The index
on the coefficients
and
denotes the direction of the velocity vector. Note that, if the energy of the particle is below the barrier height,
becomes imaginary and the wave function is exponentially decaying within the barrier. Nevertheless, we keep the notation
even though the waves are not propagating anymore in this case. Here we assumed
. The case
is treated below.The coefficients
have to be found from the boundary conditions of the wave function at
and
. The wave function and its derivative have to be continuous everywhere, soInserting the wave functions, the boundary conditions give the following restrictions on the coefficients
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
larger than the barrier height
would always pass the barrier, and a classical particle with
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
(incoming particle),
(reflection),
(no incoming particle from the right), and
(transmission). We then eliminate the coefficients
from the equation and solve for
and The result is:
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,
there is a non-zero probabilityfor 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 casewhere
Equally surprising is that for energies larger than the barrier height,
, the particle may be reflected from the barrier with a non-zero probability The transmission and reflection probabilities are in fact oscillating with
. The classical result of perfect transmission without any reflection (
,
) is reproduced not only in the limit of high energy
but also when the energy and barrier width satisfy
, where
(see peaks near
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
is [1] 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
as
approaches
. For this purpose the ratio
is defined, which is used in the function
:
In the last equation
is defined as follows:
These definitions can be inserted in the expression for
which was obtained for the case
.
Now, when calculating the limit of
as x approaches 1 (using L'Hôpital's rule), \limxf(x)=\limx
| \sinh(v0\sqrt{1-x |
)}{(1-x)} |
=\limx
\sqrt{1-x}}=v0\cosh(0)=v0
also the limit of
as
approaches 1 can be obtained:
By plugging in the above expression for
in the evaluated value for the limit, the above expression for T is successfully reproduced.Remarks and applications
. 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
while keeping
constant.See also
References
- Book: Griffiths, David J.. Introduction to Quantum Mechanics (2nd ed.) . Prentice Hall . 2004 . 0-13-111892-7.
- Book: Cohen-Tannoudji, Claude . Bernard . Diu . Franck . Laloë . transl. from the French by Susan Reid Hemley. Quantum mechanics.. limited . 1996. Wiley. Wiley-Interscience. 978-0-471-56952-7. 231–233. etal.
Notes and References
- Book: McQuarrie DA, Simon JD . Physical Chemistry - A molecular Approach. 1997. University Science Books . 978-0935702996 . 1st.