Editing Quantum harmonic oscillator (section)

==''N''-dimensional isotropic harmonic oscillator==
{{anchor|N-dimensional harmonic oscillator}}
The one-dimensional harmonic oscillator is readily generalizable to {{math|''N''}} dimensions, where {{math|1=''N'' = 1, 2, 3, ...}}. In one dimension, the position of the particle was specified by a single [[coordinate system|coordinate]], {{math|''x''}}. In {{math|''N''}} dimensions, this is replaced by {{math|''N''}} position coordinates, which we label {{math|''x''<sub>1</sub>, ..., ''x''<sub>''N''</sub>}}. Corresponding to each position coordinate is a momentum; we label these {{math|''p''<sub>1</sub>, ..., ''p''<sub>''N''</sub>}}. The [[canonical commutation relations]] between these operators are
<math display="block">\begin{align}
{[}x_i , p_j{]} &= i\hbar\delta_{i,j} \\
{[}x_i , x_j{]} &= 0                  \\
{[}p_i , p_j{]} &= 0
\end{align}</math>

The Hamiltonian for this system is
<math display="block"> H = \sum_{i=1}^N \left( {p_i^2 \over 2m} + {1\over 2} m \omega^2 x_i^2 \right).</math>

As the form of this Hamiltonian makes clear, the {{math|''N''}}-dimensional harmonic oscillator is exactly analogous to {{math|''N''}} independent one-dimensional harmonic oscillators with the same mass and spring constant. In this case, the quantities {{math|''x''<sub>1</sub>, ..., ''x''<sub>''N''</sub>}} would refer to the positions of each of the {{math|''N''}} particles. This is a convenient property of the {{math|''r''<sup>2</sup>}} potential, which allows the potential energy to be separated into terms depending on one coordinate each.

This observation makes the solution straightforward. For a particular set of quantum numbers <math>\{n\}\equiv
\{n_1, n_2, \dots, n_N\}</math> the energy eigenfunctions for the {{math|''N''}}-dimensional oscillator are expressed in terms of the 1-dimensional eigenfunctions as:

<math display="block">\langle \mathbf{x}|\psi_{\{n\}}\rangle = \prod_{i=1}^N\langle x_i\mid \psi_{n_i}\rangle</math>

In the ladder operator method, we define {{math|''N''}} sets of ladder operators,

<math display="block">\begin{align}
a_i &= \sqrt{m\omega \over 2\hbar} \left(x_i + {i \over m \omega} p_i \right), \\
a^{\dagger}_i &= \sqrt{m \omega \over 2\hbar} \left( x_i - {i \over m \omega} p_i \right).
\end{align}</math>

By an analogous procedure to the one-dimensional case, we can then show that each of the {{math|''a<sub>i</sub>''}} and {{math|''a''<sup>†</sup><sub>''i''</sub>}} operators lower and raise the energy by {{math|''ℏω''}} respectively. The Hamiltonian is
<math display="block">H = \hbar \omega \, \sum_{i=1}^N \left(a_i^\dagger \,a_i + \frac{1}{2}\right).</math>
This Hamiltonian is invariant under the dynamic symmetry group {{math|''U''(''N'')}} (the unitary group in {{math|''N''}} dimensions), defined by
<math display="block">
U\, a_i^\dagger \,U^\dagger = \sum_{j=1}^N  a_j^\dagger\,U_{ji}\quad\text{for all}\quad
U \in U(N),</math>
where <math>U_{ji}</math> is an element in the defining matrix representation of {{math|''U''(''N'')}}.

The energy levels of the system are
<math display="block"> E = \hbar \omega \left[(n_1 + \cdots + n_N) + {N\over 2}\right].</math>
<math display="block">n_i = 0, 1, 2, \dots \quad (\text{the energy level in dimension } i).</math>

As in the one-dimensional case, the energy is quantized. The ground state energy is {{math|''N''}} times the one-dimensional ground energy, as we would expect using the analogy to {{math|''N''}} independent one-dimensional oscillators. There is one further difference: in the one-dimensional case, each energy level corresponds to a unique quantum state. In {{math|''N''}}-dimensions, except for the ground state, the energy levels are ''degenerate'', meaning there are several states with the same energy.

The degeneracy can be calculated relatively easily.  As an example, consider the 3-dimensional case: Define {{math|1=''n'' = ''n''<sub>1</sub> + ''n''<sub>2</sub> + ''n''<sub>3</sub>}}. All states with the same {{math|''n''}} will have the same energy.  For a given {{math|''n''}}, we choose a particular {{math|''n''<sub>1</sub>}}. Then {{math|1=''n''<sub>2</sub> + ''n''<sub>3</sub> = ''n'' − ''n''<sub>1</sub>}}. There are {{math|''n'' − ''n''<sub>1</sub> + 1}} possible pairs {{math|{{mset|''n''<sub>2</sub>, ''n''<sub>3</sub>}}}}.  {{math|''n''<sub>2</sub>}} can take on the values {{math|0}} to {{math|''n'' − ''n''<sub>1</sub>}}, and for each {{math|''n''<sub>2</sub>}} the value of {{math|''n''<sub>3</sub>}} is fixed. The degree of degeneracy therefore is:
<math display="block">g_n = \sum_{n_1=0}^n n - n_1 + 1 = \frac{(n+1)(n+2)}{2}</math>
Formula for general {{math|''N''}} and {{math|''n''}} [{{math|''g''<sub>''n''</sub>}} being the dimension of the symmetric irreducible {{math|''n''}}-th power representation of the unitary group {{math|''U''(''N'')}}]:
<math display="block">g_n = \binom{N+n-1}{n}</math>
The special case {{math|''N''}} = 3, given above, follows directly from this general equation.  This is however, only true for distinguishable particles, or one particle in {{math|''N''}} dimensions (as dimensions are distinguishable). For the case of {{math|''N''}} bosons in a one-dimension harmonic trap, the degeneracy scales as the number of ways to partition an integer {{math|''n''}} using integers less than or equal to {{math|''N''}}.

<math display="block">g_n = p(N_{-},n)</math>

This arises due to the constraint of putting {{math|''N''}} quanta into a state ket where <math display="inline">\sum_{k=0}^\infty k n_k = n  </math>   and <math display="inline"> \sum_{k=0}^\infty  n_k = N </math>, which are the same constraints as in integer partition.

===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.