Rectangular potential barrier

The Finite potential barrier is a textbook problem of quantum mechanics. The problem consists of solving the time-independent Schrödinger equation for a particle with a finite size barrier potential in one dimension.

Scattering at a finite potential barrier of height $V_{0}$. The amplitudes and direction of left and right moving waves are indicated. In red, those waves used for the derivation of the reflection and transmission amplitude. $E>V_{0}$ for this illustration.

Calculation

The time-independent Schrödinger equation for the wave function $\psi (x)$ reads

$H\psi (x)=[-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+V(x)]\psi (x)=E\psi (x),$

where $H$ is the Hamiltonian $\hbar$ is the (reduced) Planck constant, $m$ is the mass, $E$ the energy of the particle and

$V(x)=V_{0}[\Theta (x)-\Theta (x-a)]$

is the barrier potential with height $V_{0}>0$ and width $a$. $\Theta (x)=0,\;x<0;\;\Theta (x)=1,\;x>0$ is the Heaviside step function. The barrier is positioned between $x=0$ and $x=a$. Without changing the results, any other shifted position was possible. The first term in the Hamiltonian, $-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\psi$ is the kinetic energy.

The barrier divides the space in three parts ($x<0,0<x<a,x>0$). In any of these parts the potential is constant meaning the particle is quasi-free, and the solution of the Schrödinger equation can be written as a superposition of left and right moving waves (see free particle)

$\psi _{L}(x)=A_{r}e^{ik_{0}x}+A_{l}e^{-ik_{0}x}\quad x<0$,

$\psi _{C}(x)=B_{r}e^{ik_{1}x}+B_{l}e^{-ik_{1}x}\quad 0<x<a$, and

$\psi _{R}(x)=C_{r}e^{ik_{0}x}+C_{l}e^{-ik_{0}x}\quad x>a$

where the wave vectors are related to the energy via

$k_{0}={\sqrt {2mE}}/\hbar$, and

$k_{1}={\sqrt {2m(E-V_{0})}}/\hbar$.

The index r/l on the coefficients A and B denotes the direction of the velocity vector. Note that if the energy of the particle is below the barrier height, $k_{1}$ becomes imaginary and the wave function is exponentially decaying within the barrier. Nevertheless we keep the notation r/l even though the waves are not propagating anymore in this case.

The coefficients $A,B,C$ have to be found from the boundary conditions of the wave function at $x=0$ and $x=a$. The wave function and its derivative have to be continuous everywhere, so.

$\psi _{L}(0)=\psi _{C}(0)$,

${\frac {d}{dx}}\psi _{L}(0)={\frac {d}{dx}}\psi _{C}(0)$,

$\psi _{C}(a)=\psi _{R}(a)$,

${\frac {d}{dx}}\psi _{C}(a)={\frac {d}{dx}}\psi _{R}(a)$.

Inserting the wave functions, the boundary conditions give the following restrictions on the coefficients

$A_{r}+A_{l}=B_{r}+B_{l}$

$k_{0}(A_{r}-A_{l})=k_{1}(B_{r}-B_{l})$,

$B_{r}e^{iak_{1}}+B_{l}e^{-iak_{1}}=C_{r}e^{iak_{0}}+C_{l}e^{-iak_{0}}$,

$k_{1}(B_{r}e^{iak_{1}}-B_{l}e^{-iak_{1}})=k_{0}(C_{r}e^{iak_{0}}-C_{l}e^{-iak_{0}})$.

Transmission and reflection

At this point, it is instructive to compare the situation to the classical case. In both cases, the particle behaves as a free particle outside of the barrier region. A classical particle with energy $E$ larger than the barrier height $V_{0}$ does not feel the barrier at all, while a classical particle with $E<V_{0}$ incident on the barrier would always get reflected.

However, a classical particle having a finite energy cannot pass the infinitely high potential barrier and will be reflected from it. To study the quantum case, let us consider the following situation: a particle incident on the barrier from the left side ($A_{r}$). It may be reflected ($A_{l}$) or transmitted ($C_{r}$).

To find the amplitudes for reflection and transmission for incidence from the left, we put in the above equations $A_{r}=1$ (incoming particle), $A_{l}=r$ (reflection), $C_{l}$=0 (no incoming particle from the right) and $C_{r}=t$ (transmission). We then eliminate the coefficients $B_{l},B_{r}$ from the equation and solve for $r,t$.

The result is:

$t={\frac {4e^{-ia(k_{0}-k_{1})}}{e^{2iak_{1}}(k_{0}-k_{1})^{2}-(k_{0}+k_{1})^{2}}}$

$r={\frac {(k_{0}^{2}-k_{1}^{2})\sin(ak_{1})}{2ik_{0}k_{1}\cos(ak_{1})+(k_{0}^{2}+k_{1}^{2})\sin(ak_{1})}}.$

Due to the mirror symmetry of the model, the amplitudes for incidence from the right are the same as those from the left.

Analysis of the obtained expressions.

$E<V_{0}$

The surprising result is that for energies less than the barrier height, $E<V_{0}$ there is a non-zero probability

$T=|t|^{2}={\frac {1}{1+{\frac {V_{0}^{2}\sin ^{2}(k_{1}a)}{4E(E-V)}}}}$

for the particle to be transmitted through the barrier. This effect which differs from the classical case is called quantum tunneling. The transmission is exponentially suppressed with the barrier width which can be understood from the functional form of the wave function: outside of the barrier it oscillates with wave vector $k_{0}$, while within the barrier it is exponentially damped over a distance $1/k_{1}$. If the barrier is much larger than this decay length, the left and right part are virtually independent and tunneling is consequently suppressed.

$E>V_{0}$

Equally surprising is that for energies larger than the barrier height, $E>V_{0}$, the particle may be reflected from the barrier with a non-zero probability

$R=|r|^{2}=1-T.$

This reflection probability is in fact oscillating with $k_{1}a$ and only in the limit $E\gg V_{0}$ approaches the classical result $r=0$, no reflection. Note that the probabilities and amplitudes as written are for any energy (above/below) the barrier height.

Remarks, Application

The calculation presented above may at first seem unrealistic and hardly useful. However it has proved to be a suitable model for a variety of real-life systems. One such example are interfaces between two conducting materials. In the bulk of the materials, the motion of the electrons is quasi free and can be described by the kinetic term in the above Hamiltonian with an effective mass $m$. Often the surfaces of such materials are covered with oxide layers or are not ideal for other reasons. This thin, non-conducting layer may then be modeled by a barrier potential as above. Electrons may then tunnel from one material to the other giving rise to a current.

The operation of a scanning tunneling microscope (STM) relies on this tunneling effect. In that case the barrier is due to the air between the tip of the STM and the underlying object. Since the tunnel current depends exponentially on the barrier width, this device is extremely sensitive to height variations on the examined sample.

The above model is one-dimensional while the space around us is three-dimensional. So in fact one should solve the Schrödinger equation in three dimensions. On the other hand, many systems only change along one coordinate direction and are translationally invariant along the others. The Schrödinger equation may then be reduced to the case considered here by an ansatz for the wave function of the type: $\Psi (x,y,z)=\psi (x)\phi (y,z)$.

For another, related model of a barrier see Delta potential barrier (QM).