Basic postulates of quantum statistical mechanics

In today’s article Basic postulates of quantum statistical mechanics ,we are going to discuss about the limitation of classical statistics and  Basic postulates of quantum statistical mechanics in detail along with suitable examples.

Limitation of classical statics-

  1. In classical, weins law is applicable only for shorter wavelength and fails for longer wavelength.
  2. Reyleigh law– it is applicable only for larger wavelength and fails for shorter wavelength.

Basic postulates of quantum statistical mechanics

Ev→ Uv ∝ vˆ2 and Eλ→Uλ ∝ 1/λˆ4

Where v→ ∞ or λ→0 (shorter wavelength)

According to Rayleigh’s theory

Uv and Uλ→∞

But in experiment Uv and Uλ →0.

This failure of classical physics is known as ultraviolet (uv) and catastrophe (sudden damage)

3. Whole spectrum of black body radiation could not explained by classical statistics (λ =0 to ∞ and v =0 to ∞)

4. Classical statistics could not explain finite value of energy at fixed temperature. We have,

U = ∫0 to ∞ (Uv dv)

  1. Where U= total energy or emitted energy.

Uvdv = 8πvˆ2 KTdv/ cˆ3 (Rayleigh’s law in term of frequency)

U = 8πKT ∫0 to ∞ (vˆ2)dv /cˆ3

8πKT [vˆ3]0 ˆ∞

I.e. U→∞

But in experiment Uv increase upto maximum value after that it decreases with frequency, which doesn’t agree with experiment result.

5. Classical statistics can explain only the micro properties of gas like distribution of velocity, distribution of speed, temperature, pressure and transport phenomena viscosity. Diffusion, temperature condensation. This theory cannot explain few phenomenon like photo eletric effect, thermonic emission, specific heat at low temperature, electron diffraction etc.

There phenomenon can be explained by,

B- E( base einstein fermi)

Postulate of Quantum statistics

1. In Quantum mechanics the state of the system is completely determine by the wave function Ψ(r,t) the wave function Ψ has following properties-

  • In general Ψ is a complex quantity.
  • It is continuous, single valued, and finite.
  • If Ψ* is complex conjugate of Ψ then ΨΨ* dτ = (Ψ)ˆ2 dτ. Which represent the probability that the system will be found within the volume dτ and time t.
  • For a material particle ∫-∞ to +∞ Ψ*Ψ dτ = 1

Basic postulates of quantum statistical mechanics

2. An operator is define as an quantity which operating on a system and converts it into another function.

ÂΨ(r,t) = Φ(r, t)

Where Â is operator and Ψ(r,t) is called first function while Φ(r, t) is called second function.

Operator must have following conditions-

  1. It must be linear Â(u+v) = Â(u) + Â(v)
  2. It must be hermitian. ∫u* Âvdτ = ∫ (Âu)*v* dτ

3. Eigen value-

whenan operator operates on an function and gives same function after operation with a constant then this constant quantity is called eigen value.

ÂΨ = (λ)Ψ

Where Â is called operator

λ is called eigen value

Ψ is called eigen function

And this whole equation is called eigen equation.

4. Expectation or Average value-

When a given system is in a state Ψ, the expectation values of the observable quantity (A) having operation (Â) is given-

A‾ = <A> = ∫Ψ* (Â)Ψdτ / ∫Ψ*Ψ dτ

This gives average value of A.

Example: expectation value of momentum,

Ax‾ = <Ax> = ∫Ψ* (-ihδ/δx) δτ/ Ψ*Ψ δτ

5. The time variation of a quantum mechanical state function Ψ(r,t) is determined by the time dependent schrodinger equation-

HˆΨ(r,t) = EˆΨ

HˆΨ(r,t) = ihδΨ (r,t)/ δt

And, Ψ(r,t) = Ψ(r)eˆ -E/h

6. If a particle exits within the limit then we have,

∫-∞ to +∞ (ΨiΨj* dτ = 1)

i =j

This condition is called normalisation condition and the wave function is called normalised wave function.

But if, ∫-∞ to +∞  (ΨiΨj* dτ = 0)

i≠ j

This condition is called orthogonal condition.

Thus we conclude,

∫ (ΨiΨj* dτ = 1)   when i= j

∫(ΨiΨj* dτ = 0)      when i≠j

This is called orthogonal condition.

Example- <x> = ∫Ψ* xΨdτ  {∴ xˆ= x} and {(Ψ*Ψ dτ = 1)}

7. Symmetric and antisymmetric wave function-

There are two types of solution of Ψ of schrodinger equation-

  1. Symmetric wave function -( Ψs) A wave function is said to be symmetric if the interchange of any pair of particles among its argument leaves the wave function unchanged.
  2. Antisymmetric wave function (ΨA)- A wave function is said to be antisymmetric wave function if the interchange of any pair of particles among its argument changes the sign of the wave function.

If Â is an exchange operator then we have,

ÂΨs (1,2) = Ψs (2,1)

ÂΨA (1,2) = -ΨA(2,1)

Case1 the identical particles having intregal spin quantum number are described by symmetric wave function.

ÂΨs (1,2,3,4….n) = Ψs (1,2,3….n)

The particles described by symmetric wave function are known as boson particles or Boson and obey bose Einstein statistics.

Case2. The identical particles having half odd intregal spin quantum number are described by antisymmetric wave function.

ÂΨ(1,2….n) = -ΨA(1,2…n)

These type of particles obey fermi Dirac statistics these are called fermiones.

Conclusion

In this article we have studied

  •  Antisymmetric wave function
  • Classical statistics
  •  Eigen value
  •  Expectation or average value
  •  Expectation value of psi
  •  Failures of classical mechanics
  •  Limitations of classical statistics
  •  Orthogonal condition
  •  Orthonormal condition
  •  postulates of quantum mechanics
  •  Postulates of quantum statistics
  •  Quantum statistics
  •  Rayleigh law
  •  Symmetric wave function
  •  Wein’s law

Etc in Detail and easiest way possible.

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