Bose Gas and Non Interacting Bose Gas and it's derivation Examples

1. Topic: Non Interacting Bose Gas
By Saadia Shaukat
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2. Non-Interacting Bose Gas
➢ Bose Gas
➢ Bosons
➢ Characteristics of Bosons
➢ Non-Interacting Bosons
➢ How to calculate the energy of the system.
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3. Bose Gas:
➢ A gas formed of non interacting particles with zero or integral spin, obeying
Bose-Einstein statistics. Bose gases include the gas of photons, as well as the
gas of quasi-particles (with integral spin).
➢ An ideal Bose gas is a quantum-mechanical phase of matter, analogous to a
classical ideal gas. It is composed of bosons, which have an integer value of
spin, and obey Bose–Einstein statistics.
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4. Bose Einstein Gas:
In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of
matter (also called the fifth state of matter) which is typically formed when a
gas of bosons at low densities is cooled to temperature very close to absolute
zero (-273.15 °C, -459.67 °F).
Gas of Rubidium atoms
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5. Non-Interacting Boson:
Non interacting particles are not to interact with other particles but may
be affected by fields.
Boson are particles that have integer spin.
The important difference in this case is that bosons can from Bose Einstein
Condensates but fermions can not be due to Pauli Exclusion principle.
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6. Bosons
All the particles with integral spin (S=0,1,2,3,….) are called Bosons. They represent
symmetrical wave function.
➢ Higgs Boson the only particle which have zero spin (S=0).
➢ For spin 1 (S=1) we have photon.
➢ For spin 2 (S=2) we only have graviton particle.
➢ However, particles having spin ( S=3,4,…) are not known yet.
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7. Bosons:
Higgs Boson Photon
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8. Characteristics of Bosons:
➢ They have integral spin. (S=0, 1, 2,3….)
➢ They do not obey the Pauli exclusion principle.
➢ They obey Bose Einstein Statistics.
➢ They are described by symmetric wave functions.
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9. Bose Particles or Bosons:
➢ Particles with this property are named after the Indian Physicist Satyendra
Nath Bose (1924).
➢ Satyendra Nath Bose [1 January 1894 – 4 February 1974) was an Indian
mathematician and physicist specializing in theoretical physics.
➢ He is best known for his work on quantum mechanics in the early 1920s,
collaborating with Albert Einstein in developing the foundation for Bose–
Einstein statistics and the theory of the Bose–Einstein condensate.
➢ The statistical mechanics of these particles was analyzed by Einstein
(1924,1925).
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10. Non Interacting Bose Particle:
Consider two Quantum particles in a box.
Label the possible states of one quantum particle in the box by a label α.
And the other one with the label of β.
The wave function must be in the form of :
𝝍 𝒙𝟏, 𝒙𝟐 = 𝝓𝜶 𝒙𝟏 𝝓𝜷 𝒙𝟐 + 𝝓𝜷 𝒙𝟏 𝝓𝜶 𝒙𝟐
𝛟𝛂 𝐱𝟏 Energy Eigenstate of first particle
𝛟𝛃 𝐱𝟐 Energy Eigenstate of second particle
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11. Non Interacting Bose Particle:
If we are working with three particles then:
𝝍 𝒙𝟏, 𝒙𝟐, 𝑿𝟑 = 𝝓𝜶 𝒙𝟏 𝝓𝜷 𝒙𝟐 𝝓𝜸(𝑿𝟑) + 𝝓𝜶(𝒙𝟏)𝝓𝜸(𝒙𝟐)𝝓𝜷(𝑿𝟑)
= 𝝓𝜷 𝒙𝟏 𝝓𝜸 𝒙𝟐 𝝓𝜶(𝑿𝟑) + 𝝓𝜷(𝒙𝟏)𝝓𝜶(𝒙𝟐)𝝓𝜸(𝑿𝟑)
= 𝝓𝜸 𝒙𝟏 𝝓𝜶 𝒙𝟐 𝝓𝜷(𝑿𝟑) + 𝝓r(𝒙𝟏)𝝓𝜷(𝒙𝟐)𝝓𝜶(𝑿𝟑)
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12. Non Interacting Bose Gas:
➢ If there are N particles.
➢ Then the particle states are α,β,r,…… .
➢ There is exactly one symmetric combination.
➢ This is an energy eigenstate for N bosons.
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13. Non interacting Bose Gas:
we’ll use the canonical ensemble:
An ensemble of copies of the system. All with the same N,T .
(Hence E varies amongst the copies in the ensemble.)
Constant with a heat bath at temperature T.
< 𝑬 >= 𝚺𝑬𝒊. 𝑷𝝍𝒊
𝚺𝑬𝒊 Energy of the N particles
𝑷𝝍𝒊 Prob. Of finding N-particle state in the ensemble.
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14. Conti…
We know that 𝑬𝒊 = σ𝜶 𝝐𝜶(𝒏𝜶(𝝍𝒊)
By putting the value of 𝑬𝒊 in above expression we get:
= σ𝒊 σ𝜶 𝝐𝜶(𝒏𝜶(𝝍𝒊) . 𝑷𝝍𝒊
= σ𝜶 𝝐𝜶 σ𝒊(𝒏𝜶(𝝍𝒊) . 𝑷𝝍𝒊
We know that:
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15. Conti…
< 𝒏𝜶 >= σ𝒊(𝒏𝜶(𝝍𝒊) . 𝑷𝝍𝒊
Mean value of the occupation number of particle
state α in ensemble of N particle states.
There for we can put 𝑛𝛼 in equation.
So,
< 𝐄 >= σ𝛂 𝛜𝛂 < 𝐧𝛂 >
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