- #1
Jano L.
Gold Member
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How would you define density matrix for an ensemble of identical harmonic oscillators in thermal equilibrium?
For example, consider N atoms in a crystalline lattice. I would like to find density matrix to calculate the average dipole moment of the ensemble and also its standard deviation.
Standard references propose that [itex]\rho[/itex] is defined as
[tex]
\rho = \sum_{k}p_k |\psi_k\rangle \langle \psi_k|
[/tex]
where [itex]p_k[/itex] is the probability (expected frequency) of the quantum state [itex] |\psi_k\rangle[/itex] of the oscillator. The two obvious questions are:
1) which states [itex]|\psi_k\rangle[/itex] should we choose to enter the sum (it seems we cannot choose all of them) and
2) what are the probabilities [itex]p_k[/itex]?
In case of thermal equilibrium, the standard procedure is
1) to choose the states [itex]|\psi_k\rangle[/itex] as the eigenvectors of the Hamiltonian operator:
[tex]
H|\psi_k \rangle = \epsilon_k | \psi_k \rangle,
[/tex]
and
2) the probabilities are according to Boltzmann's formula:
[tex]
p_k = \frac{e^{-\epsilon_k/kT}}{Z}.
[/tex]
But, what is the reason behind this choice? Why can we forget the other quantum states of the oscillator (or, why does the probability of any other state vanish) ?
For example, consider N atoms in a crystalline lattice. I would like to find density matrix to calculate the average dipole moment of the ensemble and also its standard deviation.
Standard references propose that [itex]\rho[/itex] is defined as
[tex]
\rho = \sum_{k}p_k |\psi_k\rangle \langle \psi_k|
[/tex]
where [itex]p_k[/itex] is the probability (expected frequency) of the quantum state [itex] |\psi_k\rangle[/itex] of the oscillator. The two obvious questions are:
1) which states [itex]|\psi_k\rangle[/itex] should we choose to enter the sum (it seems we cannot choose all of them) and
2) what are the probabilities [itex]p_k[/itex]?
In case of thermal equilibrium, the standard procedure is
1) to choose the states [itex]|\psi_k\rangle[/itex] as the eigenvectors of the Hamiltonian operator:
[tex]
H|\psi_k \rangle = \epsilon_k | \psi_k \rangle,
[/tex]
and
2) the probabilities are according to Boltzmann's formula:
[tex]
p_k = \frac{e^{-\epsilon_k/kT}}{Z}.
[/tex]
But, what is the reason behind this choice? Why can we forget the other quantum states of the oscillator (or, why does the probability of any other state vanish) ?