Normalization condition in one dimension

P (x = − \infty, + \infty) = \int_{− \infty}^{\infty} | Ψ (x, t) |^{2} d x = 1

Probability of finding a particle in a narrow interval of position in one dimension

(x, x + d x)

P (x, x + d x) = Ψ^{*} (x, t) Ψ (x, t) d x

Expectation value of position in one dimension

〈 x 〉 = \int_{− \infty}^{\infty} Ψ^{*} (x, t) x Ψ (x, t) d x

Heisenberg’s position-momentum uncertainty principle

Δ x Δ p \geq \frac{ℏ}{2}

Heisenberg’s energy-time uncertainty principle

Δ E Δ t \geq \frac{ℏ}{2}

Schrӧdinger’s time-dependent equation

- \frac{ℏ^{2}}{2 m} \frac{\partial^{2} Ψ (x, t)}{\partial x^{2}} + U (x, t) Ψ (x, t) = i ℏ \frac{\partial Ψ (x, t)}{\partial t}

General form of the wave function for a time-independent potential in one dimension

Ψ (x, t) = ψ (x) e^{− i ω t}

Schrӧdinger’s time-independent equation

- \frac{ℏ^{2}}{2 m} \frac{d^{2} ψ (x)}{d x^{2}} + U (x) ψ (x) = E ψ (x)

Schrӧdinger’s equation (free particle)

- \frac{ℏ^{2}}{2 m} \frac{\partial^{2} ψ (x)}{\partial x^{2}} = E ψ (x)

Allowed energies (particle in box of length L)

E_{n} = n^{2} \frac{π^{2} ℏ^{2}}{2 m L^{2}}, n = 1, 2, 3, . . .

Stationary states (particle in a box of length L)

ψ_{n} (x) = \sqrt{\frac{2}{L}} sin \frac{n π x}{L}, n = 1, 2, 3, . . .

Potential-energy function of a harmonic oscillator

U (x) = \frac{1}{2} m ω^{2} x^{2}

Schrӧdinger equation (harmonic oscillator)

- \frac{ℏ^{2}}{2 m} \frac{d^{2} ψ (x)}{d x^{2}} + \frac{1}{2} m ω^{2} x^{2} ψ (x) = E ψ (x)

The energy spectrum

E_{n} = (n + \frac{1}{2}) ℏ ω, n = 0, 1, 2, 3, . . .

The energy wave functions

ψ_{n} (x) = N_{n} e^{− β^{2} x^{2} / 2} H_{n} (β x), n = 0, 1, 2, 3, . . .

Potential barrier

U (x) = {\begin{matrix} 0, & when x < 0 \\ U_{0}, & when 0 \leq x \leq L \\ 0, & when x > L \end{matrix}

Definition of the transmission coefficient

T (L, E) = \frac{| ψ_{tra} (x) |^{2}}{| ψ_{in} (x) |^{2}}

A parameter in the transmission coefficient

β^{2} = \frac{2 m}{ℏ^{2}} (U_{0} - E)

Transmission coefficient, exact

T (L, E) = \frac{1}{{cosh}^{2} β L + {(γ / 2)}^{2} {sinh}^{2} β L}

Transmission coefficient, approximate

T (L, E) = 16 \frac{E}{U_{0}} (1 - \frac{E}{U_{0}}) e^{− 2 β L}