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The quantum mechanics of many-particle systems, focusing on the implications of particle indistinguishability for fermions and bosons. Fermions, with half-integer spin, have antisymmetric wavefunctions under particle exchange, while bosons, with integer spin, have symmetric wavefunctions. The document also discusses the antisymmetry of wavefunctions under particle exchange and the implications for the spectrum of lowest excited states.
Typology: Lecture notes
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Until now, our focus has largely been on the study of quantum
mechanics of individual particles.
However, most physical systems involve interaction of many (ca.
23
!) particles, e.g. electrons in a solid, atoms in a gas, etc.
In classical mechanics, particles are always distinguishable – at least
formally, “trajectories” through phase space can be traced.
In quantum mechanics, particles can be identical and
indistinguishable, e.g. electrons in an atom or a metal.
The intrinsic uncertainty in position and momentum therefore
demands separate consideration of distinguishable and
indistinguishable quantum particles.
Here we define the quantum mechanics of many-particle systems,
and address (just) a few implications of particle indistinguishability.
But which sign should we choose?
ψ(x 1
, x 2
) = ψ(x 2
, x 1
) bosons
ψ(x 1
, x 2
) = −ψ(x 2
, x 1
) fermions
All elementary particles are classified as
fermions or bosons:
(^1) Particles with half-integer spin are fermions and their
wavefunction must be antisymmetric under particle exchange.
e.g. electron, positron, neutron, proton, quarks, muons, etc.
2 Particles with integer spin (including zero) are bosons and their
wavefunction must be symmetric under particle exchange.
e.g. pion, kaon, photon, gluon, etc.
Within non-relativistic quantum mechanics, correlation between spin
and statistics can be seen as an empirical law.
However, the spin-statistics relation emerges naturally from the
unification of quantum mechanics and special relativity.
The rule that fermions have half-integer spin and
bosons have integer spin is internally consistent:
e.g. Two identical nuclei, composed of n nucleons
(fermions), would have integer or half-integer spin
and would transform as a “composite” fermion or
boson according to whether n is even or odd.
We could achieve antisymmetrization for particles 1 and 2 by
subtracting the same product with 1 and 2 interchanged,
ψ a
(1)ψ b
(2)ψ c
(3) $→ [ψ a
(1)ψ b
(2) − ψ a
(2)ψ b
(1)] ψ c
However, wavefunction must be antisymmetrized under all possible
exchanges. So, for 3 particles, we must add together all 3!
permutations of 1, 2 , 3 in the state a, b, c with factor −1 for each
particle exchange.
Such a sum is known as a Slater determinant:
ψ abc
ψ a
(1) ψ b
(1) ψ c
ψ a
(2) ψ b
(2) ψ c
ψ a
(3) ψ b
(3) ψ c
and can be generalized to N, ψ i 1 ,i 2 ,···iN
(1, 2 , · · · N) = det(ψ i
(n))
ψ abc
ψ a
(1) ψ b
(1) ψ c
ψ a
(2) ψ b
(2) ψ c
ψ a
(3) ψ b
(3) ψ c
Antisymmetry of wavefunction under particle exchange follows from
antisymmetry of Slater determinant, ψ abc
(1, 2 , 3) = −ψ abc
Moreover, determinant is non-vanishing only if all three states a, b,
c are different – manifestation of Pauli’s exclusion principle: two
identical fermions can not occupy the same state.
Wavefunction is exact for non-interacting fermions, and provides a
useful platform to study weakly interacting systems from a
perturbative scheme.
When Hamiltonian is spin-independent, wavefunction can be
factorized into spin and spatial components.
For two electrons (fermions), there are four basis states in spin
space: the (antisymmetric) spin S = 0 singlet state,
|χ S
1
2
1
2
and the three (symmetric) spin S = 1 triplet states,
|χ
1
T
1
2
〉, |χ
0
T
1
2
1
2
〉) , |χ
− 1
T
1
2
For a general state, total wavefunction for two electrons:
Ψ(r 1
, s 1
; r 2
, s 2
) = ψ(r 1
, r 2
)χ(s 1
, s 2
where χ(s 1
, s 2
) = 〈s 1
, s 2
|χ〉.
For two electrons, total wavefunction, Ψ, must be antisymmetric
under exchange.
i.e. spin singlet state must have symmetric spatial wavefunction;
spin triplet states have antisymmetric spatial wavefunction.
For three electron wavefunctions, situation becomes challenging...
see notes.
The conditions on wavefunction antisymmetry imply spin-dependent
correlations even where the Hamiltonian is spin-independent, and
leads to numerous physical manifestations...
Although, after hydrogen, helium is simplest
atom with two protons (Z = 2), two neutrons,
and two bound electrons, the Schr¨odinger
equation is analytically intractable.
In absence of electron-electron interaction, electron Hamiltonian
(0)
=
2 ∑
n=
pˆ
2
n
2 m
, V (r ) = −
4 π& 0
Ze
2
r
is separable and states can be expressed through eigenstates, ψ n"m
of hydrogen-like Hamiltonian.
(0)
=
2 ∑
n=
pˆ
2
n
2 m
In this approximation, ground state wavefunction involves both
electrons in 1s state! antisymmetric spin singlet wavefunction,
g.s.
〉 = (| 100 〉 ⊕ | 100 〉)|χ S
Previously, we have used perturbative theory to determine how
ground state energy is perturbed by electron-electron interaction,
(1)
=
4 π& 0
e
2
|r 1
− r 2
What are implications of particle statistics on spectrum of lowest
excited states?
|ψ p,o
(| 100 〉 ⊗ | 2 %m〉 ±| 2 %m〉 ⊗ | 100 〉) |χ
ms
S,T
Despite degeneracy, since off-diagonal matrix elements between
different m, % values vanish, we can invoke first order perturbation
theory to determine energy shift for ortho- and parahelium,
p,o
n"
= 〈ψ p,o
(1)
|ψ p,o
e
2
4 π& 0
d
3
r 1
d
3
r 2
|ψ 100
(r 1
)ψ n" 0
(r 2
) ± ψ n" 0
(r 1
)ψ 100
(r 2
2
|r 1
− r 2
(+) parahelium and (-) orthohelium.
N.B. since matrix element is independent of m, m = 0 value
considered here applies to all values of m.
p,o
n"
e
2
4 π& 0
d
3
r 1
d
3
r 2
|ψ 100
(r 1
)ψ n" 0
(r 2
) ± ψ n" 0
(r 1
)ψ 100
(r 2
2
|r 1
− r 2
Rearranging this expression, we obtain
p,o
n"
n"
n"
where diagonal and cross-terms given by
n"
e
2
4 π& 0
d
3
r 1
d
3
r 2
|ψ 100
(r 1
2
|ψ n" 0
(r 2
2
|r 1
− r 2
n"
e
2
4 π& 0
d
3
r 1
d
3
r 2
ψ
∗
100
(r 1
)ψ
∗
n" 0
(r 2
)ψ 100
(r 2
)ψ n" 0
(r 1
|r 1
− r 2
|ψ p,o
(| 100 〉 ⊗ |n%m〉 ±| n%m〉 ⊗ | 100 〉) |χ
ms
S,T
p,o
n"
n"
n"
Finally, noting that, with S = S 1
2
2
1
2
2
1
2
2
− S
2
1
2
2
1 / 2 triplet
− 3 / 2 singlet
the energy shift can be written as
p,o
n"
n"
2
1
2
n"
From this result, we can conclude that electron-electron interaction
leads to effective ferromagnetic interaction between spins.
Similar phenomenology finds manifestation in metallic systems as
Stoner ferromagnetism.