6-2. A quantum system has the following energy level diagram. Notice that the temperature is indicated

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1 Chapter 6 Concept Tests 6-1. In a gas of hydrogen atoms at room temperature, what is the ratio of atoms in the 1 st excited energy state (n=2) to atoms in the ground state(n=1). (Actually H forms H 2 molecules, but let s pretend.) P(n 2) P(n 1)? A) greater than 1 B) about 1 C) less than 1, but not extremely small, between.1 and.5 D) much less than 1, but not infinitesimal, between.1 and 1-6 E) infinitesimal, less than 1-1 Hint: kt room 1/4 ev, for H atom, E n = 13.6eV/n 2 Answer: infinitesimal. The ratio is 4 e - E/kT, where E is about 1 ev (energy difference between ground state and 1 st excited state in hydrogen) and kt room =.25 ev. So the ratio is about 4 e A quantum system has the following energy level diagram. Notice that the temperature is indicated g(e) E 6 3 kt i = Which is true? A) P(i=3) > P(i=2) B) P(i=3) < P(i=2) C) P(i=3) = P(i=2) Answer: P(i=3) = P(i=2) States with the same energy have the same probability, according to the Boltzmann relation.

2 A) P(i=5) > P(i=2) B) P(i=5) < P(i=2) C) P(i=5) = P(i=2) Answer: P(i=5) < P(i=2) States of higher energy are always less probable than states of lower energy. A) P(E= 1 ) > P(E= ) B) P(E= 1 ) < P(E= ) C) P(E= 1 ) = P(E= ) D) Can t tell! Answer: P(E= 1 ) > P(E= ) There are 4 states with E= 1 and only one state with E=. Also, kt is large compared to E = 1, so each of the 1 states is only slightly less probable than the state Does the partition function Z E i /kt e change with temperature? i A) Z increases as T increases B) Z decreases as T increases C) Z could increase, decrease, or stay constant, as T increases, depending on details of the spectrum of states. Answer: Yes, Z increases as T increases. Z is the number of thermally-active microstates.

3 6-4. A quantum system has the following energy level diagram. The ground state energy is set at zero. The first excited energy level has energy. Notice that the temperature is indicated. g(e) E kt i = Roughly, what is the value of Z, the partition function? A) 2.8 B) 3 C) 1 D) 2 E) impossible to tell without more information Answer: Roughly the partition function is equal to the number of thermally excited states, which is roughly the number of states with energies E < or = kt, which, in this case, is 1. This is true when the zero of energy is set to the ground state energy. (The exact answer, for the energy level diagram shown is Z = 7.6) Roughly, what is the probability that the system will be found in the ground state? A) 1 B) 1/3 C) 1/1 D) 1/1 Answer: 1/1. Roughly, each of the states with energy < kt has probability 1/Z. (The exact answer is.13.)

4 6-5. Consider the degeneracy g(e) of a large system, such as a mole of an ideal gas, or a kilogram of copper (modeled as an Einstein solid). How does g(e) depend on E? A) g(e) is a rapidly rising function of E B) g(e) is roughly independent of E C) g(e) decreases with increasing E Answer: g(e) is a very rapidly rising function of E 6-6.Consider a particle in a system with three evenly spaced non-degenerate energy levels, as seen in the figure at right. The probability that the particle is in the n th level is P(n). Is the ratio of the probabilities P3 P 2 than, or equal to the ratio of the probabilities greater than, less P 2 P1? n = 3 n = 2 n = 1.1 ev.5 ev. ev Energy A) greater B) less C) equal Answer: equal, the ratio of the probabilities depends only on the difference of the energies: P(3)/P(2) = exp[-(e 3 -E 2 )/kt]. Since (E 3 -E 2 ) = (E 2 -E 1 ), the ratios P(3)/P(2) and P(2)/P(1) are equal.

5 6-7.This is the energy-level spectrum of a A) 1D simple harmonic oscillator B) a particle in a 1D box C) a hydrogen atom D) a two-state paramagnet E) a quantum rotor E All levels Non-degenerate Answer: 1D simple harmonic oscillator: 6-8.The density of states (E) for a 1D harmonic oscillator is A) E B) E 2 C) E 1/2 D) E = constant E) None of these Answer: E = constant

6 6-9. A quantum system has the following energy level diagram. g(e) E kt 1 i = Which is larger, P(i=3) or P(E= 1 )? A) P(i=3) = P(E= 1 ) B) P(i=3) > P(E= 1 ) C) P(i=3) < P(E= 1 ) Answer: P(i=3) < P(E= 1 )

7 6-1. The number of states in the volume of the shell shown of thickness dn and radius n (assume n>> dn >>1 ) is n z n dn n y n x 1 A) 4 2 n dn B) n dn 83 C) something else Answer: n dn 6-11.A particle (moving in 1D) has position and momentum indicated. Where is the particle in phase space? x p (D) (E) (A) (B) (C) x Answer: (C)

8 6-12.A 1D simple harmonic oscillator is oscillating back and forth. As the system evolves in time, its phase space points traces out which trajectory? x p (A) (B) (C) clockwise x (D) CCW (E) More than one of these is valid. Answer: (C) The energy of a simple harmonic oscillator is E = (1/2)mv 2 + (1/2)kx 2 = p 2 /(2m) + (1/2)kx 2.The is the equation of an ellipse in x-p space. Think about why the direction of evolution of the system is clockwise in phase space.

