Editing Quantum harmonic oscillator (section)

==Applications==
===Harmonic oscillators lattice: phonons===
{{see also|Canonical quantization}}

The notation of a harmonic oscillator can be extended to a one-dimensional lattice of many particles. Consider a one-dimensional quantum mechanical ''harmonic chain'' of ''N'' identical atoms. This is the simplest quantum mechanical model of a lattice, and we will see how [[phonon]]s arise from it. The formalism that we will develop for this model is readily generalizable to two and three dimensions.

As in the previous section, we denote the positions of the masses by  {{math|''x''<sub>1</sub>, ''x''<sub>2</sub>, ...}}, as measured from their equilibrium positions (i.e.  {{math|1=''x<sub>i</sub>'' = 0}} if the particle {{mvar|i}} is at its equilibrium position). In two or more dimensions, the {{math|''x<sub>i</sub>''}} are vector quantities. The [[Hamiltonian (quantum mechanics)|Hamiltonian]] for this system is

<math display="block">\mathbf{H} = \sum_{i=1}^N {p_i^2 \over 2m} + {1\over 2} m \omega^2 \sum_{\{ij\} (nn)} (x_i - x_j)^2 \,,</math>
where {{mvar|m}} is the (assumed uniform) mass of each atom, and  {{math|''x<sub>i</sub>''}} and  {{math|''p<sub>i</sub>''}} are the position and [[momentum]] operators for the ''i'' th atom and the sum is made over the nearest neighbors (nn).  However, it is customary to rewrite the Hamiltonian in terms of the [[normal modes]] of the [[wavevector]] rather than in terms of the particle coordinates so that one can work in the more convenient [[Fourier space]].
[[File:Superposition of three oscillating dipoles.gif|thumb|239x239px|Superposition of three oscillating dipoles- illustrate the time propagation of the common wave function for different n,l,m]]
We introduce, then, a set of {{mvar|N}} "normal coordinates" {{math|''Q<sub>k</sub>''}}, defined as the [[discrete Fourier transform]]s of the {{mvar|x}}s, and {{mvar|N}} "conjugate momenta"  {{mvar|Π}} defined as the Fourier transforms of the {{mvar|p}}s,
<math display="block">Q_k = {1\over\sqrt{N}} \sum_{l} e^{ikal} x_l</math>
<math display="block">\Pi_{k} = {1\over\sqrt{N}} \sum_{l}  e^{-ikal} p_l \,.</math>

The quantity {{math|''k<sub>n</sub>''}} will turn out to be the [[Wavenumber|wave number]] of the phonon, i.e. 2''π''  divided by the [[wavelength]]. It takes on quantized values, because the number of atoms is finite.

This preserves the desired commutation relations in either real space or wave vector space
[[File:Superposition of three oscillating dipoles 2.gif|thumb|243x243px|Another illustration of the time propagation of the common wave function for three different atoms emphasizes the effect of the angular momentum on the distribution behavior]]
<math display="block"> \begin{align} 
\left[x_l , p_m \right]&=i\hbar\delta_{l,m} \\ 
\left[ Q_k , \Pi_{k'} \right] &={1\over N} \sum_{l,m} e^{ikal} e^{-ik'am}  [x_l , p_m ] \\
 &= {i \hbar\over N} \sum_{m} e^{iam(k-k')} = i\hbar\delta_{k,k'} \\
\left[ Q_k , Q_{k'} \right] &= \left[ \Pi_k , \Pi_{k'} \right] = 0 ~.
\end{align}</math>

From the general result
<math display="block"> \begin{align} 
\sum_{l}x_l x_{l+m}&={1\over N}\sum_{kk'}Q_k Q_{k'}\sum_{l} e^{ial\left(k+k'\right)}e^{iamk'}= \sum_{k}Q_k Q_{-k}e^{iamk} \\ 
\sum_{l}{p_l}^2 &= \sum_{k}\Pi_k \Pi_{-k}   ~,
\end{align}</math>
it is easy to show, through elementary trigonometry, that the potential energy term is
<math display="block"> {1\over 2} m \omega^2 \sum_{j} (x_j - x_{j+1})^2= {1\over 2} m \omega^2\sum_{k}Q_k Q_{-k}(2-e^{ika}-e^{-ika})= {1\over 2} m \sum_{k}{\omega_k}^2Q_k Q_{-k} ~ ,</math>
where
<math display="block">\omega_k = \sqrt{2 \omega^2 (1 - \cos(ka))} ~.</math>

