WICB Wentzel KramersBrillouin Approximation

Also semiclassical see Landau inLifshitz Quantum

f LIMIT thx x F 4W

write 4 eir zitmtirtfmkrf E.VN74 41744 nonlinear

54 1 Ertl art 4

Semiclassicalt smallhi r rot Chili 6 t

Otho Em To'Y E Vex ro pix classicalmomentum

CX Ifdx pox classical actionP 2mlE Vcx

µc tilt Jd M oscillateswith wavenumber kcx7 pcxVk

Validity tho I LL lo't dChik'HdxK1 dildx KIKIT

12Th VCHslowly varying0th To 2 ro'T o or E large

G To 2To p 2p Vexlarge

T X Lhp X small slow Decay

Given E can quicklysketch4 But what is E

CX3

fiY.nmatchiX feEoa

constant forceSchrodingerEg

decayAicx wkB WKB oscillatoryAdecayfunction I

on

iec

phase 4 Match at b C Aet

c Aei't

qbcxI cost fjpdxtyI

Match at a Ci AEtci A.ec

l4rfacxIfj cosltifpdx IIa

acx Lb HAKI 4,5 1

either C C and phasesdiffer by 25Mor C C and phasesdiffer by 2mi 1 I

Let C CDnc pdx Ie I'fffpdxtIy1 nT

F bapdx ht f IT pdx f h Bohr Sommerfeld

p quantizationzeropointduetomatching

Additionalapplicationsno Zero point becauseI Rotational motion Lado Uh no Airymatching

2 Energy level spacingcompare level n and htt A pdX 2Th

aEf dx

V 2E 2p A Ef deVAE _h UCE A E T c periodof

oscillation

Examples

tarmoniator w k independent of E

energy levels equally spaced

Squareweli WCEl VIL TEA E RE GENRE die rn

En m2