Quantum canonical ensemble

The quantum canonical ensemble is the quantum extension of the canonical ensemble. It is defined through the density matrix

ρ^=e−βH^\hat{\rho}=e^{-\beta \hat{H}}

where H^\hat{H} is the Hamiltonian of the system and β=1/kBT\beta=1/k_{B}T, with kBk_{B} the Boltzmann constant and TT the temperature. The elements of the matrix in the energy eigenstate basis are ρn=e−βEn\rho_{n}=e^{-\beta E_{n}}, where EnE_{n} are the energy eigenvalues.

Partition function

The partition function is

QN=Tr(ρ^)=∑ne−βEnQ_{N}=\text{Tr}(\hat{\rho})=\sum_{n}e^{-\beta E_{n}}

The sum notation holds only if we represent ρ^\hat{\rho} in the energy eigenstate basis, which is generally a bad choice as it's almost never possible to diagonalize H^\hat{H}. It is possible in an ideal gas, however.

All properties derived from the partition function have identical expressions to the classical canonical.

Internal energy

The internal energy

U=⟨H^⟩=−∂∂βln⁡QN,var(U)=∂2∂β2ln⁡QN=−∂U∂βU=\langle \hat{H} \rangle =-\frac{ \partial }{ \partial \beta } \ln Q_{N},\qquad\text{var}(U)=\frac{ \partial ^{2} }{ \partial \beta ^{2} }\ln Q_{N}=-\frac{ \partial U }{ \partial \beta }

Helmholtz free energy

The Helmholtz free energy is

A=−kBTln⁡QNA=-k_{B}T\ln Q_{N}

Entropy

Since we know the state density, we can determine entropy from the information-theoretical one as

S=−kB∑np(n)ln⁡p(n)=kBβQN∑nEne−βEn+kBln⁡QNS=-k_{B}\sum_{n} p(n) \ln p(n)=\frac{k_{B}\beta}{Q_{N}}\sum_{n}E_{n}e^{-\beta E_{n}}+k_{B}\ln Q_{N}

since p(n)=ρn/Z=e−βEn/Zp(n)=\rho_{n}/Z=e^{-\beta E_{n}}/Z. It can equivalently be expressed as

S=kB∂∂T(Tln⁡QN)S=k_{B}\frac{ \partial }{ \partial T }(T\ln Q_{N})

Occupation numbers

The occupation numbers can be derived from the partition function as

⟨n⟩=−1β∂∂Eln⁡QN\langle n \rangle =- \frac{1}{\beta}\frac{ \partial }{ \partial E } \ln Q_{N}

This is because the number can be derived from a sum over all possible states

⟨n⟩=1QN∑{n}ne−βEn\langle n \rangle = \frac{1}{Q_{N}}\sum_{{\{ n \}}}ne^{-\beta E_{n}}

The above form can seen by noticing that this sum is essentially the derivative of ln⁡QN\ln Q_{N} up to scaling constant.

Examples

> We want to find the partition function for a system of $N$ of these particles. If the particles are not coupled (that is, they do not interact), the total partition function is just the product of each particle's partition function: > $$Z=\prod_{i=1}^{N} Z_{i}=2^{N}e^{-N\beta E/2}\cosh^{N}\left( \frac{\beta E}{2} \right)

The average energy is

> The last step just comes from applying the exponential definition of the $\tanh$ and simplifying things. This result is identical to the one found with the [[microcanonical ensemble]]. Notably, this was way easier, as we didn't have to use combinatorics at all.