Distinguishable and Identical Particles

The probability P(θ) to detect particle at angle θ depends on the dispersion amplitude f(θ) for
distinguishable particles: | non distinguishable particles:
Pd(q) = |f(θ)|2 + |f(p-q)|2
for instance, nitrogen and hydrogen atoms. The detector should be sensitive to N and O atoms in order to distinguish them. Though we can't distinguish N and O atoms it's principally possible however! Hence the probabilities and amplitudes are not summed. |
Pi(θ) = |f(θ) + f(p-q)|2
for instance α-particles (helium nuclei, He++) Because there are two possibilities to register α-particle by our detector namely particle is dispersed on angles θ or p - q we can't distinguish between incident particle and particle of target.
For dispersion angle θ = π/2 it's obviously f(θ) = f(p - q), i.e.:
Pd(θ=π/2) = ½f(θ=π/2)½2 +½f(θ=π/2)½2 = 2½f(θ=π/2)½2 | Pi(θ=π/2) =½f(π/2) + f(π/2)½2 = 4½f(π/2)½2

But unfortunately it's more complicated if we observe dispersion of identical electrons. It will be the following in this case:

Pi(Elektronen) = ½f(θ) − f(p-q)½2

And also

for θ = π/2: Pi (π/2) = 0

To the best of our knowledge two electrons are identical particles that's why scientists tried (certainly, in the past) to find special feature that could help them to identify each electron. This internal electron characteristic is called Spin! (Stern-Gerlach Experiment). We now know that there are two "sorts" of electron, i.e. spin can take only two states which we will designate as Spin up and Spin down. Two colliding electrons with spins up and correspondingly with spins down have the same features in diffraction experiment, i.e. there is no special identification characteristic according to which one can identify each electron and therefore it's necessary:

Pi = ½f(θ) − f(p-q)½2.

If an electron with Spin up collide with an electron with Spin down, it's two distinguishable particles and that's why one can write down

Pd = ½f(θ)½2+½f(p-q)½2

An electron has spin s = 1/2 (both projections are possible ms = +1/2 and ms = -1/2 for spin up and spin down). Spin values can vary for other particles but it's multiple of 1/2 and therefore all particles are divided into 2 classes:

Bose particles (integer spin): f(θ) + f(p - q) [α-particles, photons]
Fermi particles (half-integer spin): f(θ) − f(p - q) [electrons, muons]

If we first have a look at two Fermi particles going in one direction then they have dispersion amplitudes f(θ1) and f(θ2). In order to obtain the Fermi particle probability one should subtract two dispersion amplitudes from each other:

Pi = ½f(θ1) − f(θ2)½2

If both particles move in the same direction then it should be θ1 = θ2 or finally

Pi = 0 !!!

Hence two electrons with the same spin can't situate at the same state. This is the well-known Pauli exclusion principle (Wolfgang Pauli).

We meet with spin conception later again when we will study atomic and molecular spectra. However we now return again to our double slit experiment in order to describe sudden behavior of a particle using more comprehensive mathematics. Moreover we will meet again with strange angle brackets and wavefunctions.

The Stern-Gerlach experiment schematic layout: the atomic beam comes through homogeneous magnetic field. One observes fragmentation of incident beam into 2 components which are designated as spin up and spin down. We would observe uniformly filled area (as it is shown on the left part of this figure).