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Matthew Schwartz Statistical Mechanics, Spring 2021
Lecture 10: Quantum Statistical Mechanics
1 Introduction
So far, we have been considering mostly classical statistical mechanics. In classical mechanics, energies are continuous, for example momenta ~p and hence kinetic energy E = p~
2 2 m can be any real number. Positions ~q can also take any continuous values in a volume V = L^3. Thus the number of states , the entropy S, the partition function Z, all involve
R
dqdp which is formally infinite. We regulated these infinities by artificially putting limits q and p on the positions and momenta that are allowed, and we found that physical predictions we made did not depend on p and q. Quantum statistical mechanics will allow us to remove these arbitrary cutoffs.
In quantum mechanics there are two important changes we must take into account. First, position, momentum and energy are in general quantized. We cannot specify the position and
momentum independently, since they are constrained by the uncertainty principle pq 6 ~ 2. For
example, the momenta modes of a particle in a box of size L are quantized as pn = (^2) L ~n, with n an integer. There is no extra degree of freedom associated with position. When summing over states, we sum over n = L p 2 ~n only, not over q. However, in the continuum limit we can replace the sum
over n with an integral, and suggestively write L =
R
dq. That is, the quantum sum gets replaced by an integral as X
n
Z
dn =
L
Z
dp =
Z
dqdp h
Recall that in classical statistical mechanics when we first computed in the microcanonical ensemble, we had to arbitrarily break up position and momentum into sizes q and p. Eq. (1) justifies the replacement pq! h that we previously inserted without proof in the Sackur- Tetrode equation and elsewhere.^1 Note that the integral measure has a factor of h not ~. We'll do conversions between sums and integrals like in Eq. (1) in more detail in deriving the density of states over the next several lectures.
Quantization is important when we are not in the continuum limit. One regime where quantum effects are clearly important is when the temperature is of order the lowest energy of the system, kBT " 0. For example, classically, each vibrational mode of a molecule contributes NkB to the heat capacity, but at low temperature, the measured heat capacity does not show this contribution. In quantum statistical mechanics, the vibrational contribution to the heat capacity is cut off at tem- peratures below kBT. ~! 0 , with! 0 the vibrational frequency, in agreement with measurements. We discussed this effect in Lecture 7.
The second quantum effect we need to account for has to do with identical particles. In quantum mechanics, identical particles of half-integer spin, like the electron, can never occupy the same state, by the Pauli exclusion principle. These particles obey Fermi-Dirac statistics. Identical particles of integer spin, like the photon, can occupy the same state but the overall wavefunction must be symmetric in the interchange of the two particles. Integer-spin particles obey Bose-Einstein statistics.
- Eq. (1) makes the appearance of dq seem like a trick. This is an artifact of using momentum eigenstates for our basis. If we used something symmetric in p and q like coherent states, the product dpdq would appear more naturally.
We discussed indistinguishability before in Lecture 6, in the context of the second law of thermodynamics and the Gibbs paradox. In that lecture, we found that we could decide if we want to treat particles as distinguishable or indistinguishable. If we want to treat the particles as distinguishable, then we must include the entropy increase from measuring the identity of all the particles to avoid a conflict with the second law of thermodynamics. Alternatively, if we never plan on actually distinguishing them, we can treat them as indistinguishable. We do
this by adding a factor of (^) N^1! to the number of states , i.e. instead of V N^ we take 1 N! V^
N (^) ; then there is automatically no conflict with the second law of thermodynamics. This
kind of classical indistinguishable-particle statistics, with the N! included, is known as Maxwell- Boltzmann statistics. Quantum identical particles is a stronger requirement, since it means the multiparticle wavefunction must be totally symmetric or totally antisymmetric. In a classical system, the states are continuous, so there is exactly zero chance of two particles being in the same state. Thus, the difference among Fermi-Dirac, Bose-Einstein and Maxwell-Boltzmann statistics arises entirely from situations were a single state has a nonzero change of being multiply occupied.
