Quantum Mechanics
Schrodinger equation
Quantum Mechanics
Propagators : Pg
Quantum Simple Harmonic Oscillator QSHO
Quantum Mechanics
Simulation With GNU Octave
© The scientific sentence. 2010


Manyelectron atom:
Identical and noninteracting particles
1. Two nonidentical and noninteracting particles:
Consider a system that contain two identical particles of mass m as Helium
atom. The instantaneous state of the system is: ψ(r1, r2,t) that we
can, if these particles are non identical and noninteracting ,
express by:
ψ(r1,r2,t) = ψ(r1,t) ψ(r2,t)
With
ψ_{E}(r1, r2) =
ψ_{E1}(r1) ψ_{E2}(r2), and
E = E_{1} + E_{2}
ψ(r1,r2,t)^{2} dr1 dr2 = the probability of finding a first
particle between r1 and r1 + dr1 and the second particle
between r2 and r2 + dr2, at a time t.
What about if they are identical and noninteracting particles
2. Two identical and noninteracting particles:
If the two particles of the system are identical we can write
that the probability to find a first particle, at a time t, at a position
between r and r + dr is the same for the second particle, because the two
particles are indistinguishable.
In other words, since the two particle are identical, the probability
to find a first particle between r1 and r1 + dr1 and the second particle
between r2 and r2 + dr2 is equal to the probability of finding the
first particle between r2 and r2 + dr2 and the second particle
between r1 and r1 + dr1
2.1 Symmetry of of the system's wavefunction
The first probability is ψ(r1,r2,t)^{2}dr1 dr2, and the second
is ψ(r2,r1,t)^{2}dr2 dr2. Therefore:
ψ(r1,r2,t)^{2} = ψ(r2,r1,t)^{2}
Hence
ψ(r1,r2,t) = ε ψ(r2,r1,t) exp {i θ}
With ε = ± 1.
We conclude that for a system that containing
two noninteracting identical particles, under label interchange, the wavefunction
of each one is either:
symmetric, that is:
ψ(r1,r2,t) = + ψ(r2,r1,t)
or
antisymmetric, that is:
ψ(r1,r2,t) =  ψ(r2,r1,t)
Any particles that have a symmetric wavefunction obey BoseEinstein Statistics, and
are called bosons (s photons) . The particles that have an antisymmetric wavefunction obey
FermiDirac Statistics, and are called fermions (as electrons and protons).
For a system of two particles, the wavefunction ψ(r2,r1,t) that
describe this system is no longer just the product of its
components ψ(r1,t) and ψ(r2,t) of each particle. It is
a linear combination of this product:
ψ(r2,r1,t) = c1 ψ(r1,t) ψ(r2,t) + c2 ψ(r2,t) ψ(r1,t)
With
ψ(r1,r2,t) = ψ(r1,r2) exp{ iE/ħ}
ψ(r1,t) = ψ(r1) exp{ iEa/ħ}
ψ(r2,t) = ψ(r2) exp{ iEb/ħ}
With E = Ea + Eb
That gives with the stationary wavefunction:
ψ(r2,r1) = c1 ψ(r1) ψ(r2) + c2 ψ(r2) ψ(r1)
(1)
For the stationary state, at the position r1, we can have
either the particle 1 with energy Ea, or the particle 2
with energy Eb, and at the position r2, we have
either the particle 1 with energy Ea, or the particle 2
with energy Eb. Indeed, since the particles are identical,
we cannot be sure which particle has energy Ea, and which has
energy Eb. We say one particle has energy Ea, and the other Eb.
That is, we have either:
ψ(r1,Ea) ψ(r2,Eb) or ψ(r1,Eb) ψ(r1,Ea)
The above equation is, instead, written as:
ψ(r2,r1,E) = c1 ψ(r1,Ea) ψ(r2,Eb) + c2 ψ(r1,Eb) ψ(r2,Ea)
ψ(r2,r1,E) = c1 ψ(r1,Ea) ψ(r2,Eb) + c2 ψ(r1,Eb) ψ(r2,Ea)
(2)
The normalisation of this function is written as:
1 = ψ(r1,r2)^{2} = <ψ^{*}(r1,r2)ψ(r1,r2)>
= ∫ dr1 dr2 ψ^{*}(r1,r2)ψ(r1,r2)
from  ∞ to + ∞
= < c1^{*}ψ^{*}(r1,Ea) ψ(r2,Eb) + c2^{*} ψ^{*}(r2) ψ^{*}(r2,Eb)  c1 ψ(r1) ψ(r2) + c2 ψ(r2) ψ(r1) >
=
< c1^{*} c1 ψ^{*}(r1,Ea) ψ^{*}(r2,Eb) 
ψ(r1,Ea) ψ(r2,Eb)> +
< c1^{*} c2 ψ^{*}(r1,Ea) ψ^{*}(r2,Eb) 
ψ(r1,Eb) ψ(r2,Ea) > +
< c2^{*} c1 ψ^{*}(r1,Eb) ψ^{*}(r2,Ea) 
ψ(r1,Ea) ψ(r2,Eb)> +
< c2^{*} c2 ψ^{*}(r1,Eb) ψ^{*}(r2,Ea) 
c2 ψ(r1,Eb) ψ(r2,Ea)>
The four wavefunctions ψ(ri,Ej) (i = 1, 2, and j = a, b)
are orthogonal. It remains:
c1^{2} + c2^{2} = 1
The outcomes are equally likely to occur, then:
c1^{2} = c2^{2}
Therefore:
c1^{2} = 1/2 , and c1^{2} = ± 1/(2)^{1/2}
The equality (2) becomes:
ψ(r2,r1,E) = 1/(2)^{1/2} [ψ(r1,Ea) ψ(r2,Eb) &plusmin; ψ(r1,Eb) ψ(r2,Ea)]
+ for the bosons and  for the fermions.
Therefore:
For a system of two identical and noninteracting fermions with an antisymmetric wavefunction:
ψ(r1,r2,t) =  ψ(r2,r1,t)
The stationary wavefunction of the whole system is:
ψ(r1,r2,E) = 1/(2)^{1/2} [ψ(r1,Ea) ψ(r2,Eb)  ψ(r1,Eb) ψ(r2,Ea)]
For a system of two identical and noninteracting bosons with an antisymmetric wavefunction:
ψ(r1,r2,t) = +ψ(r2,r1,t)
The stationary wavefunction of the whole system is:
ψ(r1,r2,E) = 1/(2)^{1/2} [ψ(r1,Ea) ψ(r2,Eb) + ψ(r1,Eb) ψ(r2,Ea)]
For a system of two nonidentical (distinguishable) and noninteracting particles,
the stationary wavefunction of the system is simply:
ψ(r1,r2,E) = ψ(r1,Ea) ψ(r2,Eb)
2.2 Pauli exclusion principle
If Ea = Eb for the fermions, then their total wavefunction ψ(r2,r1,E)
becomes zero. That is, there is no state!. We must then exclude this
possibility and say that it is impossible for the two fermions
to occupy the same singleparticle stationary state.
That is the Pauli exclusion principle.
3. Noninteracting
and identical multiparticles system
The symmetry of a wavefunction of a system with two noninteracting
and identical particles can be extended to systems containing more than two identical particles. For a system containing N identical and noninteracting fermions, the antisymmetric stationary wavefunction of the system is written as the
Slater determinant:

