We start with Hamiltons equations.
We start with some probability distribution in the phase space of the Hamiltonian .
The probability distribution evolves by the continuity equation.
Here
In equilibrium the probability distribution at every point in phase space shouldn't change.
Simple chain rule.
Substitute in Hamiltons equations.
Next, we say that the probability distribution is a function only of the total energy, so . This means that
That assumption creates a solution to the continuity equation. Now in the microcanonical ensemble (NVE) we say the total energy is known, so the only possibility for is a uniform distribution over the selected energy level.
This equation is the principle of equal a priori for the equilibrium in a closed system.
Define the number of states accessable by a system with degrees of freedom at energy to be the below.
Next split the degrees of freedom into the "bath" and "system". Let the degrees of freedom for the bath be and the degrees of freedom for the system .
In the Canonical ensemble the combined energy of the bath and system should be constant. The combination of the two makes a microcanonical ensemble.
The total accessable states with an energy should be the total number combined number of states the bath and system can be in that sum to that energy.
And in the microcanonical ensemble all microstates with the correct energy have an equal probability density.
We can calculate the probability density of just the microstate for the system.
Take the natural log of both sides
Then we expand the logarithm to its Taylor series.
We set , , and we get the following expression.
By definition if the bath was a microcanonical ensemble the entropy would be
And by definition .
Then if we assume the heat capacity of the bath is infinite so that we can remove the extra terms, and the temperature just becomes a constant so .
Then exponentiate both sides.
And the denominator is just a normalizing factor so the final expression is below.
We assume a time evolution of a system of the Schrodinger equation for some Hamiltonian operator.
To measure an observable when the stat is known is
but if there is a probability distrubition over some states then it is below.
If we have an orthanormal basis so that we can insert it into the equation.
If we define an operator
then the expected value of any observable is
Next we find the time evolution of .
Then substitute the Schrodinger equation
So it must be that for equilibrium they commute
Similar to the classical case, if we have the be only a function of the energy for each eigenvector of the Hamiltonian, the system will be in equilibrium. Since observables are Hermetian, an orthogonal eigenbases must exist. Each operator will be defined in the eigenbasis.
We can show they commute
So the quantum microcanonical ensemble must have a density matrix of the form
where the 's are eigenvectors of the Hamiltonian. Since the energy is known, it must be a uniform distribution over all eigenstates with that energy
We assume the total Hamiltonian is the form
And we assume the coupling energy is none. From the microcannonical ensemble the total density matrix is
And the density matrix of just the system is the partial trace
Where are the eigenbasis of . Measuing a system microstate gives.
And the values of are only the eigenvectors of eigenvalue , so it would just be the ones where the bath has the complement energy.
Take the ln of both sides
Then Taylor series again
We are assuming that the number of states is dense enough that the derivative exists. Substitute the definition of entropy
Then the definition of temperature
Assume heat capacity is infinte
then exponentiate both sides to get
and the extra term is just a normalizing constant so
And since is any eigenstate it is just the operator exponential.