2 Bosons and fermions
The wavefunction (x; s) of an electron depends on the coordinate x and the spin s = 1
- When there are two electrons, the wavefunction depends on two coordinates and two spins. We can write it as (x 1 ; s 1 ; x 2 ; s 2 ). Thus j (x 1 ; s 1 ; x 2 ; s 2 )j^2 gives the probability of finding one electron at x 1 with spin s 1 and one electron at x 2 with spin s 2. Every electron is identical and therefore^2
j (x 1 ; s 1 ; x 2 ; s 2 )j^2 = j (x 2 ; s 2 ; x 1 ; s 1 )j^2 (2)
If the modulus of two complex numbers is the same, they can only differ by a phase. We thus have
(x 1 ; s 1 ; x 2 ; s 2 ) = (x 2 ; s 2 ; x 1 ; s 1 ) (3)
for some phase = ei^ with j j = 1. If we swap the particles back, we then find
(x 1 ; s 1 ; x 2 ; s 2 ) = (x 2 ; s 2 ; x 1 ; s 1 ) = ^2 (x 1 ; s 1 ; x 2 ; s 2 ) (4)
So that ^2 = 1 and = 1. We call particles with = 1 bosons and those with = − 1 fermions.
Particles come with different spins s = 0; 12 ; 1 ; 32 ; . A beautiful result from quantum field theory is that particles with integer spins, s = 0; 1 ; 2 ; are bosons and particles with half-integrer spins
s = 12 ; 32 ; are fermions. This correspondance is known as the spin-statistics theorem.
As an aside, I'll give a quick explanation of where the spin-statistics theorem comes from. First we need to know what spin means. The spin determines how the state j i changes under rotations. For example, say a particle is localized at x = 0, so (~x ) = hxj i = ^3 (x), and has spin pointing in the z direction, so jz = s. Now rotate it around the x axis by an angle . Doing so means it will pick up a phase determined by the spin
j i! eis^ j i (5)
In other words, the spin s is the coefficient of in the phase. For example, write the electric field vector E~^ as a complex number z = Ex + iEy. Then under a rotation by angle , z! eiz, so s = 1 according to Eq. (5). This tells us that photons (the quanta of the electric field) have spin 1.
- Electrons are identical by definition, since if every electron were not identical, we could give the different electrons different names. For example, there's a particle called the muon which is similar to the electron but heavier. Thus the electron/muon wavefunction will have (x 1 ; x 2 ) =/ (x 2 ; x 1 ). Unless there are an infinite number of possible quantum numbers that we can use to tell all electrons apart, some of them will be identical and satisfy j (x 1 ; s 1 ; x 2 ; x 2 )j^2 = j (x 2 ; s 2 ; x 1 ; s 1 )j^2. The identical particles for which quantum statistics applies are, by definition, these identical ones.
2 Section 2
The (^) 2!^1 is the identical particles factor. Note that we treat the two particles as different when constructing the partition function, then divide by N !:
For Bose-Einstein statistics, the possible states and canonical partition function are
" 1 " 2 " 3 AA A A A A AA A A AA
; ZBE = e−^2 "^1 + e−^2 "^2 + e−^2 "^3 + e−"^1 −"^2 + e−"^1 −"^3 + e−"^2 −"^3 (7)
Note that we treat the particles as identical, so AA = BA = AB is one state. No N! is necessary.