The Hamiltonian may be written in wave vector space as
<math display="block">\mathbf{H} = {1\over {2m}}\sum_k \left(
{ \Pi_k\Pi_{-k} } + m^2 \omega_k^2 Q_k Q_{-k} 
\right) ~.</math>

Note that the couplings between the position variables have been transformed away; if the {{mvar|Q}}s and {{mvar| Π}}s were [[Hermitian operator|hermitian]] (which they are not), the transformed Hamiltonian would describe {{mvar|N}} ''uncoupled'' harmonic oscillators.

The form of the quantization depends on the choice of boundary conditions; for simplicity, we impose ''periodic'' boundary conditions, defining the {{math|(''N'' + 1)}}-th atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is

<math display="block">k=k_n = {2n\pi \over Na}
\quad \hbox{for}\ n = 0, \pm1, \pm2, \ldots , \pm {N \over 2}. </math>

The upper bound to {{mvar|n}} comes from the minimum wavelength, which is twice the lattice spacing {{mvar|a}}, as discussed above.

The harmonic oscillator eigenvalues or energy levels for the mode {{math|''ω<sub>k</sub>''}} are 
<math display="block">E_n = \left({1\over2}+n\right)\hbar\omega_k   \quad\hbox{for}\quad n=0,1,2,3,\ldots</math>

If we ignore the [[zero-point energy]] then the levels are evenly spaced at 
<math display="block">0 , \ \hbar\omega , \ 2\hbar\omega , \ 3\hbar\omega , \ \cdots </math>

So an '''exact''' amount of [[energy]] {{math|''ħω''}},  must be supplied to the harmonic oscillator lattice to push it to the next energy level. In analogy to the [[photon]] case when the [[electromagnetic field]] is quantised, the quantum of vibrational energy is called a [[phonon]].

All quantum systems show wave-like and particle-like properties. The particle-like properties of the phonon are best understood using the methods of [[second quantization]] and operator techniques described elsewhere.<ref name="Mahan">{{cite book| last=Mahan |first=GD |title=Many particle physics|publisher= Springer|location=New York | isbn=978-0306463389 |year=1981}}</ref>

In the [[continuum limit]], {{math|''a'' → 0}}, {{math|''N'' → ∞}}, while {{math|''Na''}} is held fixed. The canonical coordinates {{math|''Q<sub>k</sub>''}} devolve to the decoupled momentum modes of a scalar field, <math>\phi_k</math>, whilst the location index {{mvar|i}} (''not the displacement dynamical variable'') becomes the parameter {{mvar|x}} argument of the scalar field, <math>\phi (x,t)</math>.

===Molecular vibrations===
{{main|Molecular vibration}}
* The vibrations of a [[diatomic molecule]] are an example of a two-body version of the quantum harmonic oscillator. In this case, the angular frequency is given by <math display="block">\omega = \sqrt{\frac{k}{\mu}} </math> where <math>\mu = \frac{m_1 m_2}{m_1 + m_2}</math> is the [[reduced mass]] and <math>m_1</math> and <math>m_2</math> are the masses of the two atoms.<ref>{{Cite web | title=Quantum Harmonic Oscillator | website=Hyperphysics | access-date=24 September 2009 | url=http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html}}</ref>
* The [[Hooke's atom]] is a simple model of the [[helium]] atom using the quantum harmonic oscillator.
* Modelling phonons, as discussed above.
* A charge <math>q</math> with mass <math>m</math> in a uniform magnetic field <math>\mathbf{B}</math> is an example of a one-dimensional quantum harmonic oscillator: [[Landau quantization]].