The time-dependent Schrödinger equation for the wavefunction $\mathrm{\Ψ}\left(\overrightarrow{r}\,t\right)$ is:

$i \mathrm{\ℏ}\mathit{ }\frac{\partial \mathrm{\Ψ}}{\partial t}\=\widehat{H }\mathrm{\Ψ}$,

where $\widehat{H }$ is the Hamiltonian operator, and $\hslash$ is the reduced Planck's constant. For the case of a single non-relativistic particle, the Hamiltonian takes the form $-\frac{{\mathrm{\ℏ}}^2}{2 m}{\nabla }^2\+V\left(\overrightarrow{r}\,t\right)$:

$i\mathit{ }\mathrm{\ℏ}\mathit{ }\frac{\partial \mathrm{\Ψ}}{\partial t}\=\left(-\frac{{\hslash }^2}{2 m}{\nabla }^2\+V\left(\overrightarrow{r}\,t\right)\right) \mathrm{\Ψ}$,

where $m$ is the mass of the particle, $V$  is its potential energy, and ${\nabla }^2$ is the Laplacian. Simplifying to one dimension, and assuming the potential is constant in time, you get:

$i\mathit{ }\mathrm{\ℏ}\mathit{ }\frac{\partial \mathrm{\Ψ}}{\partial t}\=\left(-\frac{{\hslash }^2}{2 m}{}^{}\frac{{\partial }^2}{\partial x^2}\+V\left(x\right)\right) \mathrm{\Ψ}$.

Recalling that the modulus squared of the wavefunction, ${\left|\mathrm{\Ψ}\left(x\, t\right)\right|}^2$, is the probability density of the particle, a stationary state is a state where this probability density does not change in time, that is, the particle stays in the same state in every observable way. As such, you observe that $\mathrm{\Ψ}\left(x\, t\right)$ must take the form:

$\mathrm{\Psi }\left(x\, t\right) \= \mathrm{\ψ}\left(x\right) {\ⅇ}^{i \mathrm{\θ}\left(x\,t\right)}$,

for some real valued function $\mathrm{\θ}\left(x\,t\right)$. Including further restriction that the potential barrier $V\left(x\right)$ is a well-behaved function with no vertical asymptotes, it can be shown that $\mathrm{\θ}\left(x\,t\right)$ = $-\frac{E}{\mathrm{\ℏ}}\mathit{ }t$, where $E$ is the energy of the particle. Plugging this solution into the time-dependant Schrödinger equation produces the time-independent Schrödinger equation:

$\left(-\frac{{\mathrm{\ℏ}}^2}{2 m} \frac{{\partial }^2}{\partial x^2}\+ V\left(x\right) \right) \mathrm{\ψ}\left(x\right)\=E \mathrm{\ψ}\left(x\right)$.

or simply $\widehat{H }\mathrm{\ψ}\=E \mathrm{\ψ}$.

Consider the one-dimensional time-independent Schrödinger equation for a piecewise constant potential $V\left(x\right)$.

Assume that Region 1 of space  on the interval $\left(-\infty \, 0\right)$ has potential energy $V_1$. Region 2 has potential energy $V_2$ and is on the interval $\left(0\, a\right]\,$ and lastly, Region 3 has a potential energy $V_3$ and is on the interval $\left(a\, \infty \right)$. The total potential can be mathematically written as:

 $V\left(x\right) \&equals; \&lcub;\begin{array}{cc}{V}_1\&comma;& x\le 0\&comma;\\ V_2\&comma;& 0<x \le a\&comma;\\ V_3\&comma;& a<x\&comma;\end{array}$   

This is a very general derivation because you can vary the height of any potential well along with the central width $a$. The time-independent Schrödinger equation must be solved in the three regions and the solutions connected by junction conditions, that is, the requirement that the wavefunction and its derivative be continuous on the boundaries. If you the call the three solutions ${\mathrm{\&psi;}}_1\mathit{\&comma;}{\mathrm{\&psi;}}_2\mathit{\&comma;}{\mathrm{\&psi;}}_3$ respectively, then the junction conditions are:

 $$\begin{gathered} {\psi }_1\left(0\right)\&equals; {\psi }_2\left(0\right)\&comma; \\ {\psi }_2\left(a\right) \&equals; {\psi }_3\left(a\right)\&comma; \\ \mathit{ }\mathrm{\&psi;}{\mathit{\&apos;}}_1\left(0\right)\&equals; \mathrm{\&psi;}{\mathit{\&apos;}}_2\left(0\right)\&comma; \\ \mathit{ }\mathrm{\&psi;}{\mathit{\&apos;}}_2\left(a\right) \&equals; \mathrm{\&psi;}{\mathit{\&apos;}}_3\left(a\right)\&comma; \end{gathered}$$ 

where the primes denotes differentiation with respect to x. The solutions to the Schrödinger equation for $E\&gt;V_1$ in these three regions are:

$$\begin{gathered} {\psi }_1\left(x\right) \&equals; A\cdot e^{i\cdot k_1\cdot x}\&plus;B\cdot e^{-i\cdot k_1\cdot x}\&comma; \\ {\mathrm{\&psi;}}_2 \left(x\right) \&equals; C\cdot e^{i\cdot k_2\cdot x}\&plus;D\cdot e^{-i\cdot k_2\cdot x}\&comma; \\ {\psi }_3\left(x\right) \&equals; F\cdot e^{i\cdot k_3\cdot x}\&plus;G\cdot e^{-i\cdot k_3\cdot x}\&comma; \end{gathered}$$

with

$k_1\&equals; \sqrt{\frac{2\cdot m\cdot \left(E-V_1\right)}{{\mathrm{\&hslash;}}^2}}\&comma;$$k_2\&equals; \sqrt{\frac{2\cdot m\cdot \left(E-V_2\right)}{{\mathrm{\&hslash;}}^2}}\&comma;k_3\&equals; \sqrt{\frac{2\cdot m\cdot \left(E-{\mathrm{V}}_3\right)}{{\mathrm{\&hslash;}}^2}}\&period;$ 

Notice that if $E\&gt;V$, the wavefunction is a complex plane wave with the form $e^{i k x}$ with real-valued $k$, while if $E<V$, the form of the solution is $e^{k x}$ with a real-valued $k$. The case of $E\&equals;V$  has a different solution and is not considered here.

To extract more physically realizable quantities, it is necessary to assume that the particle or wavefunction 'originates' from only one side of the potential barrier. Physically, this is because of the assumption that there is no source of particles on the right hand side which travel in the $-x$ direction. This makes the constant $G\&equals;0$, since the function $G\cdot e^{-i\cdot k_3\cdot x}$is a plane wave traveling in the $-x$ direction.

Now by applying the junction conditions above, you can derive the constants B, C, D, F, in terms of A, the amplitude of the wave. This can be very messy, especially in our highly general case of arbitrary $V_1\&comma; V_2\&comma; V_3\&period;$ To extract physically meaningful quantities from these abstract functions, the reflection and transmission coefficients are defined as the ratio:

$$\begin{gathered} R \&equals; \frac{{\left|B\right|}^2}{{\&verbar;A\&verbar;}^2}\&comma; \\ T \&equals; \frac{{\left|F\right|}^2}{{\left|A\right|}^2}\&comma; \end{gathered}$$

so that $R \&plus; T \&equals; 1.$ This gives the amount of the probability density that is "reflected" or "transmitted", similar to that used in optics. Remember that this is a solution for the time independent Schrödinger equation, so the particle is in a stationary state. It is not traveling through the potential barrier in time, rather it leaks through the barrier as a result of the plane wave solution to the Schrödinger equation.