Infinite Square Well

$$U(x)=\{\begin{array}{ll}0,& 0\le x\le L\\ \infty ,& x<0\text{ or }x>L\end{array}$$

• $L$ is the length of the square potential

${\psi }_n(x)=\sqrt{\frac{2}{L}}\sin \left(\frac{n\pi x}{L}\right),n=1,2,3,\dots$

• $n$ is the energy level. It can’t equal $0$ because otherwise, normalization would not be satisfied, demonstrating Zero-Point Energy
• Desmos Demo
• As $n\to \infty$, the PDF approaches the classical case of a continuous uniform distribution
• Solutions for different $n$ are orthogonal with respect to the Square Integrable Vector Space

$E_n=\frac{n^2{\pi }^2{\hslash }^2}{2m{L}^2}$

• $E_n$ is the energy of the $n$th state of the function

Parameter Values

$⟨x⟩=\frac{a}{2}$

$⟨x^2⟩=a^2\left(\frac{1}{3}-\frac{1}{2(n\pi {)}^2}\right)$

$⟨p⟩=0$

$⟨p^2⟩={\left(\frac{n\pi \hslash }{a}\right)}^2$

${\sigma }_x^2=\frac{a}{2}\sqrt{\frac{1}{3}-\frac{2}{(n\pi {)}^2}}$

${\sigma }_p=\frac{n\pi \hslash }{a}$