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Particle confined to an interval (one-dimensional box) of the X-axis, in the range [0,L]. The Schrödinger equation can be solved analytically without taking approximations. To draw the graphs, we set L=1. For more details open the notebook below.
	Open the notebook in Wolfram Cloud	The Wolfram Mathematica notebook contains the solutions of the Schrödinger equation and allows you to produce all the graphs shown on this page.
	Motion of a particle in a box HD video Snapshots	Motion of a particle (wave packet) confined inside a one-dimensional box. Notice that the width of the particle increases over time (spreading of wave packets).
	${\phi }_1\left(x\right)$	The wave function (solution of the Schrödinger equation) corresponding to energy $E_1$
	${\phi }_2\left(x\right)$	The wave function (solution of the Schrödinger equation) corresponding to energy $E_2$
	${\phi }_3\left(x\right)$	The wave function (solution of the Schrödinger equation) corresponding to energy $E_3$
	$|{\phi }_1\left(x\right){|}^2$	The square modulus of the wave function gives the probability density of finding the particle at position x
	$|{\phi }_2\left(x\right){|}^2$	The square modulus of the wave function gives the probability density of finding the particle at position x
	$|{\phi }_3\left(x\right){|}^2$	The square modulus of the wave function gives the probability density of finding the particle at position x
	$|c_1\left(p\right){|}^2$	Probability density for momentum p for the particle in the state with energy $E_1$. $c_1\left(p\right)$ is defined by the following formula: $c_1\left(p\right)={\int }_{-\infty }^{+\infty }{e}^{-\frac{i}{\hslash }px}{\phi }_1\left(x\right)dx$
	$|c_2\left(p\right){|}^2$	Probability density for momentum p for the particle in the state with energy $E_2$. $c_2\left(p\right)$ is defined by the following formula: $c_2\left(p\right)={\int }_{-\infty }^{+\infty }{e}^{-\frac{i}{\hslash }px}{\phi }_2\left(x\right)dx$
	$|c_3\left(p\right){|}^2$	Probability density for momentum p for the particle in the state with energy $E_3$. $c_3\left(p\right)$ is defined by the following formula: $c_3\left(p\right)={\int }_{-\infty }^{+\infty }{e}^{-\frac{i}{\hslash }px}{\phi }_3\left(x\right)dx$
	All ${\phi }_{\mathrm{n}}\left(x\right)$	The three solutions of the Schrödinger equation (wave functions) in a single graph, highlighting the corresponding energy value $E_{\mathrm{n}}$
	All $|{\phi }_{\mathrm{n}}\left(x\right){|}^2$	The three probability density for position x in a single graph, highlighting the corresponding energy value $E_{\mathrm{n}}$
	All $|c_{\mathrm{n}}\left(p\right){|}^2$	The three probability density for momentum p in a single graph, highlighting the corresponding energy value $E_{\mathrm{n}}$