Editing Quantum mechanics (section)

== Examples ==

=== Free particle ===
{{Main|Free particle}}
[[File:Guassian Dispersion.gif|360 px|thumb|right|Position space probability density of a Gaussian [[wave packet]] moving in one dimension in free space]]
The simplest example of a quantum system with a position degree of freedom is a free particle in a single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy:
<math display=block>H = \frac{1}{2m}P^2 = - \frac {\hbar ^2}{2m} \frac {d ^2}{dx^2}. </math>
The general solution of the Schrödinger equation is given by
<math display=block>\psi (x,t)=\frac {1}{\sqrt {2\pi }}\int _{-\infty}^\infty{\hat {\psi }}(k,0)e^{i(kx -\frac{\hbar k^2}{2m} t)}\mathrm{d}k,</math>
which is a superposition of all possible [[plane wave]]s <math>e^{i(kx -\frac{\hbar k^2}{2m} t)}</math>, which are eigenstates of the momentum operator with momentum <math>p = \hbar k </math>. The coefficients of the superposition are <math> \hat {\psi }(k,0) </math>, which is the Fourier transform of the initial quantum state <math>\psi(x,0)</math>.

It is not possible for the solution to be a single momentum eigenstate, or a single position eigenstate, as these are not normalizable quantum states.{{refn|group=note|A momentum eigenstate would be a perfectly monochromatic wave of infinite extent, which is not square-integrable. Likewise, a position eigenstate would be a [[Dirac delta distribution]], not square-integrable and technically not a function at all. Consequently, neither can belong to the particle's Hilbert space. Physicists sometimes introduce fictitious "bases" for a Hilbert space comprising elements outside that space. These are invented for calculational convenience and do not represent physical states.<ref name="Cohen-Tannoudji" />{{rp|100–105}}}} Instead, we can consider a Gaussian [[wave packet]]:
<math display=block>\psi(x,0) = \frac{1}{\sqrt[4]{\pi a}}e^{-\frac{x^2}{2a}} </math>
which has Fourier transform, and therefore momentum distribution
<math display=block>\hat \psi(k,0) = \sqrt[4]{\frac{a}{\pi}}e^{-\frac{a k^2}{2}}. </math>
We see that as we make <math>a</math> smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making <math>a</math> larger we make the spread in momentum smaller, but the spread in position gets larger. This illustrates the uncertainty principle.

As we let the Gaussian wave packet evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that the position becomes more and more uncertain. The uncertainty in momentum, however, stays constant.<ref>{{cite book |title=A Textbook of Quantum Mechanics |first1=Piravonu Mathews |last1=Mathews |first2=K. |last2=Venkatesan |publisher=Tata McGraw-Hill |year=1976 |isbn=978-0-07-096510-2 |page=[https://books.google.com/books?id=_qzs1DD3TcsC&pg=PA36 36] |chapter=The Schrödinger Equation and Stationary States |chapter-url=https://books.google.com/books?id=_qzs1DD3TcsC&pg=PA36}}</ref>

=== 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]].

=== Harmonic oscillator ===
{{Main|Quantum harmonic oscillator}}

[[File:QuantumHarmonicOscillatorAnimation.gif|thumb|upright=1.35|right|Some trajectories of a [[harmonic oscillator]] (i.e. a ball attached to a [[Hooke's law|spring]]) in [[classical mechanics]] (A-B) and quantum mechanics (C-H). In quantum mechanics, the position of the ball is represented by a [[wave]] (called the wave function), with the [[real part]] shown in blue and the [[imaginary part]] shown in red. Some of the trajectories (such as C, D, E, and F) are [[standing wave]]s (or "[[stationary state]]s"). Each standing-wave frequency is proportional to a possible [[energy level]] of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have ''any'' energy.]]

As in the classical case, the potential for the quantum harmonic oscillator is given by<ref name="Zwiebach2022" />{{rp|234}}
<math display=block>V(x)=\frac{1}{2}m\omega^2x^2.</math>

This problem can either be treated by directly solving the Schrödinger equation, which is not trivial, or by using the more elegant "ladder method" first proposed by Paul Dirac. The [[eigenstate]]s are given by
<math display=block> \psi_n(x) = \sqrt{\frac{1}{2^n\, n!}} \cdot \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \cdot e^{
- \frac{m\omega x^2}{2 \hbar}} \cdot H_n\left(\sqrt{\frac{m\omega}{\hbar}} x \right), \qquad </math>
<math display=block>n = 0,1,2,\ldots. </math>
where ''H<sub>n</sub>'' are the [[Hermite polynomials]]
<math display=block>H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}\left(e^{-x^2}\right),</math>
and the corresponding energy levels are
<math display=block> E_n = \hbar \omega \left(n + {1\over 2}\right).</math>

This is another example illustrating the discretization of energy for [[bound state]]s.

