Sunday, October 5, 2014

Solutions to Part 1 – Intro. To Quantum Mechanical Operators


1) For the step potential, show that for the region x > 0    and

E < V o    with K1 = Ö (2m(Vo -  E) / h     


The general solution of the appropriate Schrodinger equation is:

y(x)  = C exp (K2 x) + D exp (-K2 x)

Show this.

 
Solution:

Take 1st and second derivatives: 
 

dy(x) / dx = CK2 exp (K2 x) – D K2 exp (-K2 x)

d2y(x) / dx2 =

CK22  exp (K2 x) + D K22 exp (-K2 x)

= K22   y(x) =  y(x)  [2m(Vo -  E) / h2]

Which yields the appropriate Schrodinger eqn. on substitution:

- h2/ 2m   [2m(Vo -  E) / h2] y(x)   + Vo y(x)  = Ey(x) 

 

2) For the transmitted and reflected flux of the step potential, show that:

 R + T = 1.

Solution:   
 
We have: R = (K1 – K2)2 / (K1 + K2)2

And:

T = 4K1K2 / (K1 + K2)2

Then:   R + T = 

(K1 – K2)2 / (K1 + K2)2  +  4K1K2 / (K1 + K2)2


=  [K1 2  - 2K1K2 + K1 2  +  4K1K2] / (K1 + K2)2


 =     [K1 2  +  2K1K2 + K1 2] / (K1 + K2)2


=   (K1 + K2)2 / (K1 + K2)2    =    1 

 

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