9 6-13. The gaussian integral e x 2 dx is A) about 1 B) about 1 C) about.1 D) about 1 Answer: Answer: About 1. The exact answer is The curve exp(-x 2 ) has the max value 1 at x = and falls to 1/e at x = 1, so the area under this curve is roughly The distribution of particle kinetic energies in an ideal gas is given below. The rms average kinetic energy is most nearly A B.5.4 C D( x) E(units x of kt) 4 Answer: (C) The rms KE is (3/2)kT, according to the Equipartition Theorem.

10 6-15.What is the probability that a molecule in an ideal gas has an energy between and 3kT A) about.1 B) about.5 C) about.99 D)None of these.5.4 D(E) x) x E(units of kt) 4 Answer: About How would you compute the average speed (mean speed) of a molecule in an ideal gas? D(v) dv is the probability that the speed of a molecule is between v and v+dv. A) v D(v)dv B) 3kT m C) Answers A and B give the same answer Answer (B) is incorrect because it is the root-mean-square average, which is not the same as (A). The root-mean-square-average is 2 v D(v)dv

11 6-17.Suppose a system is put in thermal contact with heat reservoir at temperature T and the system is not, initially, in thermal equilibrium. True(A) or False(B): as the system moves toward thermal equilibrium, its entropy goes to a maximum. Answer: False. The entropy of a system that is NOT isolated can go down. For instance, if you bring the system in contact with another system at lower temperature, its temperature and entropy will decrease. True(A) or False(B): as the system moves toward thermal equilibrium, the entropy of the Universe = (system+reservoir) goes to a maximum. Answer: True 6-18.True(A) or False(B): the entropy of a system in thermal contact with a heat bath cannot decrease. Answer: False True(A) or False(B): the entropy of an isolated system cannot decrease. Answer: True

12 6-2.A ball rolls back and forth in a valley and eventually comes to rest at the bottom of the valley. As the ball rolled to a stop, the Helmholtz free energy F = U TS of the ball A) increased B) decreased C) remained constant D) impossible to tell without more information Answer: decreased. When the entropy of the Universe (ball+valley+earth) is maximized, the Helmholtz Free energy of the system (the ball) is minimized A quantum system has non-degenerate energy levels described by E(n) = C n 2 where C is a constant and n = 1, 2,... This is the energy-level spectrum of a A) 1D simple harmonic oscillator B) a particle in a 1D box C) a hydrogen atom D) a two-state paramagnet E) a quantum rotor Answer: a particle in a 1D box

13 6-22. Consider the function Ae -E/kT, A some constant. What s the area under the curve Ae E / kt de? (Pick closest answer.) A kt 2kT E A) AkT B) 3AkT C) (1/3) AkT D) All these are off by a factor of 2 or more Answer: AkT

14 6-23. Recall F F df dt dv T V V T and df S dt p dv True (A) or False (B) : S p V T T V Answer: True A quantum system has degenerate energy levels described by where C is a constant and n = +1, 2,... This is the energy-level spectrum of a E(n) C n 2 A) 1D simple harmonic oscillator B) a particle in a 1D box C) a hydrogen atom D) a two-state paramagnet E) a quantum rotor Answer: A hydrogen atom 6-25.True(A) or False(B): for x, y, 1, 2.. e e e x y x y x,y x y Answer: True As the temperature decreases, wavepackets describing particles in a gas tend to get A) larger B) smaller C) stay constant in size Answer: larger.

15 6-27.A 1D SHO quantum system is in the high-temperature regime kt>>. What is the thermal average energy of the system? E kt A) kt B) (3/2)kT C) (1/2)kT Answer: kt [2 quadratic terms in energy for 1D SHO, E = (1/2)mv 2 + (1/2)kx 2, and each term contributes (1/2)kT to average energy.] 6-28.An Einstein Solid consists of N independent 1D SHOs. In the hi-t limit, what is the heat capacity C = de/dt of this solid? A) kt B) zero C) NkT D) k E) something else Answer: something else. E = NkT, so C = de/dt = Nk.

16 6-29.A 1D SHO quantum system is in the low-temperature regime kt<<. The thermal average energy of the system is closest to : A) B) kt C) E kt Answer: zero. This is freeze-out.

Physics 176 Topics to Review For the Final Exam

Physics 176 Topics to Review For the Final Exam Professor Henry Greenside May, 011 Thermodynamic Concepts and Facts 1. Practical criteria for identifying when a macroscopic system is in thermodynamic equilibrium: