The occupation number, also known as the occupation probability or the number of particles in a specific quantum state, can be determined using the Hamiltonian operator in quantum mechanics. The Hamiltonian operator (H) represents the total energy of the quantum system and is used to describe the time evolution and behaviour of the system.

In the context of quantum statistical mechanics, the occupation number can be calculated using the following steps:

Hamiltonian Operator (H): Start with the Hamiltonian operator (H) that describes the quantum system. The Hamiltonian includes terms for kinetic energy, potential energy, and any interactions between particles in the system.
Quantum States: Identify the quantum states of the system that you want to analyse. These states are the possible energy eigenstates of the Hamiltonian.
Eigenvalue Equation: Solve the eigenvalue equation for the Hamiltonian, H|ψ⟩ = E|ψ⟩, where |ψ⟩ is an eigenstate of the Hamiltonian, E is the corresponding eigenvalue (energy), and the energy states are normalized.
Occupation Number: The occupation number (n) of a specific energy eigenstate |ψ⟩ is the number of particles in that state. It can be calculated using the occupation number operator (n_i):
n_i = ⟨ψ|n_i|ψ⟩
where:
- n_i is the occupation number operator for the energy eigenstate |ψ⟩.
- ⟨ψ| is the bra vector, the complex conjugate transpose of |ψ⟩.
Expectation Value: Calculate the expectation value of the occupation number operator with respect to the state |ψ⟩.
Interpretation: The expectation value of the occupation number gives the average number of particles in the specific quantum state |ψ⟩.

The Hamiltonian for a system of N identical non interacting point particle is given by

In MCE, it needs to find out each of these 3 cases, the number of states 𝜏(E) of the system having an energy eigen value that lies between E and E+ ΔE . Here for simplicity, we define spinless particles. Any energy eigen value of an ideal system is a sum of single particles energies called levels and these are

In the limit V-> ∞, the possible values of P form a continum. The sum over P can sometimes be replaced by an integration

To specify the state of the system, we have to specify a set of occupation number {np} which is defined by np particles having momentum P in the state under consideration. The total energy E and the total number of particles N are given by

The allowed values of n_p are np = 0,1,2 for MB

n_p = 0,1,2 for BE

n_p = 0,1 for FD

In the limit v-> ∞, the energy levels 𝜖p = p²/2m form a continuum due to overlapping of energy levels. So it is difficult to count number of states. Hence we group many energy levels to form a cell. So,

g1 energy levels are in cell 1

g2 energy levels are in cell 2

gi energy levels are in cell i

And average energy of energy levels in cell 1 is 𝜖1. Average energy of energy levels in cell 2 is 𝜖2. Average energy of energy levels in cell 3 is 𝜖3. Average energy of energy levels in cell i is 𝜖i.

Also cell 1 occupied by n1 particles.

cell 2 occupied by n2 particles.

cell 3 occupied by n3 particles.

cell i occupied by ni particles.

where vector p runs over all energy level in ith cell. We have two condition and can be written as

Let {ni} denotes the set of number ni satisfying the condition 7 and 8 and w{ni} denotes number of microstates corresponding to occupation number of set ni. The total number of microstates in this ensemble is

Γn(E) = Σ w{ni} ....9

For Bose gas: In this case, energy level can be occupied by any number of particles which is equivalent to putting ni distinguishable ball into (gi -1) distinguishable boxes so total number of ways is

the Bose-Einstein statistics allow multiple particles to occupy the same quantum state, and the number of ways particles can occupy the energy levels depends on the degeneracy of each level. The total number of ways of distributing particles among energy levels is a crucial concept in statistical mechanics for understanding the behaviour of an ideal Bose gas.

The occupation numbers provide information about the statistical distribution of particles among different quantum states at a given temperature and chemical potential. In quantum statistical mechanics, the occupation numbers are described by the Bose-Einstein or Fermi-Dirac distribution functions, depending on the statistics (Bose-Einstein statistics for bosons and Fermi-Dirac statistics for fermions). These distribution functions give the probabilities of finding particles in different energy states at a given temperature and chemical potential, and they are derived from the occupation numbers using the principles of quantum statistics.

This note is a part of the Physics Repository.

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