The WKB approximation is a method to find approximate solutions to the Schrodinger equation. It was developed in 1926 by Wentzel, Kramer, and Brillouin. The approximation assumes the wavefunction is an exponentially varying function with amplitude and phase that change slowly compared to the de Broglie wavelength. It can be used to obtain approximate solutions and energy eigenvalues for systems where the classical limit is valid. The approximation breaks down near classical turning points where the particle's energy is equal to the potential energy. The document provides examples of using the WKB approximation to solve the time-independent Schrodinger equation in one dimension for cases where the particle's energy is both greater than and less than the potential energy.

1. Topic:
WKB Approximation

2. Presented By:
Zahid Mehmood

3. The WKB Approximation
Wentzel, Kramer's, and Brillouin develop in 1926.
It is semiclassical calculation or Approximation. In quantum mechanics in
which wavefunction is assumed an exponentially function with amplitude
and phase that slowly varies as compared to de-Broglie wavelength and
then classically expand in power in power of ℏ.
Generally, the WKB approximation or WKB method is a method for
finding approximate solutions to linear differential equations with
spatially varying coefficients.
 In Quantum Mechanics it is used to obtain approximate solutions to the
time independent Schrodinger equation in one dimension.it can solve
Schrodinger equation 1D,2D,3D.

4. Applications in Quantum Mechanics
it is useful in one dimension problem but also for three dimension problem.
It useful for finding the energy eigenvalues and wave functions of systems
for which the classical limit is valid. and WKB methods do not require the
existence of a closely related Hamiltonian that can be solved exactly.
Wkb method is particularly useful in estimating the energy eigenvalues of
the ground state and the first few excited states of a system for which one
has only a qualitative idea about the form of the wave function.
Applications for us will be in calculating bound-state energies and tunneling rates
through potential barriers.

5. Applications in Quantum Mechanics
In quantum mechanics used to obtain approximate solution of time
independent Schrodinger equation in one dimension.
Solution of this equation
𝑘=
2𝜋
𝑘
=⋌=
ℎ
𝑝
= √(2𝑚(𝐸−𝑉))/ℏ
(x)  A1eikx
 A2eikx
(x)  A1eikx
 A2eikx
𝜓 𝑥 = 𝐴𝑒±𝑖𝑘𝑥

6. If potential 𝑽(𝒙) is constant and energy 𝐸 of the particle is 𝑬 > 𝑽,
then the particle wave function has the form 𝜓 𝑥 = 𝐴𝑒±𝑖𝑘𝑥
where 𝑘=
2𝜋
𝑘
=⋌=
ℎ
𝑝
= √(2𝑚(𝐸−𝑉))/ℏ
(+) sign indicates : particle travelling to right
(-) sign indicates : particle travelling to left
• General solution : Linear superposition of the two.
• The wave function is oscillatory, with fixed wavelength, 𝜆 =2𝜋/𝑘
• The amplitude (A) is fixed.

7. If V (x) is not constant, but varies slow in comparison with the
wavelength λ in a way that it is essentially constant over many λ,
then the wave function is practically sinusoidal or Oscillatory, but
wavelength and amplitude slowly change with x.
 This is the inspiration behind WKB approximation.
In effect, it identifies two different levels of x-dependence : rapid
oscillations, modulated by gradual variation in amplitude and
wavelength.

8. If E<V and V is constant, then wave function is
𝜓 𝑥 = 𝐴𝑒±𝑖𝑘𝑥
where κ = √(2𝑚(𝑉−𝐸))/ℏ
If 𝑽(𝒙) is not constant, but varies slowly in
comparison with 1/𝜅 , the solution remain
practically exponential, except that 𝐴 and 𝜅 are
now slowly-varying function of 𝑥.
(x)  A1eikx
 A2eikx

9. Failure of this idea
There is one place where this whole program is bound to fail, and that
is in the immediate vicinity of a classical turning point, where 𝐸 ≈ 𝑉.
For here 𝜆(𝑜𝑟 1/𝜅)) goes to infinity and 𝑉(𝑥) can hardly be said to vary
“slowly” in comparison. A proper handling of the turning points is the
most difficult aspect of the WKB approximation, though the final
results are simple to state and easy to implement.

