Statistical Mechanics Newton's Laws in Principle Tell Us How Anything Works

Statistical Mechanics Newton's laws in principle tell us how anything works

But in a system with many particles, the actual computations can become complicated.

We will therefore be happy to get some 'average' or approximate behavior of the system that will be useful for practical purposes

Describing this average dynamics is the goal of thermodynamics A microscopic level fundamental understanding of thermodynamics was found later, and this field is called statistical mechanics Average energy

Suppose we have probability p i to have energy E i

Then the average energy is

P E E = i i i h i P P i i P Continuous variables: Probability for any given value is zero, but we have a probability for a range

P (v) dv v v v + dv P (v)

Area under curve P (v)dv =1 Z

v Sometimes we may give the number of particles per unit velocity range N(v)dv particles have momentum between v and v + dv

N(v)

Area under curve N(v)dv = Ntotal Z

57. Which of the following statements is (are) true for a Maxwell-Boltzmann description of an ideal gas 57. Which of the following statements is (are) true for of atoms in equilibriumGREPracticeBook at temperature T ? a Maxwell-Boltzmann description of an ideal gas of atomsI. The inaverage equilibrium velocity at temperatureof the atoms Tis ?zero. II. The distribution of the speeds of the atoms I. The average velocity of the atoms is zero. has a maximum at u = 0. II. The distribution of the speeds of the atoms III. The probability of finding an atom with zero has a maximum at u = 0. kinetic energy is zero. III. The probability of finding an atom with zero (A) Ikinetic only energy is zero. 56. A sample of N molecules has the distribution of (B) II only (A) I only 56. Aspeeds sample shown of N in moleculesthe figure hasabove. the distributionPduu is an of (C) I and II (B) II only estimate of the number of molecules with speeds (D) I and III speeds shown in the figure above. P u du is an (C) I and II (E) II and III estimatebetween ofu theand number u + du of, and molecules this number with isspeeds (D) I and III

nonzero only up to 3u , where u is constant. (E) II and III between u and u + du0 , and this number0 is 58. A monatomic ideal gas changes from an initial

Whichnonzero of only the followingup to 3u ,gives where the uvalue is constant. of a ? 0 0 58. Astate monatomic (Pi , Vi , idealTi, ngasi) tochanges a final fromstate an (P finitial , Vf , Which of theN following gives the value of a ? Tf , nf ), where Pi < Pf , Vi = Vf , Ti < Tf and (A) a state (Pi , Vi , Ti, ni) to a final state (Pf , Vf , 3u0 N nTi ,= nnf ),. Which where ofP the < Pfollowing, V = V gives, T < the T change and (A) a N f f i f i f i f (B) a 3u 0 nin =entropy n . Which of the of gas? the following gives the change 2u i f N0 in entropy3 of theTf gas? (B) a N (A) nR ln (C) a 2 u0 2 T u 3 Tif N0 (A) nR ln (C) a 2 T u3 N 3 Ti (D) a 0 (B) nR ln 2u 2 Tf 3 N0 3 Ti (D) a (B) nR ln 2 u 2 Tf (E) aN 0 5 f (C) nR ln 2 TTi (E) a N 5 f (C) nR ln 25 Ti (D) nR ln 2 Tf 5 Ti (D) nR ln 2 T (E) 0 f

(E) 0

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nonzero only up to 3u , where u is constant. (E) II and III between u and u + du0 , and this number0 is 58. A monatomic ideal gas changes from an initial

(E) 0

GO ON TO THE NEXT PAGE. -40- 46 GO ON TO THE NEXT PAGE. -40- Hot bath, temperature T The particle can get energy from the hot walls when it touches them

Basic law of Statistical Mechanics: Probability for the particle to have energy E