For Fermi-Dirac statistics, the possible states and canonical partition function are
A A
A A
A A
; ZFD = e−"^1 −"^2 + e−"^1 −"^3 + e−"^2 −"^3 (8)
Note that there are 9 two-particle states for the Maxwell-Boltzmann case, 6 for the boson case and only 3 for the fermion case. The 3 states in the fermion case are the ones where each particle has a different energy. If there were m possible energy levels instead of 3, then there would be
nfermi =
m 2
m^2 −
m (9)
possible fermion states. The number of boson two-particle states would include all the possible fermion ones plus the m states where both particles have the same energy, so
nbose = nfermi + m =
m^2 +
m (10)
The number of classical two-particle states is nclass = m^2. To compare to the fermion or boson cases, note that in the limit that kBT "i for all the energies, the Boltzmann factors all go to 1 so the partition function Z just counts the number states. Thus the effective number of states in Maxwell-Boltzmann statistics is
nMB = ZMB(kBT "j) =
nclass =
m^2 (11)
Thus we see that for 2 particles when the number of thermally accessible energies m is large, the number of two-particle states is the same in all three cases.
For N particles and m energy levels, the same analysis applies. The difference between the fermion and boson cases is due to the importance of states where more than one particle occupy the same energy level. The number of such states is always down by at least a factor of m from the number of states where all the particles have different energies. Thus when m gets very large, so that there are a large number of thermally accessible energy levels, then the distinction between bosons and fermions starts to vanish. In other words, m N almost all the microstates will have all the particles at different energies. For any particular set of N energies for the N particles there would be N! classical assignments of the particles to the energy levels, but only one quantum assignment. This explains why the N! Gibbs factor for identical particles in the Maxwell-Boltzmann statistics is consistent with quantum statistical mechanics. We will make more quantitative the limit in which quantum statistics is important in Section 4.
4 Section 2
3 Non-interacting gases
Next we compute the probability distributions for the different statistics, assuming only that the particles are non-interacting. We call the non-interacting systems gases , i.e. Bose gas and Fermi gas. Neglecting interactions allows us to write the total energy as the sum of the individual particle energies. For photons, neglecting interactions is an excellent approximation since Maxwell's equations are linear the electromagnetic fields does not interact with itself. For fermions, like electrons, even though there are electromagnetic interactions, because fermions like to stay away from each other (due to the Pauli exclusion principle), the interactions are generally weak. Thus the non-interacting-gas approximation will be excellent for many bosonic and fermionic systems, as we will see over the next 5 lectures. Let us index the possible energy levels each particle can have by i = 1; 2 ; 3 ; . The energy of level i is "i. For example, a particle in a box of size L has quantized momenta, pn = n L ~^ and energies "n = pn^2 2 m. In any microstate with a given^ N^ and^ E^ there will be some number^ ni^ of particles at level i, so
P
ni = N and
P
ni"i = E. It is actually very difficult to work with either the microcanonical ensemble (fixed N and E) or the canonical ensemble (fixed N and T ) for quantum statistics. For example, with 2 particles, the canonical partition function for a Bose gas would be
ZBE =
X
i=
1 X
j=i
1 e−^ ("i+"j)^ (12)
You can check that with only 3 energy levels, this reproduces Eq. (7). Note that we must have j > i in the second sum to avoid double counting the (i; j) and (j ; i) states. For N particles, we would need N indices and can write
ZBE =
X
i 16 i 26 6 iN
e−^ ("i^1 +"i^2 ++"iN)^ (13)
Doing the sum this way over N indices is very difficult; even in the simplest cases it is impossible to do in closed form. For Fermi-Dirac statistics the partition function can be written the same way with the sum over indices with strict ordering: i 1 < i 2 < < iN. It is also very hard to do. Conveniently, the grand canonical ensemble comes to the rescue. In the grand canonical ensemble, we sum over N and E, fixing instead and T. The grand partition function is
Z =
X
N ;E
e−^ (E^ −N)^ (14)
Since
P
ni = N and
