Editing Particle in a box (section)

=== Position and momentum probability distributions ===
{{More citations needed section|date=November 2024|talk=Talk:Particle_in_a_box#Essential_Self_Adjointness_and_Boundary_Conditions}}
In classic physics, the particle can be detected anywhere in the box with equal probability.  In quantum mechanics, however, the probability density for finding a particle at a given position is derived from the wave function as <math>P(x) = |\psi(x)|^2.</math>  For the particle in a box, the probability density for finding the particle at a given position depends upon its state, and is given by
<math display="block">P_n(x,t) = \begin{cases}
\frac{2}{L} \sin^2\left(k_n \left(x-x_c+\tfrac{L}{2}\right)\right), & x_c-\frac{L}{2} < x < x_c+\frac{L}{2},\\
0, & \text{otherwise,}
\end{cases}</math>

Thus, for any value of ''n'' greater than one, there are regions within the box for which <math>P(x)=0</math>, indicating that ''spatial nodes'' exist at which the particle cannot be found. If [[relativistic wave equations]] are considered, however, the probability density not go to zero at the nodes (apart from the trivial case <math>n=0</math>).<ref>{{cite journal|doi=10.1088/0143-0807/17/1/004 |title=Relativistic particle in a box |year=1996 |last1=Alberto |first1=P | last2=Fiolhais |first2=C |last3=Gil |first3=V M S |url=https://estudogeral.sib.uc.pt/bitstream/10316/12349/1/Relativistic%20particle%20in%20a%20box.pdf |journal=European Journal of Physics |volume=17 |issue=1 |pages=19–24 | bibcode=1996EJPh...17...19A | hdl=10316/12349 |s2cid=250895519 | hdl-access=free }}</ref>

In quantum mechanics, the average, or [[expectation value]] of the position of a particle is given by
<math display="block">\langle x \rangle = \int_{-\infty}^{\infty} x P_n(x)\,\mathrm{d}x.</math>

For the steady state particle in a box, it can be shown that the average position is always <math>\langle x \rangle =x_c</math>, regardless of the state of the particle.  For a superposition of states, the expectation value of the position will change based on the cross term, which is proportional to <math>\cos(\omega t)</math>.

The variance in the position is a measure of the uncertainty in position of the particle:
<math display="block">\mathrm{Var}(x) = \int_{-\infty}^\infty (x-\langle x\rangle)^2 P_n(x)\,dx = \frac{L^2}{12}\left(1-\frac{6}{n^2\pi^2}\right)</math>

The probability density for finding a particle with a given momentum is derived from the wave function as <math>P(x) = |\phi(x)|^2</math>. As with position, the probability density for finding the particle at a given momentum depends upon its state, and is given by
<math display="block">P_n(p)=\frac{L}{\pi \hbar} \left(\frac{n\pi}{n\pi+k L}\right)^2\,\textrm{sinc}^2\left(\tfrac{1}{2}(n\pi-k L)\right)</math>
where, again, <math>k = p / \hbar</math>. The expectation value for the momentum is then calculated to be zero, and the variance in the momentum is calculated to be:
<math display="block">\mathrm{Var}(p)=\left(\frac{\hbar n\pi}{L}\right)^2</math>

The uncertainties in position and momentum (<math>\Delta x</math> and <math>\Delta p</math>) are defined as being equal to the square root of their respective variances, so that:
<math display="block">\Delta x \Delta p = \frac{\hbar}{2} \sqrt{\frac{n^2\pi^2}{3}-2}</math>

This product increases with increasing ''n'', having a minimum for ''n'' = 1. The value of this product for ''n'' = 1 is about equal to 0.568 <math>\hbar</math>, which obeys the [[Heisenberg uncertainty principle]], which states that the product will be greater than or equal to <math>\hbar/2</math>.

Another measure of uncertainty in position is the [[Entropy (information)|information entropy]] of the probability distribution ''H''<sub>x</sub>:<ref name="Majernik1997">{{cite journal |last1=Majernik |first1=Vladimir |last2=Richterek |first2=Lukas | date=1997-12-01 |title=Entropic uncertainty relations for the infinite well |url=https://www.researchgate.net/publication/231121036 |journal=J. Phys. A |volume=30 |issue=4 |pages= L49|doi=10.1088/0305-4470/30/4/002 |access-date=11 February 2016| bibcode = 1997JPhA...30L..49M }}</ref>
<math display="block">H_x=\int_{-\infty}^\infty P_n(x) \log(P_n(x) x_0)\,dx =\log\left(\frac{2 L}{e \,x_0}\right)</math>
where ''x''<sub>0</sub> is an arbitrary reference length.

Another measure of uncertainty in momentum is the [[Entropy (information)|information entropy]] of the probability distribution ''H<sub>p</sub>'':
<math display="block">H_p(n)=\int_{-\infty}^\infty P_n(p) \log(P_n(p) p_0)\,dp</math>
<math display="block">\lim_{n\to\infty} H_p(n) = \log\left(\frac{4 \pi \hbar\, e^{2(1-\gamma)}}{ L\, p_0}\right)</math>
where ''γ'' is [[Euler-Mascheroni constant|Euler's constant]]. The quantum mechanical [[Uncertainty principle#Quantum entropic uncertainty principle|entropic uncertainty principle]] states that for <math>x_0\,p_0 = \hbar</math>
<math display="block">H_x+H_p(n) \ge \log(e\,\pi) \approx 2.14473...</math> ([[nat (unit)|nats]])

For <math>x_0\,p_0=\hbar</math>, the sum of the position and momentum entropies yields:
<math display="block">H_x+H_p(\infty) = \log\left(8\pi\, e^{1-2\gamma}\right) \approx 3.06974...</math>
where the unit is [[nat (unit)|nat]], and which satisfies the quantum entropic uncertainty principle.