Editing Quantum mechanics (section)

=== 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 everywhere <em>inside</em> a certain region, and therefore infinite potential energy everywhere <em>outside</em> that region.<ref name="Cohen-Tannoudji" />{{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 microscopy]].