Consider the 1-dimensional, quantum mechanical system consisting of a single structure-less particle.
How well do the classical equations of motion describe the motion of this particle?

We have Ehrenfest’s theorem, d<X>/dt = <P>/m,  d<P>/dt = -<dU(x)/dx>.
The classical equations of motion work best when the potential varies slowly with distance.

Assume the potential is a slowly varying function of distance, U = U(x), independent of time.
Then solutions of the form ΨE(x,t) = ΦE(x) exp(-iEt/ħ) can be found.
ΦE(x) is a solution of
2Φ(x)/∂x2 + k2(x) Φ(x) = 0,   E > U,
or
2Φ(x)/∂x2 - ρ2(x)Φ(x) = 0,   E < U,
where k2(x) = (2m(E - U(x))/ħ2),  ρ2(x) = (2m(U(x) - E)/ħ2).

Let us try a solution of the form Φ(x)  = A exp((i/ħ)S(x)).
[Any complex function of x can be written as Φ(x)  = A(x) exp(i S(x)/ħ).]
Substituting this solution into the time-independent Schroedinger equation we obtain
iħd2S/dx2 - (dS/dx)2 + ħ2k2(x) = 0,  or  iħd2S/dx2 - (dS/dx)2 - ħ2ρ2(x) = 0.
(For a potential square well S(x) = ±ħkx inside the well and ±iħρx outside the well.)

Assume that ħ can, in some sense, be regarded as a small quantity and that S(x) can be expanded in powers of ħ, S(x) = S0(x) + ħS1(x) + ... .
Then
iħd[dS0/dx + ħdS1/dx + ... ]/dx - [dS0/dx + ħdS1/dx + ... ]2 + ħ2k2(x) = 0,   (E > U(x)).
We assume that |dS0/dx| >> |ħdS1/dx| and collect terms with equal powers of ħ.
[Note: ħ2k2 is zeroth order in ħ, since k(x) = (2m(E - U(x))/ħ2)1/2.]

[dS0/dx]2 + ħ2k2(x) = 0  -->  S0 = ±∫x ħk(x') dx'.
id2S0/dx2  - 2(dS0/dx)(dS1/dx) = 0 --> S1 = ½i ln(k(x)).
We have used
id[S0/dx]/dx  = 2(dS0/dx)(dS1/dx),
idk/dx = 2k dS1/dx,  dS1 = (i/2)dk/dk.
Therefore Φ(x)  = A k(x)-1/2 exp(±i∫x k(x') dx'.
In the classically allowed region S0 = ±∫x ħk(x') dx' counts the oscillations of the wave function.  An increase of 2πħ corresponds to an additional phase of 2π.
Similarly, in regions where E < U(x) we have
Φ(x)  = A ρ(x)-1/2 exp(±∫x ρ(x') dx'.
This is the WKB (Wentzel, Kramers, Brillouin) approximation.

When is the  WKB approximation valid?

For our first order expansion to be accurate we need that the magnitude of higher order terms decreases rapidly.
We need  |dS0/dx| >> |ħdS1/dx| or k >> |(2k)-1dk/dx|.
The local deBroglie wavelength is λ = 2π/k.
Therefore we need |(λ /4π)(dλ/dx)| << λ,  i.e. the change in λ over a distance λ /4π is small compared to λ .
This holds when the potential varies slowly and the momentum of the particle is nearly constant over several wavelength.
Near the classical turning points the WKB solutions become invalid, because k goes to zero here.  We have to find a way to connect an oscillating solution to an exponential solution across a turning point if we want to solve barrier penetration problems or find bound states.

The WKB approximation for bound states
We want to find the wave function of a particle in a given potential well.  Assume the energy of the particle is E and that the classical turning points are x1 and x2, x1 < x2, i.e. we have a potential well with two sloping sides.

For x < x1 the wave function is of the form
Φ1(x)  = A1 ρ-1/2 exp(+∫x ρ(x') dx'.
For x > x2 the wave function is of the form
Φ3(x)  = A3 ρ-1/2 exp(-∫x ρ(x') dx'.
In the region between x1 and x2 it is of the form
Φ2(x)  = A2 k-1/2 exp(+i∫x k(x') dx' + A2' k-1/2 exp(-i∫x k(x') dx'.
At x = x1 and x = x2 the wave function Φ and its derivative have to be continuous.
How do we apply these boundary conditions?
Near x1 and x2 we expand the potential in a Taylor series expansion in x and neglect all terms of order higher than 1.
Near x1 we have U(x) = E - K1(x - x1), and near x2 we have U(x) = E - K2(x + x2), with K1 and K2 positive constants.
In the neighborhood of x1 the time-independent Schroedinger equation then becomes
2Φ(x)/∂x2 +  (2mK12)(x - x1) Φ(x) = 0,
and in the neighborhood of x2 the time-independent Schroedinger equation becomes
2Φ(x)/∂x2 -  (2mK22)(x - x2) Φ(x) = 0.