E P e kT /

23 1 Here k =1.38 10 JK is the Boltzmann constant ⇥

GRE0177 74. The Lagrangian for a mechanical system is 77. An ensemble of systems is in thermal equilibrium with a reservoir for which kT = 0.025 eV. 24 Laqbq=+ , State A has an energy that is 0.1 eV above that of state B. If it is assumed the systems obey where q is a generalized coordinate and a Maxwell-Boltzmann statistics and that the and b are constants. The equation of motion degeneracies of the two states are the same, then for this system is the ratio of the number of systems in state A to b the number in state B is (A) q = q 2 a (A) e+4 2b +0.25 (B) q = q 3 (B) e a (C) 1 2b (C) q =- q 3 (D) e-0.25 a (E) e-4 2b (D) q =+ q 3 a 78. The muon decays with a characteristic lifetime b of about 10-6 second into an electron, a muon (E) q = q 3 a neutrino, and an electron antineutrino. The muon is forbidden from decaying into an electron and Fax′ I L 12// 32 0OFax I just a single neutrino by the law of conservation of Ga′ J = M− 32// 12 0PGa J G y J M PG y J (A) charge Haz′ K NM 001QPH az K (B) mass (C) energy and momentum 75. The matrix shown above transforms the com- (D) baryon number ponents of a vector in one coordinate frame S to (E) lepton number the components of the same vector in a second coordinate frame S′. This matrix represents a 79. A particle leaving a cyclotron has a total rotation of the reference frame S by relativistic energy of 10 GeV and a relativistic momentum of 8 GeV/c. What is the rest mass (A) 30° clockwise about the x-axis of this particle? (B) 30° counterclockwise about the z-axis 2 (C) 45° clockwise about the z-axis (A) 0.25 GeV/c 2 (D) 60° clockwise about the y-axis (B) 1.20 GeV/c 2 (E) 60° counterclockwise about the z-axis (C) 2.00 GeV/c (D) 6.00 GeV/c2 76. The mean kinetic energy of the conduction (E) 16.0 GeV/c2 electrons in metals is ordinarily much higher than kT because (A) electrons have many more degrees of freedom than atoms do (B) the electrons and the lattice are not in thermal equilibrium (C) the electrons form a degenerate Fermi gas (D) electrons in metals are highly relativistic (E) electrons interact strongly with phonons

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Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. 50 E e kT

E = kT E

1 For each degree of freedom, E = kT h i 2

A state with lower energy is more likely ... First consider the 1-dimensional problem

E v P e kT /

1 E = mv2 2

mv2 P (v) e 2kT /

The most likely speed is v =0 Hot bath, temperature T 2 2 (vx+vy ) P e 2kT /

Many velocities ~ v correspond v vy to the same speed

Higher speeds are suppressed because they have more energy vx Higher speeds are enhanced because v =0 there are more possible velocities | | for higher speeds

57. Which of the following statements is (are) true for a Maxwell-Boltzmann description of an ideal gas 57. Which of the following statements is (are) true for GREPracticeBook of atoms in equilibrium at temperature T ? a Maxwell-Boltzmann description of an ideal gas of atomsI. The inaverage equilibrium velocity at temperatureof the atoms Tis ?zero. II. The distribution of the speeds of the atoms I. The average velocity of the atoms is zero. has a maximum at u = 0. II. The distribution of the speeds of the atoms III. The probability of finding an atom with zero has a maximum at u = 0. kinetic energy is zero. III. The probability of finding an atom with zero (A) Ikinetic only energy is zero. 56. A sample of N molecules has the distribution of (B) II only (A) I only 56. Aspeeds sample shown of N in moleculesthe figure hasabove. the distributionPduu is an of (C) I and II (B) II only estimate of the number of molecules with speeds (D) I and III speeds shown in the figure above. P u du is an (C) I and II (E) II and III estimatebetween ofu theand number u + du of, and molecules this number with isspeeds (D) I and III nonzero only up to 3u , where u is constant. (E) II and III between u and u + du0 , and this number0 is 58. A monatomic ideal gas changes from an initial

(E) 0

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(E) 0

GO ON TO THE NEXT PAGE. -40- 46 GO ON TO THE NEXT PAGE. -40- 1 For each degree of freedom, E = kT h i 2

1 1 v E = mv2 = kT h i h2 i 2

Hot bath, temperature T 1 1 E = mv2 + Kx2 h i h2 i h2 i

K 1 1 = kT + kT = kT 2 2 m Hot bath, temperature T

Particle in 3-d 1 1 1 3 E = mv2 + mv2 + mv2 = kT h i h2 x 2 y 2 z i 2

GRE0177 5. A three-dimensional harmonic oscillator is in thermal equilibrium with a temperature reservoir at temperature T. The average total energy of the oscillator is 1 (A) kT 2

