Free Particle in 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.
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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
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The Schrödinger Equation in Dirac Notation
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One-Dimensional Bound States
Oscillation Theorem
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Motion in a Periodic Potential
Summary of One-Dimensional Systems
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Charged Particles in an Electromagnetic Field
WKB Approximation
The Heisenberg Picture: Equations of Motion for Operators
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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

A free particle is a specific case when V_0=0\! of the motion in a uniform potential V(r) = V0, so it is more useful to consider a particle moving in a uniform potential. We will make use of these results in the next section to discuss the spherical potential well. 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_ld_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_lj_l(\rho)+B_ln_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).

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