In quantum mechanics, the Wentzel-Kramers-Brillouin or simply the WKB
approximation is a technique to approximate the wave function of
the one-dimensional time-independent Schrödinger equation. It is an example
of a semiclassical approximation, because it tries to find a
balance between classical and quantum physics.
In classical mechanics, a particle travelling in a potential
along a path has a total energy as follows, which we
The left-hand side of the rearranged version is simply the momentum squared,
so we define the magnitude of the momentum accordingly:
Note that this is under the assumption that ,
which is always true in classical mechanics,
but not necessarily in quantum mechanics.
We rewrite the Schrödinger equation:
If were constant, and by extension too, then the solution
This form is reminiscent of the generator of translations. In practice,
and vary with , but we can still salvage this solution
by assuming that varies slowly compared to the wavelength
, where is the
wavenumber. The solution then takes the following form:
is an unknown function, which intuitively should be related
to . The purpose of the integral is to accumulate the change of
from the initial point to the current position .
Let us write this as an indefinite integral for convenience:
Where is the initial point of the definite integral.
For simplicity, we absorb the constant into .
We can now clearly see that:
Next, we insert this ansatz for into the Schrödinger equation
Dividing out and rearranging gives us the following, which is
Next, we expand this as a power series of . This is why it is
called semiclassical: so far we have been using full quantum mechanics,
but now we are treating as a parameter which controls the
strength of quantum effects:
The heart of the WKB approximation is its assumption that quantum effects are
sufficiently weak (i.e. is small enough) that we only need to
consider the first two terms, or, more specifically, that we only go up to
, not or higher. Inserting the first two terms of this
expansion into the equation:
Where we have discarded all terms containing . At order
, we then get the expected classical result for :
While at order , we get the following quantum-mechanical
Therefore, our approximated wave function currently looks like
We can reduce the latter exponential using integration by substitution:
In the WKB approximation for , the solution is thus
What if ? In classical mechanics, this is just not allowed; a ball
cannot simply go through a potential bump without the necessary energy.
On the other hand, in quantum physics, particles can tunnel through barriers.
Luckily, the only thing we need to change for the WKB approximation
is to let the momentum take imaginary values:
And then take the absolute value in the appropriate place in front of :
In the classical region (), the wave function oscillates, and
in the quantum-physical region () it is exponential.
Note that for the approximation breaks down,
because of the appearance of in the denominator.
- D.J. Griffiths, D.F. Schroeter,
Introduction to quantum mechanics, 3rd edition,
- R. Shankar,
Principles of quantum mechanics, 2nd edition,