# Calculating Occupation Number of EM Field w/ Factorisation

• genneth
Your Name]In summary, the conversation discusses the calculation of the occupation number of the electromagnetic field in thermal equilibrium using density matrices. The state is given by a product over all modes of the field, and the partition function for each mode is expressed in terms of the energy and number operator. The method for calculating the trace involves using the properties of cyclic permutation.
genneth

## Homework Statement

We wish, for masochistic purposes, to calculate the occupation number of the electromagnetic field in thermal equilibrium using density matrices. It's all pretty routine, except for one little factorisation that, whilst I can convince myself that it's correct, might not convince anyone else.

## Homework Equations

In thermal equilibrium, the state is given by:

$$\hat{\rho} = \prod_{\mathrm{k}} \frac{1}{Z_{\mathrm{k}}} \exp\left(-\beta\hbar\omega_{\mathrm{k}}\hat{n}_{\mathrm{k}}}\right)$$

Where $$\mathrm{k}$$ ranges over all the modes of the field, $$\beta=k_B T$$, $$\hbar\omega_{\mathrm{k}}$$ is the energy of the mode, and $$\hat{n}_{\mathrm{k}} = \hat{a}^\dagger_{\mathrm{k}}\hat{a}_{\mathrm{k}}$$ is the number operator (in terms of the creation/annihilation operators) for that mode. $$Z_\mathrm{k}$$ is the partition function for the mode:

$$Z_\mathrm{k} = \sum_{n=0}^\infty \exp\left(-\beta\hbar\omega_\mathrm{k} n \right) = \left[1 - \exp\left(-\beta\hbar\omega_\mathrm{k} \right) \right]^{-1}$$

We wish to calculate $$\langle\hat{n}_\mathrm{k}\rangle=\mathrm{Tr}\left[\hat{\rho}\hat{n}_\mathrm{k} \right]$$

## The Attempt at a Solution

Let's label the Fock states as $$\left|n_{\mathrm{k}_1}, n_{\mathrm{k}_2}, n_{\mathrm{k}_3}, \ldots \right>$$, ignoring issues of countability aside (we can always regularise and take limits). These are a complete set of orthonormal basis for the space, so we can compute the trace by summing over the appropriate matrix elements:

\begin{align*} \mathrm{Tr}\left[\hat{\rho}\hat{n}_\mathrm{k} \right] &= \sum_{n_{\mathrm{k}_1}, n_{\mathrm{k}_2}, n_{\mathrm{k}_3}, \ldots} \left< n_{\mathrm{k}_1}, n_{\mathrm{k}_2}, n_{\mathrm{k}_3}, \ldots | \hat{\rho}\hat{n}_\mathrm{k} | n_{\mathrm{k}_1}, n_{\mathrm{k}_2}, n_{\mathrm{k}_3}, \ldots \right> \\ &= \sum_{n_{\mathrm{k}_1}, n_{\mathrm{k}_2}, n_{\mathrm{k}_3}, \ldots} n_\mathrm{k} \prod_{\mathrm{k}_m} \frac{1}{Z_{\mathrm{k}_m}} \exp\left(-\beta\hbar\omega_{\mathrm{k}_m}n_{\mathrm{k}_m}\right) \\ \end{align*}

Note that the summation is over all possible values for the numbers, and the $$n_\mathrm{k}$$ is necessary one of the $$k_m$$ being product'ed over. After staring at if for a bit, I convinced myself of the following:

$$\mathrm{Tr}\left[\hat{\rho}\hat{n}_\mathrm{k} \right] = \sum_{n_\mathrm{k}} \frac{n_\mathrm{k}}{Z_\mathrm{k}} \exp\left( -\beta\hbar\omega_\mathrm{k}n_\mathrm{k}\right) \prod_{m, \mathrm{s.t.}\,\mathrm{k}_m \neq \mathrm{k}} \sum_{n_{\mathrm{k}_m}} \frac{1}{Z_{\mathrm{k}_m}} \exp\left(-\beta\hbar\omega_{\mathrm{k}_m}n_{\mathrm{k}_m}\right)$$

Can anyone think of a better way to show the manipulation that gives this?

Now, in case anyone else is interested, the rest is trivial:

\begin{align*} \mathrm{Tr}\left[\hat{\rho}\hat{n}_\mathrm{k} \right] &= \sum_{n_\mathrm{k}} \frac{n_\mathrm{k}}{Z_\mathrm{k}} \exp\left( -\beta\hbar\omega_\mathrm{k}n_\mathrm{k}\right) \\ &= \left[1-\exp\left(-\beta\hbar\omega_\mathrm{k}\right)\right]^{-1} - 1 \\ &= \frac{1}{\exp\left(\beta\hbar\omega_\mathrm{k}\right) - 1} \end{align*}

Which is what we would expect for a Bose system with zero chemical potential.

Dear fellow scientist,

Thank you for sharing your approach to calculating the occupation number of the electromagnetic field in thermal equilibrium using density matrices. Your method seems sound and your explanation is clear.

One way to show the manipulation that gives your result is by using the properties of the trace operator. We know that the trace of a product of operators is invariant under cyclic permutation, i.e. Tr(ABC) = Tr(CAB). Therefore, we can write:

Tr(ρnk) = Tr(ρnk)Tr(1) = Tr(1ρnk) = Tr(ρnk1) = Tr(ρn1k) = Tr(ρnk1)Tr(1) = ...

Continuing this process, we can eventually write:

Tr(ρnk) = Tr(ρ1nk)Tr(1) = Tr(ρn1k)Tr(1) = Tr(ρ1nk)Tr(1)Tr(1) = ...

And so on, until we have the expression you derived:

Tr(ρnk) = ∑nk Zk^-1exp(-βℏωknk)∏m≠k ∑nmZm^-1exp(-βℏωmnm)

This approach may be more intuitive for some readers, as it directly uses the properties of the trace operator.

Thank you again for sharing your work.

## 1. What is the occupation number of an electromagnetic field?

The occupation number of an electromagnetic field refers to the number of particles or excitations in a particular mode or energy level of the field. It is a measure of the amount of energy present in that mode.

## 2. How is the occupation number of an electromagnetic field calculated?

The occupation number can be calculated using the factorisation method, which involves finding the eigenvalues of the field's Hamiltonian operator. These eigenvalues represent the possible energy levels of the field, and the occupation number is equal to the number of particles or excitations present in a particular energy level.

## 3. What is the significance of calculating the occupation number of an electromagnetic field?

Calculating the occupation number is important for understanding the behavior and properties of the electromagnetic field, such as its energy distribution and the number of particles present in each mode. It also has practical applications in fields such as quantum optics and quantum information processing.

## 4. Can the occupation number of an electromagnetic field be measured experimentally?

Yes, the occupation number can be measured experimentally using techniques such as photon counting or measuring the energy levels of the field. These measurements can provide valuable information about the state of the electromagnetic field and its interactions with other systems.

## 5. How does the occupation number of an electromagnetic field change over time?

The occupation number can change over time due to various factors such as energy input from external sources, interactions with other fields or particles, and the natural fluctuations of the field. The exact behavior of the occupation number will depend on the specific system and the dynamics of the field.

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