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The potential is always zero except inside the square well [-a, a] where it assumes the constant negative value $-U_0$. The particle has negative energy so it is confined inside the well. With negative energy values it is not possible to analytically solve the Schrödinger equation, or rather, it is possible to find the analytical solution but this depends on the energy parameter which can only be found numerically (after setting the value of all constants). 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.
	${\phi }_0\left(x\right)$	The wave function (solution of the Schrödinger equation) corresponding to energy $E_0$
	${\phi }_1\left(x\right)$	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 $E_2$
	$|{\phi }_0\left(x\right){|}^2$	The square modulus of the wave function gives the probability density of finding the particle at position x
	$|{\phi }_1\left(x\right){|}^2$	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
	$|c_0\left(p\right){|}^2$	Probability density for momentum p for the particle in the state with energy $E_0$. $c_0\left(p\right)$ is defined by the following formula: $c_0\left(p\right)={\int }_{-\infty }^{+\infty }{e}^{-\frac{i}{\hslash }px}{\phi }_0\left(x\right)dx$
	$|c_1\left(p\right){|}^2$	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: $c_2\left(p\right)={\int }_{-\infty }^{+\infty }{e}^{-\frac{i}{\hslash }px}{\phi }_2\left(x\right)dx$
	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 $E_{\mathrm{n}}$