Spherical Well

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

Let us now consider a spherical well potential, given by

$V(\mathbf {r} )={\begin{cases}-V_{0},&0\leq r<a\\0,&r>a.\end{cases}}$

The Schrödinger equations for these two regions are

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

for $0\leq r<a\!$ and

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

for $r>a.\!$

The general solutions are

${\begin{aligned}&{\frac {u_{l}(r)}{r}}=C_{l}j_{l}(k'r),&r\leq a,\\&{\frac {u_{l}(r)}{r}}=A_{l}j_{l}(kr)+Bn_{l}(kr),&r>a,\end{aligned}}$

where $k'={\sqrt {\frac {2m(E+V_{0})}{\hbar ^{2}}}}\!$ and $k={\sqrt {\frac {2mE}{\hbar }}}.\!$

Let us now consider bound states for the special case, $l=0.\!$ In this case, the centrifugal barrier drops out and the equations become

${\begin{cases}\left({\frac {-\hbar ^{2}}{2m}}{\frac {d^{2}}{dr^{2}}}-V_{0}\right)u_{0}=Eu_{0},&0\leq r<a\\{\frac {-\hbar ^{2}}{2m}}{\frac {d^{2}u_{0}}{dr^{2}}}=Eu_{0},&r>a.\end{cases}}$

The solution for this case is

${\begin{cases}u_{0}(r)=Ae^{ikr}+Be^{-ikr},&0\leq r<a\\u_{0}(r)=Ce^{\kappa r}+De^{-\kappa r},&r>a,\end{cases}}$

where $\kappa ={\frac {\sqrt {-2mE}}{\hbar }}.$

Using the boundary condition, $u(0)=0,\!$ we find that $A=-B.\!$ The wave functions for $0\leq r<a\!$ thus reduces to

$u_{0}(r)=2iA\sin(kr)=\alpha \sin(kr),\!$

where $\alpha =2iA.\!$

For $r>a\!$, we know that $D=0\!$ since, as $r\rightarrow \infty ,\!$ the wavefunction must go to zero. Therefore, for the region in which $r>a,\!$

$u_{0}(r)=Ce^{-\kappa r}.\!$

Using the conditions that at $r=a,\!$ the wave functions and their derivatives must be continuous yields the following equations:

$\alpha \sin {ka}=Ce^{-\kappa a}\!$

and

$k\alpha \cos {ka}=-\kappa Ce^{-\kappa a}.\!$

Dividing the second equation by the first, we obtain

$-k\cot {ka}=\kappa ,\!$

which is just the solution for the odd states in a one-dimensional square well.

This, combined with the fact that

$\kappa ^{2}+k^{2}={\frac {2mV_{0}}{\hbar ^{2}}},$

shows that no bound state exists if $V_{0}<{\frac {\pi ^{2}\hbar ^{2}}{8ma^{2}}}.$