Physics 214b-2008     Practice Questions for Final Exam

IMPORTANT:  The actual exam will be truly cumulative, including questions specifically on the material for exams 1 and 2.  However, the practice questions below cover only the material we have dealt with since exam 2.

You should be able to do each of these problems (including all sub-parts) in about 25 minutes. (Exception: you should be able to do problem 6 in about 60 minutes.)  If it takes you longer, you have not mastered the associated material thoroughly enough.  Note:  Of course, these problems don’t cover exactly the same topics as the questions on the real exam.  Therefore, doing well on these practice problems does not insure that you are adequately prepared for the exam.  However, doing poorly on these problems does mean that you need to study more.

 

Some integrals you may or may not need:

                                

         

                  

                                  

For the following, the parameter a is assumed >0:

                  

 

                   

 

Normalized spherical harmonics:

              

 

 

Text Box: Normalized radial functions:

 

 

 

k = 8.999 Nm2/C2         eo = 8.85-12 C2/Nm2 = 1/(4pk)             Charge of electron = -e, where e = 1.60-19 C

 

c = 3.008 m/s                mo = 4p-7 N/A2                                    Mass of electron = me = 9.11-31 kg

                                                                                                           

h = 6.63-34 J s                = 1.05-34 J s                                    Mass of proton = mp = 1.67-27 kg

 

kB = 1.38-23 J/K            NA = 6.022 atoms/mol                          = 5.29-11 m

           


 

 

1.  (Adapted from Townsend 6.22).  A particle of mass m is trapped in an infinite spherical well with

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Let R(r) be the radial part of the energy eigenfunction.  As for the Hydrogen atom, it is helpful to make the change of variables .  Since we don’t want the wavefunction to diverge anywhere, we must have R(r) finite everywhere.  This means that we must have .  Find the normalized groundstate energy eigenfunction for this potential, and the corresponding energy.

 

2.  Townsend problem 6.17, parts a-d only.             

 

3.  Calculate the expectation value for the potential energy for an electron in the  state of a Li++ ion.  (Li stands for Lithium; the Li++ ion has 3 protons and 4 neutrons in the nucleus, surrounded by a single electron.)  Hint: You should find that the expectation value of the potential energy is twice the total energy.

 

 

4.  An electron in a hydrogen atom is in the spin state , i.e. “spin up” for the z-axis.  Calculate  (the expectation value for the x-component of the spin angular momentum) and  (the expectation value for the  square of the x-component of the spin angular momentum). 

 

 

5.  There is no practice problem 5.  But problem 6 makes up for this omission, since it’s twice as long as a typical practice problem.