Key Equations

Key Equations

Normalization condition in one dimension	$P(x=\text{−}\infty ,\text{+}\infty )=\underset{\text{−}\infty }{\overset{\infty }{{\displaystyle \int }}}|\text{Ψ}(x,t){|}^{2}dx=1$
Probability of finding a particle in a narrow interval of position in one dimension $\left(x,x+dx\right)$	$P(x,x+dx)={\text{Ψ}}^{*}\left(x,t\right)\text{Ψ}\left(x,t\right)dx$
Expectation value of position in one dimension	$\langle x\rangle =\underset{\text{−}\infty }{\overset{\infty }{{\displaystyle \int }}}{\text{Ψ}}^{*}\left(x,t\right)x\text{Ψ}\left(x,t\right)dx$
Heisenberg’s position-momentum uncertainty principle	$\text{Δ}x\text{Δ}p\ge \frac{\hslash }{2}$
Heisenberg’s energy-time uncertainty principle	$\text{Δ}E\text{Δ}t\ge \frac{\hslash }{2}$
Schrӧdinger’s time-dependent equation	$-\frac{{\hslash }^2}{2m}\frac{{\partial }^2\text{Ψ}\left(x,t\right)}{\partial x^2}+U\left(x,t\right)\text{Ψ}\left(x,t\right)=i\hslash \frac{\partial \text{Ψ}\left(x,t\right)}{\partial t}$
General form of the wave function for a time-independent potential in one dimension	$\text{Ψ}\left(x,t\right)=\psi \left(x\right)e^{\text{−}i\omega t}$
Schrӧdinger’s time-independent equation	$-\frac{{\hslash }^2}{2m}\frac{d^2\psi \left(x\right)}{d{x}^2}+U\left(x\right)\psi \left(x\right)=E\psi \left(x\right)$
Schrӧdinger’s equation (free particle)	$-\frac{{\hslash }^2}{2m}\frac{{\partial }^2\psi (x)}{\partial x^2}=E\psi (x)$
Allowed energies (particle in box of length L)	$E_n=n^2\frac{{\pi }^2{\hslash }^2}{2m{L}^2},n=1,2,3,...$
Stationary states (particle in a box of length L)	${\psi }_n(x)=\sqrt{\frac{2}{L}}\text{sin}\frac{n\pi x}{L},n=1,2,3,...$
Potential-energy function of a harmonic oscillator	$U(x)=\frac{1}{2}m{\omega }^2{x}^2$
Schrӧdinger equation (harmonic oscillator)	$-\frac{{\hslash }^2}{2m}\frac{d^2\psi (x)}{d{x}^2}+\frac{1}{2}m{\omega }^2{x}^2\psi (x)=E\psi (x)$
The energy spectrum	$E_n=\left(n+\frac{1}{2}\right)\hslash \omega ,n=0,1,2,3,...$
The energy wave functions	${\psi }_n(x)=N_n{e}^{\text{−}{\beta }^2{x}^2\text{/}2}{H}_n(\beta x),n=0,1,2,3,...$
Potential barrier	$U(x)=\{\begin{array}{cc}\hfill 0,& \text{when}x<0\hfill \\ \hfill U_0,& \text{when}0\le x\le L\hfill \\ \hfill 0,& \text{when}x>L\hfill \end{array}$
Definition of the transmission coefficient	$T(L,E)=\frac{|{\psi }_{\text{tra}}(x){|}^2}{|{\psi }_{\text{in}}(x){|}^2}$
A parameter in the transmission coefficient	${\beta }^2=\frac{2m}{{\hslash }^2}(U_0-E)$
Transmission coefficient, exact	$T(L,E)=\frac{1}{{\text{cosh}}^2\beta L+{(\gamma \text{/}2)}^2{\text{sinh}}^2\beta L}$
Transmission coefficient, approximate	$T(L,E)=16\frac{E}{U_0}\left(1-\frac{E}{U_0}\right)e^{\text{−}2\beta L}$