Let us define  z = -(2mK12)1/3(x - x1).
Then we obtain ∂2Φ/∂z2 -  zΦ = 0 near x1.
The solutions of this equation which vanish asymptotically as  z --> ∞ or x --> ∞  are the Airy functions.  They are defined through
Ai(z) = π-10 cos(s3/3 + sz)ds
which for large |z| has the asymptotic form
Ai(z) ≈ (2 √π z1/4)-1 exp(-2z3/2/3),  (z > 0),
and
Ai(z) = (√π (-z)1/4)-1 sin(2(-z)3/2/3 + π/4),  (z < 0).
If the energy is high enough the linear approximation to the potential remains valid over many wavelength.
The Airy functions can therefore be the connecting wave functions through the turning point at x1.

If we define z = (2mK22)1/3(x - x2) then we find ∂2Φ/∂z2 -  zΦ = 0 near x = x2 and the Airy functions can also be the connecting wave functions through the turning point at x2

In the neighborhood of x1 we have k2 = -ρ2 =  (2mK12)1/3(x - x1) = -(2mK12)2/3z
Therefore ∫x1x ρ dx' = (2mK12)1/3x1x √z dx' = -∫0z √z' dz' = -2z3/2/3.
Similarly ∫x1x k dx' = (2mK12)1/3x1x √(-z) dx' = -∫0z √(-z') dz' = -2(-z)3/2/3.
By comparing this with the asymptotic forms of the Airy functions we note that
Φ1(x)  = A1 ρ-1/2 exp(+∫x1x ρ(x') dx'   (x < x1)
must continue on the right side as
Φ2(x)  = 2A1 k-1/2 sin(∫x1x kdx'  +  π/4)    (x > x1).

In the neighborhood of x2 we similarly find that
Φ3(x)  = A3 ρ-1/2 exp(-∫x2x ρ(x') dx'   (x > x2)
must continue in region 2 as
Φ2(x)  = 2A3 k-1/2 sin(∫xx2 (kdx'  +  π/4)   (x < x2).
Both expressions for Φ2(x) are approximations to the same eigenfunction.  We therefore need
2A1 k-1/2 sin(∫x1x (kdx ' +  π/4) =  2A3 k-1/2 sin(∫xx2 (kdx' +  π/4).
Writing ∫x1x = x1x2 - ∫xx2, we require
sin[∫x1x2 k dx' -  ∫xx2 kdx' + π/4)] = (A3/A1)sin[∫xx2 k dx' + π/4].
This condition is only satisfied if  ∫x1x2 k dx =  (n + ½)π, n = integer.
This can be rewritten as  ∫x1x2 p dx =  (n + ½)h/2,  or  ∫cylce p dx  = (n + ½)h.
Here ∫cylce denote an integral over one complete cycle of the classical motion.
The WKB method for bound states therefore leads to the Wilson- Sommerfeld quantization rule except that n is replaced by (n + ½).
It leads to a quantization of the classical action J = ∫cylce p dq.

If we let n = 0, 1, 2, ..., then n counts the number of zero of the wave function in the well.  We may also let n = 1, 2, 3, ... , and write ∫x1x2 k dx =  (n - ½)π.  Then n counts the number of anti-nodes of the wave function in the well.  If our potential well has one or two vertical walls, the results of the WKB approximation differ only in the number that is subtracted from n = 1, 2, 3, ... , respectively.
For one vertical wall we have ∫x1x2 k dx =  (n - ¼)π and for two vertical walls we have ∫x1x2 k dx =  nπ.
Since the WKB approximation works best for large n in the semi-classical regime, this distinction is more in appearance than in substance.

Problem:

Use the WKB approximation to derive the energy levels of a particle of mass m confined to the one-dimensional potential U(x) = F|x|.

Solution:

• Concepts:
The WKB approximation
• Reasoning:
We are instructed to use the WKB approximation.  The WKB approximation requires that  ∫xminxmax pdx = (n + ½)(h/2) or ∫cylce p dx  = ∫cylce ħk dx  = (n + ½)h, n = 0, 1, 2, ... .   Here ∫cylce denotes an integral over one complete cycle of the classical motion and  k2 = (2m/ħ2)(E - U(x)).
• Details of the calculation:
For stationary bound states we want ∫xminxmax pdx = (n + ½)πħ, n = 0, 1, 2, ... .
In our problem k2(x) = 2m(E - F|x|)/ħ2.
We need  (2m)1/2xminxmax dx (E - F|x|)1/2 = 2(2m)1/2x0xmax dx (E - Fx)1/2 = (n + ½)πħ.
Here Fxmax = E,  xmax = E/F.
Therefore
-2(2m)1/2 (2/(3F)) (E - F|x|)3/2|x0xmax =  4(2m)1/2/(3F))E3/2  = (n + ½)πħ.
E = [3π(n + ½)/(4√2)]2/3(Fħ)2/3/m1/3.
For n = 1 (the first excited state), for example, we have
E = 1.84 (Fħ)2/3/m1/3.

Penetration of a potential barrier
The WKB method leads to the following expression for the transmission coefficient:
T ≈ exp[-2∫x1x2 dx (2m(U(x) - E)/ħ2)1/2].
To the same approximation R = 1 - T.