The particles meet three conditions:

• They are distinguishable from each other (this fact affects entropy)
• They obey the laws of quantum mechanics.
• Furthermore, they do not interact with each other.
Our study objective will be an isolated thermodynamic system, formed by N particles with energy E=cte and volume V=cte.
Suppose that each molecule can be in a set of quantum states with energies . Calling to the number of particles that are in the quantum state with energy , the total energy of the system will be: MACROSTATE

Statistical mechanics is the link between the microscopic world studied by quantum mechanics, and the macroscopic world studied by classical thermodynamics.

The state of a macroscopic system, made up of a large number of particles, is defined by macroscopic variables called state functions, which depend only on the initial and final conditions but not on the path followed.
In a pure substance in equilibrium, the state of the system (macrostate) is defined by three variables, pressure, temperature and number of moles. It is not necessary to specify the volume because there is an equation of state that relates the four thermodynamic variables. For example, in the case of an ideal gas PV=nRT.
In this case, the macrostate of the system is specified by three state variables, since any other thermodynamic magnitude can be obtained from them. Thus, U=U(P,T,n)=U(V,T,n)=....
Giving the macrostate of a system consists, therefore, in specifying the smallest possible number of independent variables that determine its thermodynamic state.

MICROSTATE

In classical mechanics, the microscopic state of a system or microstate is obtained by specifying the coordinates and speeds of all the particles that compose it at a given instant of time.

For a classical system of N particles (1,2,3,.....N), 3N coordinates are needed to specify the position of all particles: $x_1,y_1,z_1, x_2,y_2,z_2... ...x_N,y_N,z_N$. In addition, another 3N coordinates are needed to specify their velocities, $v_{x1},v_{y1},v_{z1},v_{x2},v_{y2},v_{z2},......v_{ xN},v_{yN},v_{zN}$. Therefore, 6N coordinates are needed to describe the system in classical mechanics and calculate its properties. For example, the total energy of the system is the sum of its kinetic energy, which depends on the speeds, and its potential energy, which depends on the position of the particles. It can be shown that the energy of the system depends on the number of particles and the volume available for motion E=E(N,V).
In quantum systems the state of the system is given by the wave function, dependent on the quantum numbers n,l,m and $m_s$, which must be specified for each particle, which means 4N quantum numbers for a system of N particles.
The energy of the system is calculated from the Schrödinger equation once the wave function is known. As in the classical case, the energy depends on N and V.