{l+1}d_{l+1}^{\dagger },$ it follows that

$d_{l}^{\dagger }u_{l}(\rho )=c_{l}u_{l+1}(\rho ).$

For $l=0,$

$-{\frac {d^{2}}{d\rho ^{2}}}u_{0}(\rho )=u_{0}(\rho ),$

whose solution is

$u_{0}(\rho )=A\sin {\rho }-B\cos {\rho }.\!$

The raising operator may now be applied to this state in order to find the solutions for higher values of $l.\!$ By repeated application of this operator, we obtain the wave function for all values of $l:\!$

$f_{l}(\rho )={\frac {u_{l}(\rho )}{\rho }}=A_{l}j_{l}(\rho )+B_{l}n_{l}(\rho ),$

where $j_{l}(\rho )\!$ is a spherical Bessel function and $n_{l}(\rho )\!$ is a spherical Neumann function, or spherical Bessel functions of the first and second kinds, respectively.

Properties of the Spherical Bessel and Neumann Functions

Explicit forms of the first few spherical Bessel and Neumann functions:

$j_{0}(z)={\frac {\sin(z)}{z}}$ $j_{1}(z)={\frac {\sin(z)}{z^{2}}}-{\frac {\cos(z)}{z}}$ $j_{2}(z)=\left({\frac {3}{z^{3}}}-{\frac {1}{z}}\right)\sin(z)-{\frac {3}{z^{2}}}\cos(z)$

$n_{0}(z)=-{\frac {\cos(z)}{z}}$ $n_{1}(z)=-{\frac {\cos(z)}{z^{2}}}-{\frac {\sin(z)}{z}}$ $n_{2}(z)=-\left({\frac {3}{z^{3}}}-{\frac {1}{z}}\right)\cos(z)-{\frac {3}{z^{2}}}\sin(z)$

We may also define spherical Hankel functions of the first and second kind in terms of the spherical Bessel and Neumann functions:

$h_{\ell }^{(1)}=j_{\ell }(z)+in_{\ell }(z)$

and

$h_{\ell }^{(2)}=j_{\ell }(z)-in_{\ell }(z)$

The asymptotic forms of the spherical Bessel and Neumann functions as $z\rightarrow \infty$ are

$j_{\ell }(z)={\frac {\sin(z-{\frac {\ell \pi }{2}})}{z}}$

and

$n_{\ell }(z)={\frac {\cos(z-{\frac {\ell \pi }{2}})}{z}}.$

The first few zeros of the spherical Bessel function for $l=0\!$ and $l=1\!$ are

$l=0:3.142,6.283,9.425,12.566,\ldots$

and

$l=1:4.493,7.725,10.904,14.066,\ldots$

The derivatives of the spherical Bessel and Neumann functions are given by

$j'_{l}(z)={\frac {l}{z}}j_{l}(z)-j_{l+1}(z)$

and

$n'_{l}(z)={\frac {l}{z}}n_{l}(z)-n_{l+1}(z).$