Quantum Field Theory I

32 CHAPTER 1. INTRODUCTIONS
Born-Oppenheimerapproximation treatsnucleiandelectronsondifferent
footing. The intuitive argument is this: nuclei are much heavier, therefore they
would typically move much slower. So one can perhaps gain a good insight by
solving first the electron problem for nuclei at fixed positions (these positions,
collectivelydenoted as ⃗ R, areunderstoodasparametersofthe electronproblem)
H e ( ⃗ R)ψ n (⃗r) = ε n ( ⃗ R)ψ n (⃗r)
H e ( ⃗ R) = − ∑ i
∆ i
+V +W
2m
and only afterwardsto solvethe nucleus problem with the electron energyε n ( ⃗ R)
understood as a part of the potential energy on the nuclei
H N Ψ m ( ⃗ R) = E m Ψ m ( ⃗ R)
H N = − ∑ i
∆ i
2M +U +ε n( ⃗ R)
The natural question as to which electron energy level (which n) is to be used in
H N is answered in a natural way: one uses some kind of mean value, typically
over the canonical ensemble, i.e. the ε n ( ⃗ R) in the definition of the H N is to be
replaced by ¯ε( ⃗ R) = ∑ n ε n( ⃗ R)exp{−ε n ( ⃗ R)/kT}.
The formal derivation of the above equations is not important for us, but
we can nevertheless sketch it briefly. The eigenfunctions Φ m ( ⃗ R,⃗r) of the full
Hamiltonian H are expanded in terms of the complete system (in the variable⃗r)
of functions ψ n (⃗r): Φ m ( ⃗ R,⃗r) = c mn ( ⃗ R)ψ n (⃗r). The coefficients of this expansion
are, of course, ⃗ R-dependent. Plugging into HΦm = E m Φ m one obtains the
equation for c mn ( ⃗ R) which is, apart of an extra term, identical to the above
equation for Ψ m ( ⃗ R). The equation for c mn ( ⃗ R) is solved iteratively, the zeroth
iteration ignores the extra term completely. As to the weighted average ¯ε( ⃗ R)
the derivation is a bit more complicated, since it has to involve the statistical
physics from the beginning, but the idea remains unchanged.
The Born-Oppenheimer adiabatic approximation is nothing else but the zeroth
iteration of this systematic procedure. Once the zeroth iteration is solved,
one can evaluate the neglected term and use this value in the first iteration.
But usually one contents oneself by checking if the value is small enough, which
should indicate that already the zeroth iteration was good enough. In the solid
state physics one usually does not go beyond the zeroth approximation, not
even check whether the neglected term comes out small enough, simply because
this is too difficult.
In the QM of molecules, the Born-Oppenheimer approximation stands behind
our qualitative understanding of the chemical binding, which is undoubtedly
oneofthe greatestandthe most far-reachingachievementsofQM.Needless
to say, this qualitative understanding is supported by impressive quantitative
successes of quantum chemistry, which are based on the same underlying approximation.
In the solid state physics, the role of this approximation is more
modest. It servesonlyasaformaltoolallowingforthe separationofthe electron
and nucleus problems. Nevertheless, it is very important since it sets the basic
framework and paves the way for the other two celebrated approximations.