10. Let's now solve the Schrödinger equation using WKB
Approximation
𝑑2ψ
𝑑𝑥2 2m
+ Vx = 𝐸𝜓
can be rewritten in the following way:
𝑑2ψ
𝑑𝑥2 = −𝑝2 𝜓/ℏ2
ℏ2
𝜓 ; 𝑝 (𝑥) = 2m(E − 𝑣 𝑥)
is the classical formula for the momentum of a particle with
total energy 𝐸 and potential energy 𝑉(𝑥). Let’s assume 𝐸 > 𝑉(
𝑥), so that 𝑝(𝑥) is real. This is the classical region , as classically
the particle is confined
to this range of 𝑥.

13. The function 𝜓
In general, 𝜓 is some complex function;
we can express it in terms of its amplitude 𝐴(𝑥), and its phase, 𝜙(𝑥) –
both of which are real :
𝜓 𝑥 = 𝐴𝑥𝑒 𝑖ϕx

14. In this approach, the wave function is expanded
in powers of ℏ .
Let, the wave function be:
𝜓 = 𝐴𝑒
𝑠(𝑥)
ℏ
𝒅𝟐𝒚
𝒅𝒙𝟐
𝛙+ 𝑘2 𝜓 = 0
(ds/dx)2 − 𝑖ℏ
d2
s
dx2 − 𝑘2ℏ2 = 0
Expanding S(x) in powers of ℏ2:
𝑆(𝑥) = 𝑆0(𝑥) + 𝑆1(𝑥) ℏ + 𝑆2(𝑥) ℏ2
/2+ ⋯
(
𝑑𝑆0
𝑑𝑥
)2
−𝑘2
ℏ2
+ 2
𝑑𝑆0
𝑑𝑥
𝑑𝑆1
𝑑𝑥
− 𝑖
𝑑2 𝑆2
𝑑𝑥2 ℏ = 0
Where 𝑘2
ℏ2
= 2𝑚[𝐸 − 𝑉(𝑥)]

15. For the above equation to be valid, the coefficient
of each power of ℏ must vanish separately
(
𝑑𝑆0
𝑑𝑥
)2−𝑘2ℏ2=0
𝑑𝑆0
𝑑𝑥
= ±𝑘ℏ 𝑠 𝑜 𝑥 = ±∫ 𝑘ℏ𝑑𝑥

16. 2
𝑑 𝑥
2
and, 2 𝑑 𝑆0 𝑑 𝑆1
− 𝑖 𝑑 𝑆0
𝑑 𝑥
𝑑s1
=
𝑑 𝑥 𝑑 𝑥
0; using the value of s (x)
𝑖 𝑖 𝑆or, e = 𝑘−1/2
𝑖
2𝑘
𝑑𝑘
𝑑𝑥
𝑠1=
𝑖
2
𝑙𝑛𝑘
𝜓 = 𝐴𝑒
𝑠(𝑥)
ℏ
= 𝐴𝑒
𝑖
ℏ 𝑆0(x)+ 𝑆1(x) ℏ]
= 𝐴𝑒
𝑖
ℏ
𝑠 𝑜
𝑒 𝑖𝑆1
= A
1
√𝑘
𝑒±𝑖∫ 𝑘𝑑𝑥
Or 𝜓 =
𝐴
√𝑘
𝑒±𝑖∫ 𝑘𝑑𝑥 where A is a constant.

17. For E < 𝑉x: the Schrodinger equation is ,
2𝑑2 𝜓
𝑑𝑥2 = 𝑘2 𝜓=0 𝑘 =
(2𝑚(𝐸−𝑉))
ℏ
2
Proceeding in a similar fashion , the solution can be
obtained as
𝜓
𝐵
√𝑘
𝑒( ±∫ 𝑘ℏ𝑑𝑥
where B is a normalization constant
.