Quantum non-interacting systems#

Occupation number representatiton of energy#

Preparing ground for the quantum statistics we adopt an occupation number representation of a microstate. Occupation number or orbital representation explicitly tracks the occupation of single particle eigenvalues. In this representation a microstate is identified with the occupancy vector \([n_1, n_2, ... n_N]\) specifying number of particles populating every fixed energy level.

\[E = \sum_i n_i \epsilon_i\]
\[N = \sum_i n_i\]

Boltzmann distribution#

  • The number of states in the occupation representation is given by the multinomial factor:

\[\Omega(E) = \frac{N!}{n_1! n_2! n_3! ... }\]
\[Z_N = \sum_{n_1, n_2, ... n_N} \Omega(n_1, n_2, ...) e^{-\beta (n_1 \epsilon_1 +n_2 \epsilon_2 + ...)} = \sum_{n_1, n_2, ... n_N} \frac{N!}{n_1! n_2!... n_N!} [e^{-\beta \epsilon_1}]^{n_1} [e^{-\beta \epsilon_2}]^{n_2} ...\]

  • The last expression is seen as a a polynomial expression leading to the factorization of partition function as we would expect for the non-interacting nature of the system.

\[Z =(e^{-\beta \epsilon_1}+e^{-\beta \epsilon_2}+ ...)^N= Z^N_1 \]
\[Z_1 =\sum_j e^{-\beta \epsilon_j} \]

  • Thus, we see that the factor of \(N!\) needs to br introduced to correct for multiple counting of states.

\[Z = \frac{1}{N!}Z^N_1\]

  • This way of handling partition functions is called Maxwell-Boltzmann statistics which is widely applicable in the “classical regime”.

  • Boltzmann derived a distribution bearing his name by maximizing \(\Omega(E)\) subject to \(\sum_{k} n_{k}=N\) and \(\sum_k n_k E_k = N U\) constraints. This is essentially a MaxEnt derivation seen in the module on Entropy

Example of MB statistics

Consider 2 particles each having 3 energy levels \(\epsilon_1, \epsilon_2, \epsilon_3\)

\[Z_2 = \frac{1}{2!}(e^{-\beta \epsilon_1}+e^{-\beta \epsilon_2}+e^{-\beta \epsilon_3})^2\]
\[Z_2 = \frac{1}{2} \big[e^{-2\beta \epsilon_1}+e^{-2\beta \epsilon_2}+e^{-2\beta \epsilon_3}\big]+\big[e^{-2\beta (\epsilon_1+\epsilon_2)}+e^{-2\beta (\epsilon_1+\epsilon_3)}+e^{-2\beta (\epsilon_2+\epsilon_3)}\big]\]

  • Multiply occupied states are unphysically suppressed.

  • Single occupied states are nicely weighted which removes the double-counting.

Validity of the classical regime#

Average and fluctuations of occupancy numbers#

\[\langle n_i \rangle = - \frac{1}{\beta}\frac{\partial log Z}{\partial \epsilon_i}\]
\[p_i = \frac{n_i}{N} = \frac{e^{-\beta \epsilon_i}}{Z_i}\]

Thermal wavelength demarcates classical vs quantum regimes#

A sufficient condition for the validity of the classical regime is the low mean occupancy of energy levels \(\epsilon_i\).

\[\bar{n}_i \ll 1\,\,\,\,\, for\,\,\,\, all\,\,\,\, i\]

Using translational partition function for a free particle in a box \(Z_1 = V \Big(\frac{2\pi m k_B T}{h^2} \Big)^{3/2}\) we express low mean occupancy limit in terms of microsocipic parameters:

\[\bar{n}_i = N e^{-\beta \epsilon_i}/Z_1 \ll 1\]
\[\frac{N}{V} \Big(\frac{h^2}{2\pi m k_B T} \Big)^{3/2} \ll 1\]
\[n/n_Q \ll 1\]
\[\lambda^3_T \ll n = \frac{N}{V} = l^3\]

Classical limit is the limit where thermal wavelength is much smaller than interatomic distances.

Partition functions for quantum statistics#

Evaluating partition functions in the occupation number representation is easier in grand-canonical ensemble where there are no restrictions \(N=\sum_i n_i\) placed on sums.

\[Y(\beta, \mu) = \sum_{n_1, n_2, ...} e^{-\beta E} e^{\beta \mu N}\]
\[Y = \sum_{n_1} \Big[e^{-\beta(\epsilon_1-\mu)n_1}\Big]\cdot \sum_{n_2} \Big[e^{-\beta(\epsilon_2-\mu)n_2} \Big]...\]
\[Y = \prod_i Y_i\]
\[Y_i = \sum^{n_{max}}_{n=0} e^{-\beta(\epsilon_i -\mu)n}\]

Average and fluctuations of occupation numbers#

The averages and flucutations as usual are obtained by taking appropriate derivatives of partiton function.

\[\bar{n}_i = \frac{1}{\beta} \frac{\partial logY_i}{\partial \mu}\]
\[\sigma^2(n_i) = \frac{1}{\beta^2} \frac{\partial^2 logY_i}{\partial \mu^2}\]

Bosons and Fermions#

  • bosons Any one particle state has no upper bound for its occupation number. Integer spins.

