1. (22 points) ZERO-POINT ENERGY: For more info visit: http://en.wikipedia.org/wiki/Zero-point_energy or http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc4.html#c1  . 

Consider a particle with mass m moving in a  potential U= (1/2)k'x² , as in a mass-spring system. The total energy of particle is E =  p²/2m  + ½ k’x² .  We write the spring constant as k’ to avert confusion with  wave number k= 2π/λ used in this chapter.   Assume  Δp and Δx are related  by the lower limit of the Heisenberg Uncertainty Principle (equation 39.11) as:

(i) ΔpΔx = .  Here,  Δp and Δx are the minimum allowed values of the uncertainty in the momentum and position, respectively.  The  minimum energy is thus:

(ii)  E_min = (Δp)²/2m  + ½ k’(Δx)²  = kinetic energy + potential energy.

 (a) (12 POINTS) Find the minimum of energy  E in terms of  and ω by differentiating  the relationship (ii) with respect to Δx. Follow these instructions:  Use (ii)  ΔpΔx =  and solve  (i) for Δp in terms of Δx. Then  substitute into (ii) to eliminate Δp. Then use calculus. Differentiate the formula for E with respect to Δx. Find the value of Δx that gives a minimum in E. This is basically an straight Math 1 minimization  problem.  Substitute the value of Δx you get into (ii) and  find the minimum energy E in terms of  and ω

(b) (7  POINTS) For the value of Δx you got in part (a), what is the ratio of the kinetic energy to the potential energy of the particle?

(c ) (3 POINTS) Explain why the value you got for  the  minimum energy is higher than that obtained at this link:  http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc4.html#c1  .

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2.  (23 points) Consider a  finite square  well. The potential energy is U_o = 6E_1box  , where E_1box  is the ground state energy (n = 1) for the wave function of an electron  in an infinite box. Assume the  square well has U_o = 20.00 eV.  

(a) (8) How much energy does it take to ionize  an electron from the first excited  level (n = 2) ?

(b) (7) An 8.86 eV photon is absorbed by the electron in its first excited level.  As the atom returns to the ground  state ( n = 1) , what possible  energies can the  emitted photons have? Assume there can be transitions between all pairs of levels, including sub-transitions back to the ground state.
(c) (4) What will happen if a photon of energy of 10.65 eV  strikes an electron in the well  in its ground state? Explain why.
(d) (4) What will happen if a photon of energy of 14.88-eV strikes an electron in the well in  its ground state? Explain why.

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An electron in an infinite square well has a wave function given by :

                                             ,

for   and zero otherwise, where L = 0.100 nm.

(a)(3 points) The wave function is zero for what values of x  such that  0 < x < L ?

(b) (3 points) Sketch the function  on the diagram provided below.

(c) (3 points) What is the probability of finding the particle between x = 0 and x = L/6. 
(d) (5 points) What is the probability of finding the particle between x = 5L/6 and  x = L ?

(e) (3 points) What is the probability of finding the particle between x = 0 and  x = 2L/6 ?

(f) (3 points)  What is the momentum of the  particle? If you  want to use the mass for the calculation. Assume the particle is an  electron  of mass m = 9.11x10^-31 kg.
(g) (3 points) What is the kinetic energy of the particle? Assume                     electron  mass  m = 9.11x10^-31 kg.

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4. ( 22  POINTS) A particle is confined within a box with perfectly rigid wall at x = 0 and x = L. Although the magnitude of the instantaneous force exerted on the ball infinite and the time over which it acts is zero, the impulse (that involves a product of force and time) is both finite and quantized. In Chapter 8 we equated impulse to the change in momentum.

(a) (11 points) Using the definition that relates  force to the derivative  of the potential energy U, show that the magnitude of the force is infinite in magnitude.  Hint: Review section 7.4.

(b) (11  points) Show that the impulse exerted by the wall at x = 0 is:
 
and the impulse exerted by the wall at x = L is:
 , where L is the well width.

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5. (22) A harmonic oscillator consists of a  0.020 kg mass on a spring. Its frequency is 1.50 Hz and the mass has a speed of 0.360 m/s as it passes through the equilibrium position x = 0.
(a) ( 11 POINTS) What’s the value of the quantum number n  for its energy level?
(b) ( 11 POINTS) What’s the difference in energy E_n and _En+1 ? Is the difference detectable? Explain.

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6. ( 22 POINTS) In section 40.1 it was shown that for the ground level of a harmonic oscillator ,
 .
Using the same method as in sec. 40.1, show that for general n,