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The probability P(θ) to detect particle at angle θ depends on the dispersion amplitude f(θ) for | |||
distinguishable particles: | non distinguishable particles: | ||
for instance, nitrogen and hydrogen atoms.
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for instance α-particles (helium nuclei, He++)
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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:
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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]
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.
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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). |