P
ni"i = E the grand partition function can also be written as
Z =
Y
single particle states i
X
ni=
1 e−ni^ ("i−)^ (15)
where ni is the number of particles in state i. To see that this agrees with Eq. (14) consider that each term in the product over i picks one factor from each term in the sum over ni. Since all possible values of "i and ni are included once, we reproduce the sum over all possible N and E with
P
ni = N and
P
ni"i = E. For example, if ni = 0; 1 ; 2 ; are allowed (as in a Bose system), then Eq. (15) means (with = = 1 for simplicity):
Z = 1 + e−"^1 + e^2 −^2 "^1 + + e^2 −^2 "^2 + e^2 −"^1 −"^2 + + e−"^3 + + e^9 −^6 "^1 −^2 "^2 −"^3 + (16) =(1 + e−"^1 + e^2 −^2 "^1 + )(1 + e−"^2 + e^2 −^2 "^2 + )(1 + e−"^3 + e^2 −^2 "^3 + ) (17)
The second line shows all terms with N = 2 and energies " 1 ; " 2 ; " 3 and agrees with the explicit enumeration in Eq. (7). The equivalence of Eq. (15) and Eq. (14) is important. So please make sure you understand it, working out your own checks as needed. Eq. (15) can be written more suggestively as
Z =
Y
single particle states i
Zi (18)
with
Zi =
X
possible occupancies ni
e−ni^ ("i−)^ (19)
Non-interacting gases 5
Note that if "i < then hnii is negative, which is impossible. This means that < "i for all "i, i.e. the chemical potential for any Bose system will be less than all of the energies. Often we set the ground state energy to zero in which case
< 0 for bosons
Negative chemical potential is consistent with the formula we derived in Lecture 7 for for a classical monatomic ideal gas: = kBT ln n^3 −0.39 eV < 0 with n the number density and the thermal wavelength. We'll use the chemical potential for bosons in Lectures 11 and 12.
3.2 Non-interacting Fermi gas
For fermions, no two particles can occupy the same single-particle state. So the one-state grand- canonical partition function is simply
Zi =
X
n=0; 1
e−^ ("i−)n^ = 1 + e−^ ("i−)^ (28)
The full grand-canonical partition function is then
Z =
Y
i
Zi =
Y
i
[1 + e−^ ("i−)] (29)
The grand free energy for state i is
i = −
ln Zi = −
ln[1 + e−^ ("i−)] (30)
So that the occupation number for state i is
hnii = − @i @
e ("i−)^ + 1
f (") =
This is known as the Fermi-Dirac distribution or Fermi function , f("). We see that at low temperature, states with energy greater than are essentially unoccupied: f (" > ) 0 and states with energies less than are completely filled. The chemical potential at T = 0 is called the Fermi energy , "F = (T = 0). At T = 0, all states below the Fermi energy are filled and the ones above the Fermi energy are empty. Thus the chemical potential for Fermi systems is positive at low temperature and corresponds to the highest occupied energy level of the system. We'll use the Fermi energy extensively to discuss fermionic systems in the quantum regime, in Lectures 13, 14 and 15. As the temperature increases, the chemical potential decreases. This follows from @@T = − (^) NS and S > 0. Eventually, the chemical potential will drop below all of the energy levels of the system. That is, it becomes negative, as with a Bose system or a classical gas.
3.3 Maxwell-Boltzmann statistics
Maxwell-Boltzmann statistics are the statistics of classical indistinguishable particles. The parti- tion function is computed by summing over states with any particles in any state, and the sum is divided by N! to account for indistinguishability. Although it is often an excellent approximation, no physical system actually obeys Maxwell-Boltzmann statistics exactly. Nevertheless, counting states this way turns out to be very much easier than using Bose or Fermi statistics and gives the same answer in the continuum/classical limit.
Non-interacting gases 7
For Bose or Fermi gases, we were able to write the partition function as the product of partition functions for each state, as in Eq. (18): Z =
Q
i Zi. With Maxwell-Boltzmann statistics, it's not clear yet if we can do this since we must divide by N! where N is the total number of particles, not just the number of particles in state i. To avoid this subtlety, let's instead compute the grand-canonical partition function directly from the canonical partition function. Indeed, we can always write
Z =
X
N
ZNe N^ (32)
where ZN is the canonical partition function with N particles. We could have tried this for Bose or Fermi statistics, but for those cases the canonical partition function ZN with N fixed is hard to evaluate. The canonical partition function ZN with Maxwell-Boltzmann statistics is
ZN =
N!
