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

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1 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 problem, write down all relevant equations and explain how you will approach the solution Show steps clearly Note: (c = m/s, h=666x10-34 J s = 4136x10-15 ev s : m e=9109x10-31 kg : m p=1673x10-7 kg: m n=1649x10-7 kg : k=138x10-3 J/K = 86x10-5 ev/k) 1 You have two lasers one red and one purple Each laser has a power of 5 mw The red laser produces light at 700 nm The purple laser produces light at 400 nm (a) Calculate the energy per photon emitted by each laser Which color has the higher energy? (b) How many photons are emitted per second by each laser? (c) Which laser produces more photons per unit time? (d) You shine both lasers at a surface of Calcium (which has a photoelectric work function of 87 ev) Which laser(s) would be expected to produce photoelectrons (red, purple, both or neither)? (e) What would be the kinetic energy and velocity of the resulting electrons from the previous question (in cases where photoelectrons result)?

2 In 193, a French Physicist Louis de Broglie, hypothesized that matter (particles) held properties of waves Within a few years, this hypothesis was tested by several experimental groups and upheld confirming the wave-particle duality of matter (a) If an electron and proton are moving at identical (sub-relativistic) speeds, which has the longer de Broglie wavelength? Calculate the ratio of the two de Broglie wavelengths (λe/λp) (b) If the two particles were moving at identical relativistic speeds (eg 098 c) would the answer to part (A) differ? How so or why not? (c) An electron with mass me and charge qe is accelerated through an electrostatic potential difference of V Derive the relationship for the electron s de Broglie wavelength expressed in terms of me, qe and V

3 3 Swiss schoolteacher Johann Balmer found in 1885 an empirical formula: R, n 3,4,5,6 4 n that explained the wavelengths of all four spectral lines of hydrogen in the visible part of the electromagnetic spectrum, ie 654 nm, 486 nm, 434 nm and 410 nm, where R is the Rydberg constant Mass of hydrogen nucleus: x 10-7 kg, Mass of deuterium nucleus: x 10-7 kg, (a) Is this constant R the same value for hydrogen, deuterium and tritium atoms when the reduced mass of the electron is used in the Bohr formula for the energy levels? (b) What is the wavelength difference in the red spectral lines for hydrogen and deuterium? (c) Can these different isotopes of hydrogen be resolved by experimental optical spectroscopy? If so, what would the necessary resolution be to do so; if not, why not?

4 4 The grand canonical ensemble describes the possible states of an isolated system that is in a thermal and chemical equilibrium with a reservoir The ensemble assigns a probability P to each distinct state, such that : 1 i Ei Pi Z e, where β=1/kt, E is the energy of the state, N is the number of particles in the state and Z is the grand partition function: Ni Ei Z e i Consider a lithium (Li) atom with an ionization energy I Ionization energy is the energy required to remove an electron from a neutral atom leaving behind a positively charged ion: Li Li e (a) Suppose the atom is in the presence of a dense electron gas whose chemical potential is μ For simplicity, we ignore the excited electronic states of the lithium atom and assume that both the ionized state and the ground level of the neutral lithium atom are nondegenerate For consistency we can set the energies f the neutral and ionized forms as follows: The e- bound state (Li): E = -I The ionized state (Li+): E = 0 N Show that the grand partition function Z for this system is: z1e I (b) The photosphere of the Sun is at a temperature of 6000 K and has an electron density of 60x10 19 m -3 The ionization energy of lithium is I=54 ev What fraction of lithium atoms in the suns photosphere would you expect to be in the neutral form (Li)? Treat the electrons as if they were a monoatomic ideal gas, such that: 3 kt ln V mkt N h

5 5 In 1907, Albert Einstein proposed a simplistic model of a solid employing the atomic model of matter and Planck s quantized hypothesis for energy The Einstein model of a solid imagined a solid composed of atoms in a lattice connected by independent quantum mechanical oscillators (ie, bonds) Each oscillator is restricted to integer quanta of energy (n=0, 1,, 3 ) Following this derivation, the number of ways q quanta of energy can be distributed among N oscillators is given by multiplicity : ( qn 1)! ( Nq, ) q!( N 1)! (note: aka (q+n-1) choose q ) Consider two Einstein solids A & B that are allowed to exchange 0 quanta of energy (ie, q total=0) Each solid has 10 oscillators (ie, N A=N B=10) (a) What is the total multiplicity of the combined two-solid system over long periods of time (ie, what is the total number of microstates available)? (b) What is the probability that the system is in a state such that all the 0 quanta of energy is in solid A? (c) What is the probability that the system is in a state such that the 0 quanta of energy is shared equally between the two solids A and B? (d) Thinking about the broader implications of this thought experiment, what do the results of parts (b) and (c) indicate about the behavior of two macroscopic solid systems (N>10 3 ) exchanging large quantities of energy?