statistical mechanics

Question 1

Definition of Temperature

Answer 1

1/KT =(dlng(E)/de)

where K : boltzmanns constnant converts j to k

E: energy of microsttes
g(E) :density of microstates, g(E) e = no. of microstates between E and dE

Question 2

PEEP

Answer 2

principle of equal equilibrium probability

For isolated system exploring W microstates with the same energy, each is equally probable

Question 3

Mean energy of a system equation

Answer 3

mean E = sum (EiPi)

Question 4

Energy when T is small

Answer 4

when T –> 0 E—> 0 at low T system is frozen in ground state which has energy =0

Question 5

Energy when T is large

Answer 5

when T–> infinity E–>sum of states/number of states

As high T all states have equal boltzmann factors so all equally likely and E is just 1/no. of staates (sum of states)

Question 6

Gibbs expression for the entropy of a general system

Answer 6

S= -K sumPi (ln) pi
where
pi is the probability that the system is in microstate i
K is the boltzman constant

Question 7

Entropy of isolated system

Answer 7

by PEEP
pi=1/w for each of w states
0 fo other states

S= -kw(1/w)ln(1/w) = Klnw

Question 8

Meaning of entropy (based on Gibbs and Boltzmann expression)

Answer 8

s measures (logorithmically) the no. of microstates being explored (klnW) are, more generally (gibbs) the extent to which many microstates are being explored with significant probability.

Question 9

S at low temps

Answer 9

by third law of thermodynamics s–>0 most real systems have a unique ground state, into which the system will be ‘frozen’ (boltzmann) at low T, w=1

therefore : S=KlnW=0

Question 10

pressure of a system is given by…?

Answer 10

the thermodynamic relation

P=-(df/dv)T,N

Question 11

pressure of ideal gas

Answer 11

PV=NRT

Question 12

relate pressure of a system to pressure of a ideal gas

Answer 12

P=-(df/dv)T,N =+NKT(1/V) –> PV=NKT

Question 13

Thermodynamic expression for chemical potential

Answer 13

mu = (df/dN)T,V

Question 14

chemical potential of ideal gas

Answer 14

mu = KT ln (N/V [2πhbar^2/mkT]^3/2)

Question 15

State the Pauli exclusion principle which applies to identical fermions

Answer 15

Spin S= integers are bosons, integer +1/2 is a fermion, you can’t put more than 1 fermion in each sinlge particle state

Question 16

fermi dirac equation

Answer 16

f(epsilon) = 1/ e^(epsilon-mu)/KT + 1

Question 17

what is fermi dirac

Answer 17

The Fermi-Dirac distribution that gives the mean occupation f(c) of a single-particle state of energy ε, in a system of identical non-interacting fermions at temperature T

Question 18

micro state

Answer 18

quantum state of the entire system. isolated system has fixed energy so only has access to those microstates that have same energy as one another.

Question 19

g(E) density of microstates in energy

Answer 19

g(E)dE= no. of microstates in range E–> E+dE if E is fixed to within deltaE, W=gdeltaE but for large systems g approx 10^10^23 where as deltaE approaches 10^-20 -10^-1 J so lnw approx lng as -20 –1 are negligable by comparison with 10^23

Question 20

a physical system that has g(E)=AE^n density of microstates

Answer 20

A system of n harmonics oscillators has, at high temp (KT»hbarW) on energy nKT (classically equipartition) and hence has g(E) of the form given (at high energies)

Question 21

partition function

Answer 21

Z = sum j exp (-Ei/KT)

Ei : energy of microstates

Question 22

boltmann function

Answer 22

Pi = exp(-Ei/KT)/Z

Question 23

no. of q points per unit vol of Q space (sphere)

Answer 23

(L/π)^3)^-1 = V/π^3.

becausse q points form a cubic lattice with spacing π/L

therefore no. points per unit vol of q-space is
((π/L)^3)^-1 = V/π^3. where V=L^3

Question 24

energy density in q space, sphere ignoring zero point

Answer 24

no. of modes in +ive octant of sphere radius q thickness dq

vol of +ive octant X no of q puv X no. of q modes X energy per mode

Question 25

vol of +ive octant

Answer 25

4piq^2/8 dq

Question 26

no of q per unit vol of q modes

Answer 26

v/pi^3

Question 27

graph epsilon(omega )omega for different temps

Answer 27

at low T, epsilon(omega) is exponentially small throughout visible range

At high T, epsilon(omega is large at red end of visible range but falls off

Question 28

what is a boson

Answer 28

has a interger of spin and certain atoms and photons

Question 29

what is a fermion

Answer 29

integer +1/2 of spin electron and proton

Question 30

For a system of many non-interacting identical particles, state how a fundamental principle of quantum mechanics, depending on whether the particles are bosons or fermions, affects the possible values of the number of particles in a given single-particle quantum state.

Answer 30

Pauli exclusion principle

only one or zero fermions in each single particle state, no restrictions for bosons so 0,1,2,3…

Question 31

mu in the gibbs disrtibution

Answer 31

mu: chemical potential ith microstate in which energy is Ei

Question 32

heat resevoir

Answer 32

large system (large heat capacity) whose temperature changes negligibly when energy on the energy scale ofout system is added/subtracted. allowed to exchange energy with our system

Question 33

microstate

Answer 33

quantum state of our entire system

Question 34

pi

Answer 34

is the probability that the system is in a microstate i