X
distinguishable N particlemicrostatesk
e−^ Ek^ (33)
In this sum the particles are treated as distinguishable and indistinguishability is enforced entirely through the N! factor.^3 Since each particle is independent and the total energy E is unconstrained, we can pick any state i for any particle and therefore
ZN =
N!
X
i 1
X
iN
e−^ ("i^1 ++"iN)^ (34)
Here the sum over ij is over the possible states i for particle j. Since the sums are all the same, we can simplify this as in Eq. (15):
ZN =
N!
X
i
e−^ "i
X
i
e−^ "i
N!
Z 1 N^ (35)
where the single-particle canonical partition function is
Z 1 =
X
i
e−^ "i^ (36)
The grand partition function for Maxwell-Boltzmann statistics is then
Z =
X
N
N!
Z 1 Ne N^ = exp[Z 1 e ] = exp
X
i
e−^ ("i−)
Y
i
exp[e−^ ("i−)] (37)
This has the same form as Eq. (18) after all
Z =
Y
i
Zi (38)
with
Zi = exp[e−^ ("i−)] (39)
In fact, if we write
Zi =
X
n
n!
e−n^ ("i−)^ (40)
We see that Zi is exactly the grand partition function for a single state using Maxwell-Boltzmann statistics. The grand free energy for state i is then
i = −
lnZi = −
e−^ ("i−)^ (41)
- Note that dividing by N! is a little too much. N! is the number of permutations when the particles are in different states, such as (A; B; 0) and (B; A; 0) in the table in Eq. (6). But it divides by too much when there is more than one particle in the same state, like the state (AB; 0 ; 0) in the top row in the table in Eq. (6); there is only one state like this, not two. This approximation is valid when the number of accessible states is much larger than N , as at high temperature or in the classical continuum limit. In such limits, the number of configurations with more than one particle in a state is negligible.
8 Section 3
This is, of course, just the original Maxwell-Boltzmann distribution we derived back in Lecture
- Note that at large T the exponential goes to 1 and the prefactor ^3 1 T 3/2^ goes to zero. More physically, at high T , more and more states become accessible so it is less and less likely for any state to be multiply occupied. In addition, at high T particles are moving fast and their de Broglie wavelengths shrink: they become more like classical particles. So we can have hnii 1 either when T! 0 at fixed or when T! 1 at fixed N. Another way is simply taking V! 1 to make more and more states. There are many ways to get hnii 1. Quite generally, increasing the temperature at fixed N decreases the chemical potential, since @ @T =^ −^
S N <^0. For a Bose system,^ < "i^ for all^ "i^ so^ "i^ −^ >^0 and grows with temperature. In fact, it grows with temperature faster than decreases with temperature: roughly " − (^) NS T f T ln T
for some f (e.g. S NkBln T in an ideal gas) so (" − ) f ln T and hnii (^) T^1 f at large T. Thus
we see more broadly that high temperature gives classical statistics, as you would expect, despite the fact that a superficial reading of e−^ ("i−)^ 1 makes it seem otherwise. I suspect a number of you are cursing the chemical potential at this point. Why don't we just sue the microcanonical or canonical ensembles, at fixed N , rather than the grand canonical ensemble with its unintuitive ? As I have been saying, it is extremely difficult to use the micro- canonical or canonical ensembles for Bose-Einstein of Fermi-Dirac statistics due to the challenging combinatorics of counting microstates at fixed N. As we will see, having around is not bad at all once you get used to it. is essential to understanding Bose-Einstein condensation (Lecture
- and metals (Lectures 13 and 14). We have seen that many limits (large T , small T , large V ) lead to classical statistics. The challenge is to find a regime where the quantum statistics dominate. For quantum statistics to be relavant, we need hnii / 1. Since the higher energy states are going to be less populated than the ground state, a reasonable condition is that hn 0 i 1. Using Eq. (46), the ground state with "i = 0 has hn 0 i = 1 when N V
2 mkBT h^2
^3