(B) kT 3 7. Two long, identical bar magnets are placed under (C) kT 2 a horizontal piece of paper, as shown in the figure above. The paper is covered with iron filings. (D) 3kT When the two north poles are a small distance apart and touching the paper, the iron filings (E) 6kT move into a pattern that shows the magnetic field lines. Which of the following best illustrates the 6. An ideal monatomic gas expands quasi-statically pattern that results? to twice its volume. If the process is isothermal, (A) the work done by the gas is Wi . If the process is adiabatic, the work done by the gas is Wa . Which of the following is true? (A) W = W i a (B) (B) 0 = Wi < Wa (C) 0 < Wi < Wa (D) 0 = Wa < Wi (E) 0 < W < W a i (C)

(D)

(E)

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GRE0177 5. A three-dimensional harmonic oscillator is in thermal equilibrium with a temperature reservoir at temperature T. The average total energy of the oscillator is 1 (A) kT 2

(D)

(E)

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47. A sealed and thermally insulated container of total 50. At 20∞C, a pipe open at both ends resonates at a volume V is divided into two equal volumes by frequency of 440 hertz. At what frequency does an impermeable wall. The left half of the con- the same pipe resonate on a particularly cold day tainer is initially occupied by n moles of an ideal when the speed of sound is 3 percent lower than gas at temperature T. Which of the following it would be at 20∞C ? gives the change in entropy of the system when the wall is suddenly removed and the gas expands (A) 414 Hz to fill the entire volume? (B) 427 Hz (C) 433 Hz (A) 2nR ln2 (D) 440 Hz (B) nR ln2 (E) 453 Hz 1 (C) nR ln2 2 51. Unpolarized light of intensity I0 is incident on a (D) -nR ln2 series of three polarizing filters. The axis of the second filter is oriented at 45° to that of the first (E) -2nR ln2 filter, while the axis of the third filter is oriented

GRE0177 48. A gaseous mixture of O (molecular mass 32 u) at 90° to that of the first filter. What is the intensity 2 of the light transmitted through the third filter? and N2 (molecular mass 28 u) is maintained at (A) 0 urms ()N2 constant temperature. What is the ratio (B) I0/8 urms()O2 of the root-mean-square speeds of the molecules? (C) I0/4 7 (D) I0/2 (A) 8 (E) I / 2 0 7 (B) 8

8 (C) 7

8 2 (D) F I 7 H K 8 (E) lnF I H 7K

49. In a Maxwell-Boltzmann system with two states of energies ⑀ and 2⑀, respectively, and a degeneracy of 2 for each state, the partition function is 52. The conventional unit cell of a body-centered cubic Bravais lattice is shown in the figure above. (A) e-⑀ /kT The conventional cell has volume a3 . What is (B) 2e-2⑀ /kT the volume of the primitive unit cell? -3⑀ /kT (C) 2e (A) a3/8 (D) e-⑀ /kT + e-2⑀ /kT (B) a3/4 -⑀ /kT -2⑀ /kT (C) a3/2 (E) 2[e + e ] (D) a3 (E) 2a3

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8 (C) 7

8 2 (D) F I 7 H K 8 (E) lnF I H 7K

Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. 38 Avagadro number N =6 10 23 (1 mole) ⇥ Avagadro number of H atoms weigh 1 gram

Avagadro number of He atoms weigh 4 grams

Molar mass of H = 1gm = mass of H atom times N Molar mass of He = 4 gm = mass of He atom times N

1 We define kN = R R =8.31 JK /mole

1 One degree of freedom E = kT h i 2 1 1 N degrees of freedom E = NkT = RT h i 2 2

9. The root-mean-square speed of molecules in an 12. A single-electron atom has the electron in the GREPracticeBook ideal gas of molar mass M at temperature T is 2 state. The number of allowed values of 9. The root-mean-square speed of molecules in an 12. A single-electron atom has the electron in the the quantum number m is (A)ideal gas 0 of molar mass M at temperature T is 2 state. The number of allowed values of

