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Normalization condition in one dimension | $P(x=\text{\u2212}\infty ,\text{+}\infty )=\underset{\text{\u2212}\infty}{\overset{\infty}{{\displaystyle \int}}}|\text{\Psi}(x,t){|}^{2}dx=1$ |
Probability of finding a particle in a narrow interval of position in one dimension $\left(x,x+dx\right)$ | $P(x,x+dx)={\text{\Psi}}^{*}\left(x,t\right)\text{\Psi}\left(x,t\right)dx$ |
Expectation value of position in one dimension | $\u27e8x\u27e9=\underset{\text{\u2212}\infty}{\overset{\infty}{{\displaystyle \int}}}{\text{\Psi}}^{*}\left(x,t\right)x\text{\Psi}\left(x,t\right)dx$ |
Heisenberg’s position-momentum uncertainty principle | $\text{\Delta}x\text{\Delta}p\ge \frac{\hslash}{2}$ |
Heisenberg’s energy-time uncertainty principle | $\text{\Delta}E\text{\Delta}t\ge \frac{\hslash}{2}$ |
Schrӧdinger’s time-dependent equation | $-\frac{{\hslash}^{2}}{2m}\phantom{\rule{0.2em}{0ex}}\frac{{\partial}^{2}\text{\Psi}\left(x,t\right)}{\partial {x}^{2}}+U\left(x,t\right)\text{\Psi}\left(x,t\right)=i\hslash \frac{{\partial}^{2}\text{\Psi}\left(x,t\right)}{\partial t}$ |
General form of the wave function for a time-independent potential in one dimension | $\text{\Psi}\left(x,t\right)=\psi \left(x\right){e}^{\text{\u2212}i\omega t}$ |
Schrӧdinger’s time-independent equation | $-\frac{{\hslash}^{2}}{2m}\phantom{\rule{0.2em}{0ex}}\frac{{d}^{2}\psi \left(x\right)}{d{x}^{2}}+U\left(x\right)\psi \left(x\right)=E\psi \left(x\right)$ |
Schrӧdinger’s equation (free particle) | $-\frac{{\hslash}^{2}}{2m}\phantom{\rule{0.2em}{0ex}}\frac{{\partial}^{2}\psi (x)}{\partial {x}^{2}}=E\psi (x)$ |
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 ) | ${\psi}_{n}(x)=\sqrt{\frac{2}{L}}\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\frac{n\pi x}{L},n=1,2,3,...$ |
Potential-energy function of a harmonic oscillator | $U(x)=\frac{1}{2}m{\omega}^{2}{x}^{2}$ |
Stationary Schrӧdinger equation | $-\frac{\hslash}{2m}\phantom{\rule{0.2em}{0ex}}\frac{{d}^{2}\psi (x)}{d{x}^{2}}+\frac{1}{2}m{\omega}^{2}{x}^{2}\psi (x)=E\psi (x)$ |
The energy spectrum | ${E}_{n}=\left(n+\frac{1}{2}\right)\hslash \omega ,n=0,1,2,3,...$ |
The energy wave functions | ${\psi}_{n}(x)={N}_{n}{e}^{\text{\u2212}{\beta}^{2}{x}^{2}\text{/}2}{H}_{n}(\beta x),n=0,1,2,3,...$ |
Potential barrier | $U(x)=\{\begin{array}{cc}\hfill 0,& \text{when}\phantom{\rule{0.2em}{0ex}}x<0\hfill \\ \hfill {U}_{0},& \text{when}\phantom{\rule{0.2em}{0ex}}0\le x\le L\hfill \\ \hfill 0,& \text{when}\phantom{\rule{0.2em}{0ex}}x>L\hfill \end{array}$ |
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 | $T(L,E)=16\frac{E}{{U}_{0}}\left(1-\frac{E}{{U}_{0}}\right){e}^{\text{\u2212}2\beta \phantom{\rule{0.2em}{0ex}}L}$ |
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