So we want
V N
. In other words quantum statistics are important when the thermal de Broglie wavelength is of order or bigger than the interparticle spacing
For example, at room temperature air 1.87 10 −^11 m while the average interparticle spacing in
air is
V N
kBT P
= 3 10 −^9 , so particles in air are around a factor of 100 times too far for
quantum effects to be important. If we keep the pressure at 1 atm, we would have to cool air to T = 0.57K for quantum effects to matter. For a classical monatomic ideal gas = kBT ln(^3 n). Thus in the classical regime when ^3 NV 1 the chemical potential is a large negative number, 0. As temperature is decreased at constant N and V then goes up and ^3 n increases. As ^3 n gets close to 1 then ! 0 from below (more precisely, ! " 0 with " 0 the ground state energy, but we usually set " 0 = 0). As ! 0 , whether the system is bosonic or fermionic becomes important. For a bosonic system, can get closer and closer to 0, but can never reach it. As we will see in Lecture 12, as ! 0 , bosons accumulate in the ground state, a process called Bose-Einstein condensation. For a fermionic system, goes right through zero and ends up at a positive value the Fermi energy = "F > 0 at T = 0. Thus the closeness of to 0 or equivalently the closeness of ^3 n to 1 indicates the onset of quantum statistics.
5 Summary
In this lecture we have seen how statistical mechanics is modified for systems of identical particles. There are three types of statistics we use Fermi-Dirac statistics applies to particles of half-integer spin like electrons in metals or white- dwarf stars or half-integer spin nuclei like protons or neutrons. Fermi-Dirac statistics applies not just to elementary particles, but also to atoms that have an odd number of fermions, such as 3 He, which comprises 2 protons, 1 neutron and 2 electrons. In Fermi-Dirac statistics, no two identical particles can occupy the same single-particle state.
10 Section 5
Bose-Einstein statistics applies to particles of integer spin, like photons, phonons, vibrational modes, 4 He atoms (=2 neutrons, 2 protons and 2 electron) or 95 Rb atoms (35 protons, 60 neutrons, 35 electrons). In Bose-Einstein statistics, any number of bosons can occupy the same state. Maxwell-Boltzmann statistics are a kind of phony classical statistics for which calculations are generally easier than with bosons or fermions. We treat the particles as all distinguishable, then throw in a factor of (^) N^1! to the partition function to account for indistinguishability. The main results of this lecture were the expressions for the partition functions and probability distributions for the various statistics. We observed that for ideal gases (non-interacting particles) with any statistics, the grand-canonical partition function can be written as
Z =
Y
single particle states i
Zi (48)
where Zi is the grand-canonical partition function for a single state. For the various statistics we found
Zi =
1 − e−^ ("i−)^
(Bose − Einstein) (49)
Zi = 1 + e−^ ("i−)^ (Fermi − Dirac) (50)
Zi = exp[e−^ ("i−)] (Maxwell − Boltzmann) (51)
From these, we deduce the probabilities hnii for finding ni particles in a given state i with energy
"i. The general formula is hnii = @ @
lnZi
. We found
hnii =
e ("i−)^ − 1
(Bose − Einstein) (52)
hnii =
e ("i−)^ + 1
(Fermi − Dirac) (53)
hnii = e−^ ("i−)^ (Maxwell − Boltzmann) (54)
The general rule of thumb is that quantum statistics are relevant when states have a decent chance of being multiply occupied. At room temperature, there are so many possible states for the momenta and position of the molecules, that the chance of two being in the same state is utterly negligible. That's why classical statistical mechanics works fine most of the time. The way to find out if quantum statistics are important is to look at the thermal de Broglie wavelength
h^2 2 mkBT
r (55)
When the interparticle spacing (V /N )1/3^ is smaller than then quantum statistics is important.
Summary 11
