# What are bosons?

##### 1 Answer

**Bosons** (such as photons) are quantum mechanical particles with integer spins, that are symmetric upon particle interchange. *More than one* can exist in a given quantum state, unlike fermions (such as electrons), which have half-integer spins.

In other words...

- Bosons do
follow the Pauli Exclusion Principle, but fermions do.*not* - Bosons have
#m_s = pm1# , but fermions have#m_s = pm1/2# . - Bosons that are interchanged return the
state, so it is easier to double-count the microstates in a bosonic system by accident (there are generally*same**fewer nonredundant*configurations possible).

**DISOBEYING THE PAULI EXCLUSION PRINCIPLE**

The **grand canonical partition function** for an ensemble of bosons where each state is singly-degenerate is given by:

#Xi = prod_(i=1)^(N) 1/(1-e^(-(epsilon_i - mu)//k_BT))# where

#epsilon_i# is the energy of level#i# and#mu# is the chemical potential for bosons.

It can be shown that, since

#PbarV = k_BTln Xi# #d(PbarV) = barSdT + PdbarV + barNdmu# #sum_(i=1)^(N) barn_i = barN#

the **average occupation number**

#barn_i = e^(-(epsilon_i - mu)//k_BT)/(1 - e^(-(epsilon_i - mu)//k_BT))#

When **the energy above**

*This means that you CAN have an infinite number of particles in the ground state of a boson system.*

When **the temperature** instead, it looks like:

This means that at some point (around

If

**SYMMETRY UPON PARTICLE INTERCHANGE**

Define the **wave function** for two particles (with a specific ordering) as

Then we say that

#Psi(1,2) = Psi(2,1)# ,

if both particles are bosons (such as photons). That is, the sign of the wave function *stays the same* upon bosonic particle interchange.

But we say

#Psi(1,2) = -Psi(2,1)# ,

If both particles are fermions (such as electrons). That is, the sign of the wave function *switches* upon fermionic particle interchange.

Or, consider a simple **Slater determinant** for fermions (such as electrons), which treats the wave function as having a spatial part and a spin part.

#Psi(1,2) = 1/sqrt2 |(color(green)(phi_(1s)(1)alpha(1)), color(blue)(phi_(1s)(1)beta(1))),(color(green)(phi_(1s)(2)alpha(2)), color(blue)(phi_(1s)(2)beta(2)))|#

#= 1/sqrt2 [color(green)(phi_(1s)(1)alpha(1))color(blue)(phi_(1s)(2)beta(2)) - color(blue)(phi_(1s)(1)beta(1))color(green)(phi_(1s)(2)alpha(2))]# (We represent spatial orbitals as

#phi# and spin orbitals as#alpha# (spin-up) and#beta# (spin-down). The numbers in parentheses indicate the particle number.)

Another way to depict particle interchange is to switch these two columns (one could choose instead to switch the two rows, but let's not get too complicated). **For fermions**, we would have:

#Psi(1,2) = -Psi(2,1) = -1/sqrt2 |(color(blue)(phi_(1s)(1)beta(1)), color(green)(phi_(1s)(1)alpha(1))),(color(blue)(phi_(1s)(2)beta(2)), color(green)(phi_(1s)(2)alpha(2)))|#

#= ul(-1/sqrt2 [color(blue)(phi_(1s)(1)beta(1))color(green)(phi_(1s)(2)alpha(2)) - color(green)(phi_(1s)(1)alpha(1))color(blue)(phi_(1s)(2)beta(2))])#

On the other hand, **for bosons**, the analogous determinant uses a

#Psi(1,2) = Psi(2,1) = 1/sqrt2 |(color(blue)(phi_(1s)(1)beta(1)), color(green)(phi_(1s)(1)alpha(1))),(color(blue)(phi_(1s)(2)beta(2)), color(green)(phi_(1s)(2)alpha(2)))|#

#= ul(1/sqrt2 [color(blue)(phi_(1s)(1)beta(1))color(green)(phi_(1s)(2)alpha(2)) + color(green)(phi_(1s)(1)alpha(1))color(blue)(phi_(1s)(2)beta(2))])#

In the case of bosons, particle interchange is symmetric, and that is seen in that addition is commutative.