Editing Quantum harmonic oscillator (section)

===Hamiltonian and energy eigenstates===
[[Image:HarmOsziFunktionen.png|thumb|Wavefunction representations for the first eight bound eigenstates, ''n'' = 0 to 7. The horizontal axis shows the position ''x''.]]
[[Image:Aufenthaltswahrscheinlichkeit harmonischer Oszillator.png|thumb|Corresponding probability densities.]]

The [[Hamiltonian (quantum mechanics)|Hamiltonian]] of the particle is:
<math display="block">\hat H = \frac{{\hat p}^2}{2m} + \frac{1}{2} k {\hat x}^2 = \frac{{\hat p}^2}{2m} + \frac{1}{2} m \omega^2 {\hat x}^2 \, ,</math>
where {{mvar|m}} is the particle's mass, {{mvar|k}} is the force constant, <math display="inline">\omega = \sqrt{k / m}</math> is the [[angular frequency]] of the oscillator, <math>\hat{x}</math> is the [[position operator]] (given by {{mvar|x}} in the coordinate basis), and <math>\hat{p}</math>  is the [[momentum operator]] (given by <math>\hat p = -i \hbar \, \partial / \partial x</math> in the coordinate basis). The first term in the Hamiltonian represents the kinetic energy of the particle, and the second term represents its potential energy, as in [[Hooke's law]].{{sfnp|Zwiebach|2022|pp=233-234}}

The time-independent [[Schrödinger equation]] (TISE) is,
<math display="block"> \hat H \left| \psi \right\rangle = E \left| \psi \right\rangle  ~,</math>
where <math>E</math> denotes a real number (which needs to be determined) that will specify a time-independent [[energy level]], or [[eigenvalue]], and the solution  <math>| \psi \rangle</math> denotes that level's energy [[eigenstate]].{{sfnp|Zwiebach|2022|p=234}}

Then solve the differential equation representing this eigenvalue problem in the coordinate basis, for the [[wave function]] <math>\langle x | \psi \rangle = \psi (x) </math>, using a [[spectral method]]. It turns out that there is a family of solutions. In this basis, they amount to [[Hermite polynomials#Hermite functions| Hermite functions]],{{sfnp|Zwiebach|2022|p=241}}<ref>{{cite book|first=Gregory J. |last=Gbur |author-link=Greg Gbur |title=Mathematical Methods for Optical Physics and Engineering |publisher=Cambridge University Press |year=2011 |isbn=978-0-521-51610-5 |pages=631–633}}</ref>
<math display="block"> \psi_n(x) = \frac{1}{\sqrt{2^n\,n!}}  \left(\frac{m\omega}{\pi \hbar}\right)^{1/4}  e^{
- \frac{m\omega x^2}{2 \hbar}} H_n\left(\sqrt{\frac{m\omega}{\hbar}} x \right), \qquad n = 0,1,2,\ldots. </math>

The functions ''H<sub>n</sub>'' are the physicists' [[Hermite polynomials]],
<math display="block">H_n(z)=(-1)^n~ e^{z^2}\frac{d^n}{dz^n}\left(e^{-z^2}\right).</math>

The corresponding energy levels are{{sfnp|Zwiebach|2022|p=240}}
<math display="block"> E_n = \hbar \omega\bigl(n + \tfrac{1}{2}\bigr).</math>The expectation values of position and momentum combined with variance of each variable can be derived from the wavefunction to understand the behavior of the energy eigenkets. They are shown to be <math display="inline">\langle \hat{x} \rangle = 0 </math> and <math display="inline">\langle \hat{p} \rangle = 0  </math> owing to the symmetry of the problem, whereas:

<math>\langle \hat{x}^2 \rangle = (2n+1)\frac{\hbar}{2m\omega} = \sigma_x^2 </math>

<math>\langle \hat{p}^2 \rangle =  (2n+1)\frac{m\hbar\omega}{2} = \sigma_p^2  </math>

The variance in both position and momentum are observed to increase for higher energy levels. The lowest energy level has value of <math display="inline">\sigma_x \sigma_p = \frac{\hbar}{2}   </math> which is its minimum value due to uncertainty relation and also corresponds to a gaussian wavefunction.{{sfnp|Zwiebach|2022|pp=249-250}}

This energy spectrum is noteworthy for three reasons.  First, the energies are quantized, meaning that only discrete energy values (integer-plus-half multiples of {{math|''ħω''}}) are possible; this is a general feature of quantum-mechanical systems when a particle is confined. Second, these discrete energy levels are equally spaced, unlike in the [[Bohr model]] of the atom, or the [[particle in a box]]. Third, the lowest achievable energy (the energy of the {{math|1=''n'' = 0}} state, called the [[ground state]]) is not equal to the minimum of the potential well, but {{math|''ħω''/2}} above it; this is called [[zero-point energy]]. Because of the zero-point energy, the position and momentum of the oscillator in the ground state are not fixed (as they would be in a classical oscillator), but have a small range of variance, in accordance with the [[Heisenberg uncertainty principle]].

The ground state probability density is concentrated at the origin, which means the particle spends most of its time at the bottom of the potential well, as one would expect for a state with little energy. As the energy increases, the probability density peaks at the classical "turning points", where the state's energy coincides with the potential energy. (See the discussion below of the highly excited states.) This is consistent with the classical harmonic oscillator, in which the particle spends more of its time (and is therefore more likely to be found) near the turning points, where it is moving the slowest. The [[correspondence principle]] is thus satisfied. Moreover, special nondispersive [[wave packet]]s, with minimum uncertainty,  called [[Coherent states#The wavefunction of a coherent state|coherent states]] oscillate very much like classical objects, as illustrated in the figure; they are ''not'' eigenstates of the Hamiltonian.