(A) 0 the(A) quantum 1 number m is RT (B) (B) 2 M (A) 1 RT (C) 3 (B) (D)(B) 42 M RT (E)(C) 53 (C) M (D) 4 RT (E) 5 (C) 13. A particle of mass m is confined inside a M 3RT one-dimensional box (infinite square well) of (D) 13. A particle of mass m is confined inside a M length a. The particle’s ground state energy is 3RT one-dimensional box (infinite square well) of (D) which of the following? M length a. The particle’s ground state energy is 3RT (E) (A)which of the following? M 3RT 8ma (E) (A) 2 10. Light ofM variable frequency shines on the metal (B) 8ma 2 surface of a photoelectric tube. Einstein’s theory 8ma 2 10. Light of variable frequency shines on the metal (B) 2 of the photoelectric effect predicts that the 2 (C) 8ma surface of a photoelectric tube. Einstein’s theory 2 of(A) the work photoelectric function of eff theect metal predicts is proportional that the ma 2 (C) 22 to the frequency 2 (A) work function of the metal is proportional (D) ma (B) work function of the metal is proportional 2 2ma22 to the frequencywavelength (D) 22 (B) work function of the metal is proportional a 2 (C) current in the tube is a linear function of (E) 2ma to the wavelength 2 the wavelength 22 2mc a (C) current in the tube is a linear function of (E) (D) potential difference necessary to stop the 2 emittedthe wavelength electrons is a linear function of 14. The Planck2mc length is the only combination of (D) potentialthe frequency difference above necessary the threshold to stop frequency the the factors G (Newton’s gravitational constant), (E) potentialemitted differenceelectrons is necessary a linear function to stop the of 14. The (Planck’s Planck length constant is the/ 2 only), and combination c (the speed of of the frequency above the threshold frequency emitted electrons is equal to the work thelight) factors that has G units(Newton’s of length. gravitational Which of constant), the (E) potential difference necessary to stop the (Planck’s constant / 2 ), and c (the speed of function following gives the Planck length? emitted electrons is equal to the work light) that has units of length. Which of the function 1/2 11. Characteristic X rays, appearing as sharp lines followingG gives the Planck length? (A) on a continuous background, are produced 3 1/ 2 11. Characteristic X rays, appearing as sharp lines cG when high-energy electrons bombard a metal (A) on a continuous background, are produced G3 target. Which of the following processes results (B) c when high-energy electrons bombard a metal 3 in the characteristic X rays? cG target. Which of the following processes results (B) G32 in(A) the Electrons characteristic producing X rays? erenkov radiation (C) c c (B) Electrons colliding with phonons in the metal G2 (A) Electrons producing erenkov radiation (C) (C) Electrons filling inner shell vacancies that are c (B) Electronscreated in colliding the metal with atoms phonons in the metal (D) cG (D)(C) Electrons fillingcombining inner with shell protons vacancies to form that are created in the metal atoms (D) cGG neutrons (E) (D) Electrons combining with protons to form c (E) Electrons undergoing Coulomb scattering G neutrons (E) with nuclei c (E) Electrons undergoing Coulomb scattering with nuclei

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GO ON TO THE NEXT PAGE. -8- 14 GO ON TO THE NEXT PAGE. -8- Blackbody radiation is made of photons py

E P e kT ⇠

p =0 | |

3 Wien displacement constant W =2.9 10 mK ⇥ W Wien displacement law: = Tpeak peak Double T double peak E halve wavelength

GRE0177

F hv I 2 e hv/ kT CkN= 3 A H kT K (e hv/ kT - 1)2

65. Einstein’s formula for the molar heat capacity C of solids is given above. At high temperatures, C approaches which of the following? (A) 0 hv (B) 3kN F I A H kT K

(D) 3kNA 63. The distribution of relative intensity I ()l of (E) NhvA blackbody radiation from a solid object versus the wavelength l is shown in the figure above. 66. A sample of radioactive nuclei of a certain element can decay only by g -emission and b -emission. If If the Wien displacement law constant is -3 the half-life for g -emission is 24 minutes and that mؒK, what is the approximate for b -emission is 36 minutes, the half-life for the 10 ¥ 2.9 temperature of the object? sample is (A) 10 K (A) 30 minutes (B) 50 K (B) 24 minutes (C) 250 K (C) 20.8 minutes (D) 1,500 K (D) 14.4 minutes (E) 6,250 K (E) 6 minutes

64. Electromagnetic radiation provides a means to 67. The 238U nucleus has a binding energy of about probe aspects of the physical universe. Which 7.6 MeV per nucleon. If the nucleus were to of the following statements regarding radiation fission into two equal fragments, each would have spectra is NOT correct? a kinetic energy of just over 100 MeV. From this, (A) Lines in the infrared, visible, and ultraviolet it can be concluded that regions of the spectrum reveal primarily the (A) 238U cannot fission spontaneously nuclear structure of the sample. (B) 238U has a large neutron excess (B) The wavelengths identified in an absorption (C) nuclei near A = 120 have masses greater than spectrum of an element are among those in 238 its emission spectrum. half that of U (C) Absorption spectra can be used to determine (D) nuclei near A = 120 must be bound by about which elements are present in distant stars. 6.7 MeV/nucleon (D) Spectral analysis can be used to identify the (E) nuclei near A = 120 must be bound by about composition of galactic dust. 8.5 MeV/nucleon (E) Band spectra are due to molecules.

