Editing Quantum harmonic oscillator (section)

===Example: 3D isotropic harmonic oscillator===
{{see also|Particle in a spherically symmetric potential#3D isotropic harmonic oscillator}}
[[File:2D_Spherical_Harmonic_Orbitals.png|thumb|300px|right|Schrödinger 3D spherical harmonic orbital solutions in 2D density plots; the [[Mathematica]] source code that used for generating the plots is at the top]]
The Schrödinger equation for a particle in a spherically-symmetric three-dimensional harmonic oscillator can be solved explicitly by separation of variables. This procedure is analogous to the separation performed in the [[Hydrogen-like atom#Schrödinger equation in a spherically symmetric potential|hydrogen-like atom]] problem, but with a different [[Particle in a spherically symmetric potential|spherically symmetric potential]]
<math display="block">V(r) = {1\over 2} \mu \omega^2 r^2,</math>
where {{mvar|μ}} is the mass of the particle. Because {{mvar|m}} will be used below for the magnetic quantum number, mass is indicated by  {{mvar|μ}}, instead of {{mvar|m}}, as earlier in this article.

The solution to the equation is:<ref>[[Albert Messiah]], ''Quantum Mechanics'', 1967, North-Holland, Ch XII,  § 15, p 456.[https://archive.org/details/QuantumMechanicsVolumeI/page/n239 online]</ref>
<math display="block">\psi_{klm}(r,\theta,\phi) = N_{kl} r^{l}e^{-\nu r^2}L_k^{\left(l+{1\over 2}\right)}(2\nu r^2) Y_{lm}(\theta,\phi)</math>
where
:<math>N_{kl}=\sqrt{\sqrt{\frac{2\nu^3}{\pi }}\frac{2^{k+2l+3}\;k!\;\nu^l}{(2k+2l+1)!!}}~~</math> is a normalization constant; <math>\nu \equiv {\mu \omega \over 2 \hbar}~</math>;
:<math>{L_k}^{(l+{1\over 2})}(2\nu r^2)</math> 
are [[Laguerre polynomials#Generalized Laguerre polynomials|generalized Laguerre polynomials]]; The order {{mvar|k}}  of the polynomial is a non-negative integer;
*<math>Y_{lm}(\theta,\phi)\,</math> is a [[spherical harmonics|spherical harmonic function]];
*{{mvar|ħ}} is the reduced [[Planck constant]]: <math>\hbar\equiv\frac{h}{2\pi}~.</math>

The energy eigenvalue is
<math display="block">E=\hbar \omega \left(2k + l + \frac{3}{2}\right) .</math>
The energy is usually described by the single [[quantum number]]
<math display="block">n\equiv 2k+l  \,.</math>

Because {{mvar|k}} is a non-negative integer, for every even {{mvar|n}} we have {{math|1=''ℓ'' = 0, 2, ..., ''n'' − 2, ''n''}} and for every odd {{mvar|n}}  we have {{math|1=''ℓ'' = 1, 3, ..., ''n'' − 2, ''n''}} . The magnetic quantum number {{mvar|m}} is an integer satisfying {{math|−''ℓ'' ≤ ''m'' ≤ ''ℓ''}}, so for every {{mvar|n}} and ''ℓ'' there are 2''ℓ''&nbsp;+&nbsp;1 different [[quantum state]]s, labeled by {{mvar|m}} . Thus, the degeneracy at level {{mvar|n}} is
<math display="block">\sum_{l=\ldots,n-2,n} (2l+1) = {(n+1)(n+2)\over 2} \,,</math>
where the sum starts from 0 or 1, according to whether {{mvar|n}} is even or odd.
This result is in accordance with the dimension formula above, and amounts to the dimensionality of a symmetric representation of {{math|SU(3)}},<ref>{{cite journal|last=Fradkin |first=D. M. |title=Three-dimensional isotropic harmonic oscillator and SU3. |journal=American Journal of Physics |volume=33 |number=3 |year=1965 |pages=207–211|doi=10.1119/1.1971373 }}</ref> the relevant degeneracy group.