=== Mach–Zehnder interferometer ===
[[File:Mach-Zehnder interferometer.svg|upright=1.3|thumb|right|Schematic of a Mach–Zehnder interferometer]]

The [[Mach–Zehnder interferometer]] (MZI) illustrates the concepts of superposition and interference with linear algebra in dimension 2, rather than differential equations. It can be seen as a simplified version of the double-slit experiment, but it is of interest in its own right, for example in the [[delayed choice quantum eraser]], the [[Elitzur–Vaidman bomb tester]], and in studies of quantum entanglement.<ref name=Paris1999>{{cite journal |last=Paris |first=M. G. A. |title=Entanglement and visibility at the output of a Mach–Zehnder interferometer |journal=[[Physical Review A]] |date=1999 |volume=59 |issue=2 |pages=1615–1621 |arxiv=quant-ph/9811078 |bibcode=1999PhRvA..59.1615P |doi=10.1103/PhysRevA.59.1615 |s2cid=13963928}}</ref><ref name=Haack2010>{{Cite journal |last1=Haack |first1=G. R. |last2=Förster |first2=H. |last3=Büttiker |first3=M. |title=Parity detection and entanglement with a Mach-Zehnder interferometer |doi=10.1103/PhysRevB.82.155303 |journal=[[Physical Review B]] |volume=82 |issue=15 |pages=155303 |year=2010 |arxiv=1005.3976 |bibcode=2010PhRvB..82o5303H |s2cid=119261326}}</ref>

We can model a photon going through the interferometer by considering that at each point it can be in a superposition of only two paths: the "lower" path which starts from the left, goes straight through both beam splitters, and ends at the top, and the "upper" path which starts from the bottom, goes straight through both beam splitters, and ends at the right. The quantum state of the photon is therefore a vector <math>\psi \in \mathbb{C}^2</math> that is a superposition of the "lower" path <math>\psi_l = \begin{pmatrix} 1 \\ 0 \end{pmatrix}</math> and the "upper" path <math>\psi_u = \begin{pmatrix} 0 \\ 1 \end{pmatrix}</math>, that is, <math>\psi = \alpha \psi_l + \beta \psi_u</math> for complex <math>\alpha,\beta</math>. In order to respect the postulate that <math>\langle \psi,\psi\rangle = 1</math> we require that <math>|\alpha|^2+|\beta|^2 = 1</math>.

Both [[beam splitter]]s are modelled as the unitary matrix <math>B = \frac1{\sqrt2}\begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix}</math>, which means that when a photon meets the beam splitter it will either stay on the same path with a probability amplitude of <math>1/\sqrt{2}</math>, or be reflected to the other path with a probability amplitude of <math>i/\sqrt{2}</math>. The phase shifter on the upper arm is modelled as the unitary matrix <math>P = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\Delta\Phi} \end{pmatrix}</math>, which means that if the photon is on the "upper" path it will gain a relative phase of <math>\Delta\Phi</math>, and it will stay unchanged if it is in the lower path.

A photon that enters the interferometer from the left will then be acted upon with a beam splitter <math>B</math>, a phase shifter <math>P</math>, and another beam splitter <math>B</math>, and so end up in the state
<math display=block>BPB\psi_l = ie^{i\Delta\Phi/2} \begin{pmatrix} -\sin(\Delta\Phi/2) \\ \cos(\Delta\Phi/2) \end{pmatrix},</math>
and the probabilities that it will be detected at the right or at the top are given respectively by
<math display=block> p(u) = |\langle \psi_u, BPB\psi_l \rangle|^2 = \cos^2 \frac{\Delta \Phi}{2},</math>
<math display=block> p(l) = |\langle \psi_l, BPB\psi_l \rangle|^2 = \sin^2 \frac{\Delta \Phi}{2}.</math>
One can therefore use the Mach–Zehnder interferometer to estimate the [[Phase (waves)|phase shift]] by estimating these probabilities.

It is interesting to consider what would happen if the photon were definitely in either the "lower" or "upper" paths between the beam splitters. This can be accomplished by blocking one of the paths, or equivalently by removing the first beam splitter (and feeding the photon from the left or the bottom, as desired). In both cases, there will be no interference between the paths anymore, and the probabilities are given by <math>p(u)=p(l) = 1/2</math>, independently of the phase <math>\Delta\Phi</math>. From this we can conclude that the photon does not take one path or another after the first beam splitter, but rather that it is in a genuine quantum superposition of the two paths.<ref name="vedral">{{cite book |first=Vlatko |last=Vedral |title=Introduction to Quantum Information Science |date=2006 |publisher=Oxford University Press |isbn=978-0-19-921570-6 |oclc=442351498 |author-link=Vlatko Vedral}}</ref>