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GRE0177

F hv I 2 e hv/ kT CkN= 3 A H kT K (e hv/ kT - 1)2

65. Einstein’s formula for the molar heat capacity C of solids is given above. At high temperatures, C approaches which of the following? (A) 0 hv (B) 3kN F I A H kT K

Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. 46 Approximations

E e kT

E = kT E

For x 1 ⌧ x e 1 x + ... ⇡

E E Low energy e kT 1 ⇡ kT

GRE0177

F hv I 2 e hv/ kT CkN= 3 A H kT K (e hv/ kT - 1)2

65. Einstein’s formula for the molar heat capacity C of solids is given above. At high temperatures, C approaches which of the following? (A) 0 hv (B) 3kN F I A H kT K

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GRE0177

F hv I 2 e hv/ kT CkN= 3 A H kT K (e hv/ kT - 1)2

65. Einstein’s formula for the molar heat capacity C of solids is given above. At high temperatures, C approaches which of the following? (A) 0 hv (B) 3kN F I A H kT K

Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. 46 The partition function The average energy is

i Pi Ei Ei E = with P e kT h i P i P i i / P To compute this, we define the Partition Function

Ei Z = e kT i X 1 We write = kT

E @Z Ei This gives Z = e i = e E @ i i i X X 1 @Z e Ei E = i i = E Z @ e Ei h i P i P

GRE0177 98. Suppose that a system in quantum state i has energy Ei . In thermal equilibrium, the expression

-EkTi / Â Eei i -EkT/ Â e i i represents which of the following? (A) The average energy of the system (B) The partition function (C) Unity (D) The probability to find the system with 97. A beam of light has a small wavelength spread energy Ei d l about a central wavelength l . The beam (E) The entropy of the system travels in vacuum until it enters a glass plate at an angle q relative to the normal to the plate, as 99. A photon strikes an electron of mass m that is shown in the figure above. The index of refraction initially at rest, creating an electron-positron pair. of the glass is given by n()l . The angular spread The photon is destroyed and the positron and two dq ¢ of the refracted beam is given by electrons move off at equal speeds along the initial direction of the photon. The energy of the 1 (A) dq¢ = dl photon was n 2 (A) mc dn()l (B) 2mc2 (B) dq¢ = dl dl (C) 3mc2 (D) 4mc2 1 dl (C) dq¢= dl (E) 5mc2 l dn sin q dl (D) dq¢ = sin q ¢ l tan q ¢ dn()l (E) dq¢ = dl n dl

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GRE0177 98. Suppose that a system in quantum state i has energy Ei . In thermal equilibrium, the expression

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48. A gaseous mixture of O (molecular mass 32 u) at 90° to that of the first filter. What is the intensity 2 of the light transmitted through the third filter? and N2 (molecular mass 28 u) is maintained at (A) 0 urms ()N2 constant temperature. What is the ratio (B) I0/8 urms()O2 of the root-mean-square speeds of the molecules? (C) I0/4 7 (D) I0/2 (A) 8 (E) I / 2 0 7 (B) 8

8 (C) 7

8 2 (D) F I 7 H K 8 (E) lnF I H 7K GRE0177 49. In a Maxwell-Boltzmann system with two states of energies ⑀ and 2⑀, respectively, and a degeneracy of 2 for each state, the partition function is 52. The conventional unit cell of a body-centered cubic Bravais lattice is shown in the figure above. (A) e-⑀ /kT The conventional cell has volume a3 . What is (B) 2e-2⑀ /kT the volume of the primitive unit cell? -3⑀ /kT (C) 2e (A) a3/8 (D) e-⑀ /kT + e-2⑀ /kT (B) a3/4 -⑀ /kT -2⑀ /kT (C) a3/2 (E) 2[e + e ] (D) a3 (E) 2a3

