Bound states of particle in a potential well Recall WKB breaks - - PowerPoint PPT Presentation

Bound states of particle in a potential well

Recall WKB breaks down near turning pts

These must be smoothly connected across turning points

For that, we expand the potential near turning points Substitute in Schrodinger equation near turning points

Similarly near x=x2 Define variable z as follows

Equation to solve

Solutions are Airy functions Fourier transform Inverse Fourier transform

solution

prove We are interested in asymptotic form of Airy function to fix constants C1 , C3 in region I and III solution

Asymptotic form

Region I C1 can be fixed by comparing z>>0 Airy function asymptotic form

Region II

• General solution can be written with

a phase shift as follows:

• Near x=x1

C2 = 2 C1 , a = p/4 Do a similar exercise near x= x2 Region II, x < x2 and z <<0 and region III z>>0 Region III

In region II

Asymptotic Airy function comparison with WKB wavefunction near x=x2 Wavefunction in region II obtained near x=x1 and x=x2 must be same. This will fix relation between C1 , C3

Compare with the form Solution agrees if C1= (-1)n C3

connection formula for potential barrier

• This will be useful to work out the

transmission coefficient T for a barrier.

• Please work out connection formula - In

region I and III, E> V and in region II we will have E<V.

• Oscillatory in region I and III but

exponential in region II

• For high and broad barrier what is T

Using connection formula, try to derive where Transmission coefft For high and broad barrier θ>>1

Lifetime τ of parent nucleus from αlpha decay

• τ is inversely proportional to T
• Assignment problems