Ch. 40----2, 4, 6, 8, 9, 15, 16, 20, 22, 23, 26, 28, 29, 32, 33, 34, 35 (DUE MAY 4TH)
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SAMPLE EXAMS  #2
REAL EXAM 3
REAL EXAM 4
We start with # 15 and continue to the end of the set. Note:  hbar = h/(2*pi).
15. For the so-called squared well potential of section 40.2, 2.00 eV = E1 = 0.625E1box, where E1box is the ground state of the electron confined to box given by equation 40.10, with n = 1. Find L.
16. To escape, the energy needed is hc/lambda = Uo - E1 = Uo - 0.625E1box
20. The longest wavelength lambda means the smallest change in energy: hc/lambda = E2 - E1 = 2.43E1box - 0.625E1box. Given lambda, find L. E1box is the ground state of the electron confined to box given by equation 40.10, with n = 1. Find L.
22. T = Ge-2KL , where G is within equation  40.21.   Evaluate for ratios E/Uo = 5/7, 5/9 and 5/13.
23. We use the phonetic symbol Psi to rep  wave function. Note:  hbar = h/(2*pi). 2 equations:

(A) -(hbar /2m)*d2Psi/dx2 = E*Psi, with Psi = A*eikx + B*e-ikx .

(B)  -(hbar /2m)*d2Psi/dx2 + Uo*Psi = E*Psi or
 
-(hbar /2m)*d2Psi/dx2  = (E - Uo)*Psi,  Psi =  C*eik'x . 

Substitute  appropriate Psi into each equation. Use (ik)2 = -k property to find k compared  with k' found  through the same steps.
26. See 22, applying Uo - E twice.
28. Substitute into
-(hbar /2m)*d2Psi/dx2 + (1/2)*k*x2*Psi =  E*Psi where  C*e-alpha*x2 , alpha = sqrt(mk')/(2hbar). Show E = published ground state energy.
29. (a) hc/lambda = hc/(5.8x10 -6 m) (b) hbar *w  = hbar*sqrt( k'/m) to find k'.
32. Ratios: (a) C*e-alpha*x2 , alpha = sqrt(mk')/(hbar) and x2 = A2 = (hbar/k')*w.
(b)   C*e-alpha*x2 , alpha = sqrt(mk')/(hbar) and x2 = 4A2 =4* (hbar/k')*w.   
33. (a) m = 3.82x10 -26 and  (1/2)*k'*(0.014 x10 -9)2   =  0.0075 eVx1.6x10-19 J.  Find k' and w. From that get ground state energy.
(b) hbar*w = hbar*sqrt( k'/m).
(c) hc/lambda = hbar*w
34.

-(hbar /2m)*d2Psi/dx2 + Uo*Psi = E*Psi or
 
-(hbar /2m)*d2Psi/dx2  = (E - Uo)*Psi,  Psi = A*eikx . 

Substitute  appropriate Psi into each equation. Use (ik)2 = -k property to find k in terms of E and Uo.
Set k = (2*pi)/lambda.

35. Continuous  Psi leads to
(A)  A+ B = C and continuous derivative  leads to
(B) ik*(A - B) = ik'*C, where k and k' are the wave numbers x < 0 and x > 0, respectively.  
Solve for B and C  in terms of A. Reproduce the answer in  back of book.