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8 (C) 7

Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. 38 An unusual situation

74. The Lagrangian for a mechanical system is 77. An ensemble of systems is in thermal equilibrium with a reservoir for which kT = 0.025 eV. 24 Laqbq=+ , State A has an energy that is 0.1 eV above that of state B. If it is assumed the systems obey where q is a generalized coordinate and a Maxwell-Boltzmann statistics and that the and b are constants. The equation of motion degeneracies of the two states are the same, then for this system is the ratio of the number of systems in state A to b the number in state B is (A) q = q 2 a (A) e+4 2b +0.25 (B) q = q 3 (B) e a (C) 1 2b (C) q =- q 3 (D) e-0.25 a (E) e-4 2b (D) q =+ q 3 a 78. The muon decays with a characteristic lifetime b of about 10-6 second into an electron, a muon (E) q = q 3 a neutrino, and an electron antineutrino. The muon is forbidden from decaying into an electron and Fax′ I L 12// 32 0OFax I just a single neutrino by the law of conservation of Ga′ J = M− 32// 12 0PGa J G y J M PG y J (A) charge Haz′ K NM 001QPH az K (B) mass (C) energy and momentum 75. The matrix shown above transforms the com- (D) baryon number ponents of a vector in one coordinate frame S to (E) lepton number the components of the same vector in a second coordinate frame S′. This matrix represents a 79. A particle leaving a cyclotron has a total rotation of the reference frame S by relativistic energy of 10 GeV and a relativistic momentum of 8 GeV/c. What is the rest mass (A) 30° clockwise about the x-axis of this particle? (B) 30° counterclockwise about the z-axis 2 (C) 45° clockwise about the z-axis (A) 0.25 GeV/c 2 (D) 60° clockwise about the y-axis (B) 1.20 GeV/c 2 (E) 60° counterclockwise about the z-axis (C) 2.00 GeV/c 2 (D)GRE0177 6.00 GeV/c 76. The mean kinetic energy of the conduction (E) 16.0 GeV/c2 electrons in metals is ordinarily much higher than kT because (A) electrons have many more degrees of freedom than atoms do (B) the electrons and the lattice are not in thermal equilibrium (C) the electrons form a degenerate Fermi gas (D) electrons in metals are highly relativistic (E) electrons interact strongly with phonons

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E kT ⇠

Unauthorized copying or reuse of px any part of this page is illegal. GO ON TO THE NEXT PAGE. 50 p =0 | | 1 Deriving E = kT h i 2 First consider the 1-dimensional problem

v E P e kT ⇠

1 E = mv2 2

mv2 P (v) e 2kT /

1 Adding over all possibilities : dv Z1 First consider the 1-dimensional problem p E E = i i i h i p P i i v P E p e kT / 1 E = mv2 2

mv2 p(v) e 2kT /

2 1 mv 1 2 2kT dv( 2 mv ) e 1 E = 1 = kT mv2 2 h i R 1 dv e 2kT 1 R Hot bath, temperature T

K E p e kT / m

1 1 mv2 Kx2 E = mv2 + Kx2 p e 2kT e 2kT 2 2 /

1 Adding over all possibilities of velocity : dv Z1 1 Adding over all possibilities of position : dx Z1 Hot bath, temperature T

K E p e kT / m

1 1 mv2 Kx2 E = mv2 + Kx2 p e 2kT e 2kT 2 2 /

Adding over all probabilities

mv2 Kx2 mv2 Kx2 dvdx e 2kT e 2kT = dv e 2kT dx e 2kT ZZ ✓Z ◆✓Z ◆ Hot bath, temperature T

mv2 Kx2 p e 2kT e 2kT K / 1 1 E = mv2 + Kx2 2 2 m

2 2 1 2 mv Kx 1 dv dx ( mv ) e 2kT e 2kT mv2 = 2 h2 i mv2 Kx2 R dv dx e 2kT e 2kT

2 2 R 1 mv Kx 2 2kT 2kT dv ( 2 mv ) e dx e = mv2 Kx2 R dv e 2kT R dx e 2kT 1 R R = kT 2 Hot bath, temperature T

mv2 Kx2 p e 2kT e 2kT K / 1 1 E = mv2 + Kx2 2 2 m

1 1 1 1 mv2 + Kx2 = kT + kT = kT h2 i h2 i 2 2