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(V_o -  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(V_o -  E) / h²]

Which yields the appropriate Schrodinger eqn. on substitution:

- h²/ 2m   [2m(V_o -  E) / h²] y(x)   + V_o 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