Stationary State Approximate Methods Chapter 2 Lecture 2.3 Dr. Arvind Kumar Physics Department NIT Jalandhar e.mail: iitd.arvind@gmail.com https://sites.google.com/site/karvindk2013/

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

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.

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.

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

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

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

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)

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)

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

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

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

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

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)

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

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.

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)

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

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

Airy functions plot

In overlap region ---(23)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

Using (5) in (3)

Exercise: Fig: Alpha decay