Rectangular potential barrier explained

In quantum mechanics, the rectangular (or, at times, square) potential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling (also called "quantum tunneling") and wave-mechanical reflection. The problem consists of solving the one-dimensional time-independent Schrödinger equation for a particle encountering a rectangular potential energy barrier. It is usually assumed, as here, that a free particle impinges on the barrier from the left.

Although classically a particle behaving as a point mass would be reflected if its energy is less than a particle actually behaving as a matter wave has a non-zero probability of penetrating the barrier and continuing its travel as a wave on the other side. In classical wave-physics, this effect is known as evanescent wave coupling. The likelihood that the particle will pass through the barrier is given by the transmission coefficient, whereas the likelihood that it is reflected is given by the reflection coefficient. Schrödinger's wave-equation allows these coefficients to be calculated.

Calculation

The time-independent Schrödinger equation for the wave function

\psi(x)

reads\hat H\psi(x)=\left[-\frac{\hbar^2}{2m} \frac{d^2}{dx^2}+V(x)\right]\psi(x)=E\psi(x)where

\hatH

is the Hamiltonian,

\hbar

is the (reduced)Planck constant,

m

is the mass,

E

the energy of the particle andV(x) = V_0[\Theta(x)-\Theta(x-a)]is the barrier potential with height

V0>0

and width

a

.

\Theta(x)=0,x<0;\Theta(x)=1,x>0

is the Heaviside step function, i.e.,V(x)= \begin0 &\text x < 0 \\V_0 &\text 0 < x < a \\0 &\text a < x\end

The barrier is positioned between

x=0

and

x=a

. The barrier can be shifted to any

x

position without changing the results. The first term in the Hamiltonian, -\frac \frac\psi is the kinetic energy.

The barrier divides the space in three parts (

x<0,0<x<a,x>a

). In any of these parts, the potential is constant, meaning that 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). If

E>V0

\begin\psi_L(x) = A_r e^ + A_l e^ & x<0 \\\psi_C(x) = B_r e^ + B_l e^ & 0a\endwhere the wave numbers are related to the energy via\begink_0 = \sqrt & x<0 \quad \text\quad x>a \\k_1 = \sqrt & 0

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,

k1

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. Here we assumed

EV0

. The case

E=V0

is treated below.

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\begin\psi_L(0) &= \psi_C(0) \\\left.\frac\right|_ &= \left.\frac\right|_ \\\psi_C(a) &= \psi_R(a) \\\left.\frac\right|_ &= \left.\frac\right|_.\end

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

A_r+A_l=B_r+B_lik_0(A_r-A_l)=ik_1(B_r-B_l)B_re^+B_le^ = C_re^+C_le^ik_1 \left(B_re^-B_le^\right) = ik_0 \left(C_re^-C_le^\right).

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

V0

would always pass the barrier, and a classical particle with

E<V0

incident on the barrier would always get reflected.

To study the quantum case, consider the following situation: a particle incident on the barrier from the left side It may be reflected or transmitted

To find the amplitudes for reflection and transmission for incidence from the left, we put in the above equations

Ar=1

(incoming particle),

Al=r

(reflection),

Cl=0

(no incoming particle from the right), and

Cr=t

(transmission). We then eliminate the coefficients

Bl,Br

from the equation and solve for

r

and

The result is:

t=\fracr=\frac.

Due to the mirror symmetry of the model, the amplitudes for incidence from the right are the same as those from the left. Note that these expressions hold for any energy If then so there is a singularity in both of these expressions.

Analysis of the obtained expressions

E < V0

The surprising result is that for energies less than the barrier height,

E<V0

there is a non-zero probabilityT=|t|^2= \frac

for the particle to be transmitted through the barrier, with 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 whereas within the barrier it is exponentially damped over a distance If the barrier is much wider than this decay length, the left and right part are virtually independent and tunneling as a consequence is suppressed.

E > V0

In this caseT=|t|^2= \frac,where

Equally surprising is that for energies larger than the barrier height,

E>V0

, the particle may be reflected from the barrier with a non-zero probability R=|r|^2=1-T.

The transmission and reflection probabilities are in fact oscillating with

k1a

. The classical result of perfect transmission without any reflection (

T=1

,

R=0

) is reproduced not only in the limit of high energy

E\ggV0

but also when the energy and barrier width satisfy

k1a=n\pi

, where

n=1,2,...

(see peaks near

E/V0=1.2

and 1.8 in the above figure). Note that the probabilities and amplitudes as written are for any energy (above/below) the barrier height.

E = V0

The transmission probability at

E=V0

is [1] T=\frac.

This expression can be obtained by calculating the transmission coefficient from the constants stated above as for the other cases or by taking the limit of

T

as

E

approaches

V0

. For this purpose the ratio

x = \frac

is defined, which is used in the function

f(x)

:

f(x) = \frac

In the last equation

v0

is defined as follows:

v_0 = \sqrt

These definitions can be inserted in the expression for

T

which was obtained for the case

E<V0

.

T(x) = \frac

Now, when calculating the limit of

f(x)

as x approaches 1 (using L'Hôpital's rule),

\limxf(x)=\limx

\sinh(v0\sqrt{1-x
)}{(1-x)}

=\limx

d\sinh(v0\sqrt{1-x
dx
)}{d
dx

\sqrt{1-x}}=v0\cosh(0)=v0

also the limit of

T(x)

as

x

approaches 1 can be obtained:

\limxT(x)=\limx

1
1+f(x)2
4x

=

1
1+
2
v
0
4

By plugging in the above expression for

v0

in the evaluated value for the limit, the above expression for T is successfully reproduced.

Remarks and applications

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 gap 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 space is three-dimensional. 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; they are separable. 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), which can be regarded as a special case of the finite potential barrier. All results from this article immediately apply to the delta potential barrier by taking the limits

V0\toinfty,a\to0

while keeping

V0a=λ

constant.

See also

References

Notes and References

  1. Book: McQuarrie DA, Simon JD . Physical Chemistry - A molecular Approach. 1997. University Science Books . 978-0935702996 . 1st.