If you cannot solve the whole problem, write down all relevant equations and explain how you will approach the solution. Show steps clearly.

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1 Letter ID Comprehensive Exam Session I Modern Physics (Including Stat.Mech) Physics Department- Proctor: Dr. Chris Butenhoff (Sat. Jan. 11 th, 2014) (3 hours long 9:00 to 12:00 AM) If you cannot solve the whole problem, write down all relevant equations and explain how you will approach the solution. Show steps clearly. 1. Heisenberg uncertainty principle Before discovery of the neutron, one might have argued that a nucleus of atomic number Z and mass number A is made of A protons and A-Z electrons, i.e., just enough electrons such that the net nuclear charge is +Z. (A) A typical atomic nucleus has a diameter of about m. Use the uncertainty principle to estimate the uncertainties in velocity of an electron and a proton confined to a nuclear size potential-well. A proton s rest mass m p = kg = 938 MeV/c 2 ; An electron s rest mass m e = kg = MeV/c 2 ; Planck s constant h = J s = ev s (B) Since the uncertainty in momentum is a measure of its standard deviation, then the variance (standard deviation squared) in momentum is equal to average of the squares minus the square of the averages in momentum, i.e., p 2 p 2 p 2 From this, we can infer that the average momentum itself must be at least comparable in magnitude to its uncertainty. With this in mind, calculate a lower bound on the kinetic energy of a proton and an electron confined to a m nucleus. You can express your answer in Joules or electron volts (1 ev= J)

2 2. Schrodinger equation The quantum mechanical treatment of the hydrogen atom serves as a prototype for the behavior of all atoms and therefore the building blocks of matter. The two-body problem of the hydrogen atom has yielded several analytic solutions. (A) The Schrodinger equation for an electron in the ground-state of a hydrogen atom is spherically symmetric and can be written (r,, ) 1 a 0 3 / 2 e r / a 0 where r is the radius of the electron and a 0 is the Bohr radius (0.53 Å). The probability P(r)dr of finding the electron between r and r + dr from the nucleus is maximum at a 0. How much more likely is the electron in its ground-state to be at a distance a 0 rather than at the distance of a 0 /2? (B) For the hydrogen atom, the number of states of principal quantum number n associated with energy E n is equal to n 2 (i.e., the state s degeneracy). The energies of states E n are quantized for the hydrogen atom and have the solution: e 2 E n 8 0 a 0 n 2 n=1, 2, 3, where e is the electron charge ( C) and 0 is the permittivity of free space ( C 2 /N m 2 ). Using this relationship, the ground state for the hydrogen atom is calculated as 13.6 ev below the dissociated state of the hydrogen ion (1 ev = J). Find the energies of the first two excited energy levels of the hydrogen atom.

3 3. Radioactivity and decay The radioactive decay by an unstable nuclide is a stochastic or random event, the probability of which is. The activity A of a sample of radioactive material containing N radioactive nuclides is the number of decay events dn expected to occur in short time interval dt. Since the probability of decay of a single nuclide does not change in time, it follows that activity of a sample of radioactive material obeys a first order loss process i.e., A= N. (A) Tritium is an unstable isotope of hydrogen ( 3 H) with two neutrons and a half-life of years. Imagine a sample of tritium is made in a nuclear reactor and extracted and isolated from its sample matrix in Its initial activity A was measured using a Geiger-Muller counter to be 1.0 MBq, where 1 Bq (becquerel) = 1 decay per second. It is stored in a sealed glass ampoule. When cleaning out the radiation laboratory, the sample is discovered four decades later by a graduate student. What should its activity be in 2013? (B) The beta-decay of tritium releases 18.6 kev of energy in the process. An average of 5.7 kev of this is carried away by the beta particle. What is the expected velocity of an average beta particle as it is released in the decay? Are relativistic effects important to consider for this particle? Note: 1 ev= J ; a proton s rest mass m p = kg = 938.3MeV/c 2 ; a neutron s rest mass m n = kg = 939.6MeV/c 2 ; an electron s rest mass m e = kg = MeV

4 4. Compton Scattering (A) Why are x-ray photons used in the Compton experiment, rather than visible-light photons? Look at comparing the following 3 different photons to explain this: (1) very high energy -rays from cobalt, = Å (2) x-rays from molybdenum, = Å (3) green light from a mercury lamp, = 5461 Å (C) The so-called free electrons in carbon are actually electrons with a binding energy of about 4 ev. Why may this binding energy be ignored for x-rays with o =.712 Å?

5 5. Boltzmann statistics The population of excited energy states in the hydrogen atom depends on temperature. Following Boltzmann statistics, the canonical distribution describes probability P of occupying state i: P i Z 1 e E i / kt, where Z is the partition function or sum of the Boltzmann factors given by Z e E i / kt. i Determine the probability that a hydrogen atom is in the first excited state at room temperature 298 K. How does your answer change for temperatures similar to those experienced on the surface of the sun 5800 K? The Boltzmann constant is k = J/K = ev/k.

red laser produces light at 700 nm. The purple laser produces light at 400 nm.

red laser produces light at 700 nm. The purple laser produces light at 400 nm. Letter ID Comprehensive Exam Session I Modern Physics (Including StatMech) Physics Department- Proctor: Dr Chris Butenhoff (Sat Jan 9 th, 016) (3 hours long 9:00 to 1:00 AM) If you cannot solve the whole