Isotropic Harmonic Oscillator

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

We now solve the isotropic harmonic oscillator using the formalism that we have just developed. While it is possible to solve it in Cartesian coordinates, we gain additional insight by solving it in spherical coordinates, and it is easier to determine the degeneracy of each energy level.

The radial part of the Schrödinger equation for a particle of mass $M\!$ in an isotropic harmonic oscillator potential $V(r)={\frac {1}{2}}M\omega ^{2}r^{2}$ is given by:

$-{\frac {\hbar ^{2}}{2M}}{\frac {d^{2}u_{nl}}{dr^{2}}}+\left({\frac {\hbar ^{2}}{2M}}{\frac {l(l+1)}{r^{2}}}+{\frac {1}{2}}M\omega ^{2}r^{2}\right)u_{nl}=Eu_{nl}.$

Let us begin by looking at the solutions $u_{nl}\!$ in the limits of small and large $r.\!$

As $r\rightarrow 0\!$, the equation reduces to

$-{\frac {\hbar ^{2}}{2M}}{\frac {d^{2}u_{nl}}{dr^{2}}}+{\frac {\hbar ^{2}}{2M}}{\frac {l(l+1)}{r^{2}}}u_{nl}=Eu_{nl}.$

The only solution of this equation that does not diverge as $r\rightarrow 0$ is $u_{nl}(r)\simeq r^{l+1}.$

In the limit as $r\rightarrow \infty ,$ on the other hand, the equation becomes

$-{\frac {\hbar ^{2}}{2M}}{\frac {d^{2}u_{nl}}{dr^{2}}}+{\frac {1}{2}}M\omega ^{2}r^{2}u_{nl}=Eu_{nl}$

whose solution is given by $u_{nl}(r)\simeq e^{-M\omega r^{2}/2\hbar }.$

We may now assume that the general solution to the equation is given by

$u_{nl}(r)=r^{l+1}e^{-M\omega r^{2}/2\hbar }f_{nl}(r).$

Substituting this expression into the original equation, we obtain

${\frac {d^{2}f_{nl}}{dr^{2}}}+2\left({\frac {l+1}{r}}-{\frac {M\omega }{\hbar }}r\right){\frac {df_{nl}}{dr}}+\left[{\frac {2ME}{\hbar ^{2}}}-(2l+3){\frac {M\omega }{\hbar }}\right]f_{nl}=0.$

We now use a series solution for this equation:

$f_{nl}(r)=\sum _{n=0}^{\infty }a_{n}r^{n}=a_{0}+a_{1}r+a_{2}r^{2}+a_{3}r^{3}+\ldots +a_{n}r^{n}+\ldots$

Substituting this solution into the reduced form of the equation, we obtain

$\sum _{n=0}^{\infty }\left[n(n-1)a_{n}r^{n-2}+2\left({\frac {l+1}{r}}-{\frac {M\omega }{\hbar }}r\right)na_{n}r^{n-1}+\left[{\frac {2ME}{\hbar ^{2}}}-(2l+3){\frac {M\omega }{\hbar }}\right]a_{n}r^{n}\right]=0,$

which reduces to

${\frac {2(l+1)}{r}}a_{1}+\sum _{n=0}^{\infty }\left[(n+2)(n+2l+3)a_{n+2}+\left(-{\frac {2M\omega }{\hbar }}n+{\frac {2ME}{\hbar ^{2}}}-(2l+3){\frac {M\omega }{\hbar }}\right)a_{n}\right]r^{n}=0.$

For this equation to hold, the coefficients of each of the powers of $r\!$ must vanish seperately. Doing this for the positive powers of $r\!$ yields the following recursion relation:

$(n+2)(n+2l+3)a_{n+2}=\left[-{\frac {2ME}{\hbar ^{2}}}+(2n+2l+3){\frac {M\omega }{\hbar }}\right]a_{n}$

In addition, we have an $r^{-1}\!$ term; for it to vanish, we must set $a_{1}=0.\!$ This, combined with the above recursion relation, means that the function $f_{nl}(r)\!$ contains only even powers of $r.\!$ In other words,

$f_{nl}(r)=\sum _{n=0,2,4,\ldots }^{\infty }a_{n}r^{n}=\sum _{n'=0}^{\infty }a_{n'}r^{n'}.$

By a similar argument as the one that we employed for the one-dimensional harmonic oscillator, we find that, unless the series for $f_{nl}(r)\!$ terminates, the resulting full wave function will diverge as $r\rightarrow \infty .$ Because the series must only contain even powers of $r,\!$ the resulting quantization condition on the energy is

${\frac {2M}{\hbar ^{2}}}E_{n'l}-{\frac {M\omega }{\hbar }}(4n'+2l+3)=0,\,n'=0,1,2,3,\ldots ,$

or

$E_{nl}=\left(n+{\frac {3}{2}}\right)\hbar \omega ,\,n=0,1,2,3,\ldots ,$

where $n=2n'+l.\!$

The degeneracy corresponding to the $n^{\text{th}}\!$ level may be found to be ${\tfrac {1}{2}}(n+1)(n+2).$ We see that energy levels with even $n\!$ correspond to even values of $l,\!$ while those with odd $n\!$ have odd values of $l.\!$

The total wave function of the isotropic harmonic oscillator is thus given by

$\psi _{nlm}(r,\theta ,\phi )=r^{l+1}e^{-M\omega r^{2}/2\hbar }f_{nl}(r)Y_{lm}(\theta ,\phi )=R_{nl}(r)Y_{lm}(\theta ,\phi ).$

One may show that, in fact, $f_{nl}(r)\!$ is an associated Laguerre polynomial in ${\frac {M\omega }{\hbar }}r^{2}.$