1. Stationary State Approximate Methods
Chapter 2
Lecture 2.3
Dr. Arvind Kumar
Physics Department
NIT Jalandhar
e.mail:

2. Wentzel-Kramers-Brillouin (WKB) Method
Consider the particle having total energy E
moving through a potential V(x).
Schrodinger Eq
----(1) or
-----(2)

3. For constant potential i.e. V(x) = V and E>V
i.e., we have and oscillatory wave function with
Constant wavelength l = 2pi/p.
However, if V(x) is not constant but varies slowly
in comparison of wavelength so that over a region
containing many wavelength, we can still assume V
as constant, we can use still sinusoidal type wave
function but in this case wavelength and amplitude
will be a function of x.

4. For E<V
In above Ψ is exponetial function.
Again if V(x) is not constant, but varies slowly in
comparison to then solution will still be
exponential type. A and k will be function of x.
At turning points E =V and wavelength will diverge.

5. Classical Region (E>V(x))
Schrodinger Eq
----(1) or
-----(2)

6. In classical region, p will be a real function and wave
function will be complex
phase
amplitude (3)
First and 2nd derivative
-----(4)
-----(5)

7. Using (3) and (5) in (2), we get
-----(6)
Above eq is equal to two rela equations.
For
real
part (7)
imaginary (8)
part

8. From (8)
---(9)
For Eq (7), we use approx. that A is varying slowly
so that A`` negligible. Thus, from (7) we write
---(10)
Integrating
------(11)

9. Using (9) and (11) in (3)
-----(12)
Note that the general solution will be a linear
combination of two terms one with each sign.
Note:
------(13)

10. Example (From Griffith):
Consider infinite square well potential
What will be
energy of particle
Inside the well?

11. Inside well (assuming E>V) (see Eq.(15) and 13a)
We can also write above Eq. as

12. Now, phase factor (see Eq. (13a))
At x = 0,
At x = a,

13. Thus
which is quantization condition.
For special case, flat bottom V(x) = 0
And
Thus, .

14. Case (ii) Non classical region (E<V(x))
In this case p(x) will be imaginary. In the regions
where E<V(x), solution will be
-----(14)

15. Connection formulas
At turning points (E=V(x)) classical region joins
non-classical regions and WKB approximation
breaks.
In fig. axes are shifted
Such that the right
turning point occurs
at x = 0

16. Using WKB approximation, we can write
----(15)
+Ve exponent in x>0 region rejected because
It will blow when
As WKB app breaks at turning points, we slice
The two WKB solution with a patching wave function.

17. In neighbourhood of turning point we approx.
potential by straight line (to find patching wave
function )
----(16)
We now solve Schrodinger Eq for above potential.
------(17)

18. Eq (17) is further written as
(18)
Where
(19)
Define (20)
So we can write
----(21)
Airy Equation

19. Solution of Eq. (21) as linear combination of
Airy function
------(22)

21. Airy functions plot

22. In overlap region
---(23)

23. In overlap region 2
----(24)
WKB sol. (From Eq. 15))
---(25)

24. Patching wave function in overlap region 2
(large z asymptotic limit)
----(26)
Comparing (25) and (26), we get
----(27)

25. In region (1)
----(28)
WKB sol in region (1) is
----(29)

26. Patching wave function in region (1), in large –Ve z
asymptotic limit
------(30)

27. Comparing (29) and (30), we get
---(30)
Using (30) in (27)
---(31)
Which are known as connecting formulas.

28. For turning point at point say x2, we write
WKB solution, (expression constants in D)
----(32)

29. Potential well with one vertical wall
Here,
From (32)
-----(33)

30. Example: Half Harmonic oscillator
----(34)
We have
--(35)
Where
(turning point)---(36)

31. Thus,
---(37)
From (33) and (37),
---(38)

32. Potential well with no vertical wall
For fig(a) (Eq. (32) ) (upward sloping)
-----(39)

33. For fig(b) (downward sloping)
---(40)
For fig © in region x1 < x < x2, we write
(from (39))
---(41)

34. Or from (40)
----(42)
From (41) and (42)
Thus
------(43)

35. Tunnelling Problem
Consider the potential barrier
In region x<0
-----(1)

36. In region x>a
------(2)
Tunnelling probability
(3)
In tunnelling region
(Using WKB approx) (4)

37. For wider potential barrier first term will
increase exponentially and hence C should be
small for physical results and 1st term of Eq (4)
Will be neglected
Total decrease of exponential over
non-classical region will determine relative
amplitude need in (3) i.e.
----(5)

38. Using (5) in (3)

39. Exercise:
Fig: Alpha decay