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's consider spherical well potentials,

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

The Schrödinger equations for these two regions can be written by

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

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

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

for $r>a\!$.

The general solutions are

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

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

For the $l=0\!$ term, the centrifugal barrier drops out and the equations become the following

${\begin{cases}0\leq r<a&({\frac {-\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial r^{2}}}-V_{0})u_{0}(r)=Eu_{0}(r)\\r>a&({\frac {-\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial r^{2}}})u_{0}(r)=Eu_{0}(r)\end{cases}}$

The generalized solutions are

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

Using the boundary condition, $u(r=0)=0\!$, we find that $A=-B\!$. The second equation can then be reduced to sinusoidal function where $\alpha =2iA\!$.

$u_{0}(r)=2iA\sin(kr)=\alpha \sin(kr)=\alpha \sin \left({\frac {r}{\hbar }}{\sqrt {2m(E+V_{0})}}\right)$

for $r>a\!$, we know that $D=0\!$ since as $r\!$ approaches infinity, the wavefunction does not go to zero.

$u_{0}(r)=Ce^{ik'r}+De^{-ik'r}=Ce^{-{\frac {r}{\hbar }}{\sqrt {-2mE}}}$

Matching the conditions that at $r=a\!$, the wavefunctions and their derivatives must be continuous which results in 2 equations

$\alpha \sin \left({\frac {a}{\hbar }}{\sqrt {2m(E+V_{0})}}\right)=Ce^{-{\frac {a}{\hbar }}{\sqrt {-2mE}}}$

$\alpha {\frac {\sqrt {2m(E+V_{0})}}{\hbar }}\cos \left({\frac {a}{\hbar }}{\sqrt {2m(E+V_{0})}}\right)=-{\frac {\sqrt {-2mE}}{\hbar }}Ce^{-{\frac {a}{\hbar }}{\sqrt {-2mE}}}$

Dividing the above equations, we find

$-\cot \left({\sqrt {{\frac {2m}{\hbar ^{2}}}(V_{0}-|E|)a^{2}}}\right)={\frac {\sqrt {\frac {2m|E|}{\hbar ^{2}}}}{\sqrt {\frac {2m(V_{0}-|E|)}{\hbar ^{2}}}}}$, which is the solution for the odd state in 1D square well.

Solving for $V_{0}\!$, we know that there is no bound state for

$V_{0}<{\frac {\pi ^{2}\hbar ^{2}}{8ma^{2}}}$.