  • fermions The occupation number for any one-particle state is at most one (Pauli exclusion). Half odd integer spins.

Ideal Boson systems#

\[Y_i = \sum^{n=\infty}_{n=0} = \big[ 1-e^{-\beta(\epsilon_i-\mu)} \big]^{-1}\]
\[\boxed{\bar{n}_i = \frac{\partial log Y_i}{\partial \beta \mu } = \frac{1}{e^{\beta(\epsilon_i-\mu)}-1} }\]

For the ground state energy \(\epsilon_i=0\) we have \(\bar{n}_0 = (e^{-\beta \mu}-1)^{-1}\). Thus \(\mu\) is not allowed to be positive. That is chemical ptoential must be smaller than graound state energy to maintain non-negative occupation number

Ideal Fermion systems#

\[Y_i = \sum^{n=1}_{n=0} = e^{-\beta(\epsilon_i-\mu)}+1\]
\[\boxed{ \bar{n}_i = \frac{\partial log Y_i}{\partial \beta \mu } = \frac{1}{e^{\beta(\epsilon_i-\mu)}+1}}\]

Classical Limit#

  • The Fermi-Dirac and Bose-Einstein distributions are very different from Maxwell-Boltzmann. THe difference is due to quantum interference when number density is high.

  • \(\mu \rightarrow -\infty\) System has no resistance to chemostat’s donation of particles. E.g the system is highly dilute with low ocupancy numbers \(n_i \ll 1\)

\[\bar{n_i}\rightarrow e^{\beta \mu}e^{-\beta \epsilon_i} = \frac{1}{Z}e^{-\beta \epsilon_i}\]

All distributions captured wihin one universal formula#

\[\boxed{\bar{n}_i = \frac{1}{e^{\beta(\epsilon_i-\mu)}+c }}\]

  1. \(c = -1\) Bose-Einstein distribution.

  2. \(c = +1\) Fermi-Dirac distribution.

  3. \(c = 0\) Maxwell-Boltzmann distribution.

Density states (DOS) for an ideal gas#

Let us analyze the density of states of free particles in volume V in some detail starting from fundamental equation governing state of such particles.

\[ \hat{H}\psi(r) = E \psi(r)\]

Solution for free particles are plane waves for which we need to impose boundary conditions.

\[\psi(r) = exp(i k \cdot r) = exp[i(k_x x + k_y y + k_z z)]\]

We impose periodic boundary conditions \(k_x(x + Lx) = k_x x + 2πn\)

\[k_x = \frac{2 \pi}{L_x} n_x ;\,\,\, k_y = \frac{2 \pi}{L_y} n_y ;\,\,\, k_z = \frac{2 \pi}{L_z} n_z\]

Discrete eigenvalues emerge from boundary conditions

\[\epsilon = \frac{\hbar^2}{2m} (k^2_x+k^2_y+k^2_z) \]

Now we want to count the number of modes or waves that have wavevectors between \(([k_x, k_x+dk_x], [k_y, k_y+dk_y], [k_z, k_z+dk_z])\):

\[\Delta n_x = \frac{L_x}{2 \pi} dk_x = \frac{L_x}{h} dp_x\]

Number of states within \(\Delta n_x \Delta n_y \Delta n_z\) region is given in terms of momenta values \(\rho(p) d^3p\)

\[\Delta n_x \Delta n_y \Delta n_z = \frac{L_x L_y L_z}{(2\pi)^3} d^3 k = \frac{L_x L_y L_z}{h^3}d^3p\]

  • Density of states in cartesian coordines

\[\boxed{g(p) = \frac{V}{h^3};\,\,\,\,\, g(k) = \frac{V}{(2\pi)^3}}\]

  • Azimuthally averaged density of states \(g(p)dp = \int^{\pi}_{0} d\theta \int^{2\pi}_{0} d\phi g(p)d^3 p\) results in multiplying by \(4\pi r^2\) term.

\[\boxed{g(p) = \frac{V}{h^3} 4\pi p^2;\,\,\,\, g(k) = \frac{V}{2\pi^2}k^2}\]

Another insightful way of computing \(\Omega(k)\) is by dividing the volume of a sphere with radius \(k\) \(V_k = \frac{4\pi }{3} k^3\) by the volume availible per one state \(g(k)\)

\[N(k) = \frac{4\pi k^3}{3}\cdot \frac{1}{g(k)} = \frac{Vk^3}{6\pi^2}\]
\[g(k) = \frac{dN(k)}{dk} = \frac{Vk^2}{2\pi^2}\]