Editing Schrödinger equation (section)

== Examples ==
{{See also|List of quantum-mechanical systems with analytical solutions}}

=== Particle in a box ===
[[File:Infinite potential well.svg|thumb|1-dimensional potential energy box (or infinite potential well)]]
{{Main|Particle in a box}}
The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy ''inside'' a certain region and infinite potential energy ''outside''.<ref name = "Cohen-Tannoudji">{{cite book|last1=Cohen-Tannoudji |first1=Claude |last2=Diu |first2=Bernard |last3=Laloë |first3=Franck |title=Quantum Mechanics |author-link1=Claude Cohen-Tannoudji |publisher=John Wiley & Sons |year=2005 |isbn=0-471-16433-X |translator-first1=Susan Reid |translator-last1=Hemley |translator-first2=Nicole |translator-last2=Ostrowsky |translator-first3=Dan |translator-last3=Ostrowsky}}</ref>{{rp|77–78}} For the one-dimensional case in the <math>x</math> direction, the time-independent Schrödinger equation may be written
<math display="block"> - \frac {\hbar ^2}{2m} \frac {d ^2 \psi}{dx^2} = E \psi.</math>

With the differential operator defined by
<math display="block"> \hat{p}_x = -i\hbar\frac{d}{dx} </math>
the previous equation is evocative of the [[Kinetic energy#Kinetic energy of rigid bodies|classic kinetic energy analogue]],
<math display="block"> \frac{1}{2m} \hat{p}_x^2 = E,</math>
with state <math>\psi</math> in this case having energy <math>E</math> coincident with the kinetic energy of the particle.

The general solutions of the Schrödinger equation for the particle in a box are
<math display="block"> \psi(x) = A e^{ikx} + B e ^{-ikx} \qquad\qquad E =  \frac{\hbar^2 k^2}{2m}</math>
or, from [[Euler's formula]],
<math display="block"> \psi(x) = C \sin(kx) + D \cos(kx).</math>

The infinite potential walls of the box determine the values of <math>C, D, </math> and <math>k</math> at <math>x=0</math> and <math>x=L</math> where <math>\psi</math> must be zero. Thus, at <math>x=0</math>,
<math display="block">\psi(0) = 0 = C\sin(0) + D\cos(0) = D</math>
and <math>D=0</math>. At <math>x=L</math>,
<math display="block"> \psi(L) = 0 = C\sin(kL),</math>
in which <math>C</math> cannot be zero as this would conflict with the postulate that <math>\psi</math> has norm 1. Therefore, since <math>\sin(kL)=0</math>, <math>kL</math> must be an integer multiple of <math>\pi</math>,
<math display="block">k = \frac{n\pi}{L}\qquad\qquad n=1,2,3,\ldots.</math>

This constraint on <math>k</math> implies a constraint on the energy levels, yielding
<math display="block">E_n = \frac{\hbar^2 \pi^2 n^2}{2mL^2} = \frac{n^2h^2}{8mL^2}.</math>

A [[finite potential well]] is the generalization of the infinite potential well problem to potential wells having finite depth. The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wave function is not pinned to zero at the walls of the well. Instead, the wave function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well. Another related problem is that of the [[rectangular potential barrier]], which furnishes a model for the [[quantum tunneling]] effect that plays an important role in the performance of modern technologies such as [[flash memory]] and [[scanning tunneling microscope|scanning tunneling microscopy]].

=== Harmonic oscillator ===
[[File:QuantumHarmonicOscillatorAnimation.gif|300px|thumb|right|A [[harmonic oscillator]] in classical mechanics (A–B) and quantum mechanics (C–H). In (A–B), a ball, attached to a [[Hooke's law|spring]], oscillates back and forth. (C–H) are six solutions to the Schrödinger Equation for this situation. The horizontal axis is position, the vertical axis is the real part (blue) or imaginary part (red) of the [[wave function]]. [[Stationary state]]s, or energy eigenstates, which are solutions to the time-independent Schrödinger equation, are shown in C, D, E, F, but not G or H.]]

{{Main|Quantum harmonic oscillator}}

The Schrödinger equation for this situation is
<math display="block"> E\psi = -\frac{\hbar^2}{2m}\frac{d^2}{d x^2}\psi + \frac{1}{2} m\omega^2 x^2\psi, </math>
where <math> x </math> is the displacement and <math> \omega </math> the angular frequency. Furthermore, it can be used to describe approximately a wide variety of other systems, including [[Molecular vibration|vibrating atoms, molecules]],<ref>{{cite book|title=Physical Chemistry |first=P. W. |last=Atkins |publisher=Oxford University Press |year=1978 |isbn=0-19-855148-7}}</ref> and atoms or ions in lattices,<ref>{{cite book|title=Solid State Physics |edition=2nd |first1=J. R. |last1=Hook |first2=H. E. |last2=Hall |series=Manchester Physics Series |publisher=John Wiley & Sons |year=2010 |isbn=978-0-471-92804-1}}</ref> and approximating other potentials near equilibrium points. It is also the [[Perturbation theory (quantum mechanics)#Applying perturbation theory|basis of perturbation methods]] in quantum mechanics.

The solutions in position space are
<math display="block"> \psi_n(x) = \sqrt{\frac{1}{2^n\,n!}} \ \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \ e^{
- \frac{m\omega x^2}{2 \hbar}} \ \mathcal{H}_n\left(\sqrt{\frac{m\omega}{\hbar}} x \right), </math>
where <math>n \in  \{0, 1, 2, \ldots \}</math>, and the functions <math> \mathcal{H}_n </math> are the [[Hermite polynomials]] of order <math> n </math>. The solution set may be generated by
<math display="block">\psi_n(x) = \frac{1}{\sqrt{n!}} \left( \sqrt{\frac{m \omega}{2 \hbar}} \right)^{n} \left( x - \frac{\hbar}{m \omega} \frac{d}{dx}\right)^n \left( \frac{m \omega}{\pi \hbar} \right)^{\frac{1}{4}} e^{\frac{-m \omega x^2}{2\hbar}}.</math>

The eigenvalues are
<math display="block">  E_n = \left(n + \frac{1}{2} \right) \hbar \omega. </math>

The case <math>  n = 0 </math> is called the [[ground state]], its energy is called the [[zero-point energy]], and the wave function is a [[Normal distribution|Gaussian]].<ref>{{Cite book|title=A Modern Approach to Quantum Mechanics |last=Townsend |first=John S. |publisher=University Science Books|year=2012|isbn=978-1-891389-78-8|pages=247–250, 254–5, 257, 272 |chapter=Chapter 7: The One-Dimensional Harmonic Oscillator}}</ref>

The harmonic oscillator, like the particle in a box, illustrates the generic feature of the Schrödinger equation that the energies of bound eigenstates are discretized.<ref name="Cohen-Tannoudji" />{{rp|352}}

=== Hydrogen atom ===
[[File:Hydrogen Density Plots.png|thumb|[[Wave function]]s of the [[electron]] in a hydrogen atom at different [[energy level]]s. They are plotted according to solutions of the Schrödinger equation.]]
The Schrödinger equation for the electron in a [[hydrogen atom]] (or a hydrogen-like atom) is
<math display="block"> E \psi = -\frac{\hbar^2}{2\mu}\nabla^2\psi - \frac{q^2}{4\pi\varepsilon_0 r}\psi </math>
where <math> q </math> is the electron charge, <math> \mathbf{r} </math> is the position of the electron relative to the nucleus, <math> r = |\mathbf{r}| </math> is the magnitude of the relative position, the potential term is due to the [[Coulomb's law|Coulomb interaction]], wherein <math> \varepsilon_0 </math> is the [[permittivity of free space]] and
<math display="block"> \mu = \frac{m_q m_p}{m_q+m_p} </math>
is the 2-body [[reduced mass]] of the hydrogen [[Nucleus (atomic structure)|nucleus]] (just a [[proton]]) of mass <math> m_p </math> and the electron of mass <math> m_q </math>. The negative sign arises in the potential term since the proton and electron are oppositely charged. The reduced mass in place of the electron mass is used since the electron and proton together orbit each other about a common center of mass, and constitute a two-body problem to solve. The motion of the electron is of principal interest here, so the equivalent one-body problem is the motion of the electron using the reduced mass.

The Schrödinger equation for a hydrogen atom can be solved by separation of variables.<ref>{{cite book|title=Physics for Scientists and Engineers – with Modern Physics |edition=6th |first1=P. A. |last1=Tipler |first2=G. |last2=Mosca |publisher=Freeman |year=2008 |isbn=978-0-7167-8964-2}}</ref> In this case, [[spherical polar coordinates]] are the most convenient. Thus,
<math display="block"> \psi(r,\theta,\varphi) = R(r)Y_\ell^m(\theta, \varphi) = R(r)\Theta(\theta)\Phi(\varphi),</math>
where {{math|''R''}} are radial functions and <math> Y^m_l (\theta, \varphi) </math> are [[spherical harmonic]]s of degree <math> \ell </math> and order <math> m </math>. This is the only atom for which the Schrödinger equation has been solved for exactly. Multi-electron atoms require approximate methods. The family of solutions are:<ref>{{cite book|first=David J. |last=Griffiths |author-link=David J. Griffiths |title=Introduction to Elementary Particles|url=https://books.google.com/books?id=w9Dz56myXm8C&pg=PA162 | access-date=27 June 2011|year=2008|publisher=Wiley-VCH|isbn=978-3-527-40601-2|pages=162–}}</ref>
<math display="block"> \psi_{n\ell m}(r,\theta,\varphi) = \sqrt {\left (  \frac{2}{n a_0} \right )^3\frac{(n-\ell-1)!}{2n[(n+\ell)!]} } e^{- r/na_0} \left(\frac{2r}{na_0}\right)^\ell L_{n-\ell-1}^{2\ell+1}\left(\frac{2r}{na_0}\right) \cdot Y_{\ell}^m(\theta, \varphi ) </math>
where
* <math> a_0 = \frac{4 \pi \varepsilon_0 \hbar^2}{m_q q^2} </math> is the [[Bohr radius]],
* <math> L_{n-\ell-1}^{2\ell+1}(\cdots) </math> are the [[Laguerre polynomial#Generalized Laguerre polynomials|generalized Laguerre polynomials]] of degree <math> n - \ell - 1 </math>,
* <math> n, \ell, m </math> are the [[principal quantum number|principal]], [[azimuthal quantum number|azimuthal]], and [[magnetic quantum number|magnetic]] [[quantum numbers]] respectively, which take the values <math>n = 1, 2, 3, \dots,</math> <math>\ell = 0, 1, 2, \dots, n - 1,</math> <math>m = -\ell, \dots, \ell.</math>

=== Approximate solutions ===
It is typically not possible to solve the Schrödinger equation exactly for situations of physical interest. Accordingly, approximate solutions are obtained using techniques like [[Variational method (quantum mechanics)|variational methods]] and [[WKB approximation]]. It is also common to treat a problem of interest as a small modification to a problem that can be solved exactly, a method known as [[perturbation theory (quantum mechanics)|perturbation theory]].