30
B. Sutcliffe and R.G. Woolley
for the answer (for example, loss of permutation symmetry, loss of parity—in the
case of chirality) and have no clear connection to Löwdin’s’s Coulomb Hamiltonian.
An alternative account that is based on the Coulomb Hamiltonian may however
be possible in the light of the fact that each part of it in the division made in (1.26)
has a completely continuous spectrum. As noted in Sect. 1.3.4 the formal eigenvectors of ˆ
H elec from the ground state up can exhibit extensive degeneracy. It might be
that ‘broken symmetry’ solutions corresponding to molecular structure could result
from treating the two parts as asymptotic states in a scattering or reaction process in
a manner analogous to that used in standard S-matrix theory. Such a state would be
characterised as a ‘resonance’ and would have to be long-lived to be describable as
a molecule. Only a true bound state, of infinite lifetime, such as a bound eigenstate
of the molecular Hamiltonian is really independent of how it was formed; resonant
states have histories that describe the environment of their preparation. The Potential Energy Surface would then appear only as an auxiliary concept through the involvement of the clamped-nuclei Hamiltonian in the construction of the states {ϕ m }
required for (1.36). That however remains subject to further investigation.
When Bohr introduced his quantum theory of the Rutherford atomic model of the
hydrogen atom he made a drastic change in the status of electrodynamics. Hitherto,
it had been understood 17 that charged particles affected, and were affected by, the
electromagnetic field, and that was the root cause of the failure of a dynamical classical atom (‘radiation damping’ is a strong coupling interaction). Bohr relegated the
electromagnetic field to a perturbation theoretic—weak coupling—role; the charges
interacted among themselves according to Coulomb’s law, to be treated as a strong
coupling situation, and would exist permanently in the stationary states selected by
the quantization conditions unless perturbed by an ‘external’ electromagnetic field
which produced ‘transitions’. That perturbation theory viewpoint was maintained
when quantum theory was applied to the atom and the electromagnetic field, and
largely survives to this day, to the extent that the electromagnetic field is frequently
regarded as a classical system. Such a spectroscopic viewpoint is not appropriate
in the present context; quantum electrodynamics teaches us that there is no strict
separation of charged particles and the (quantized) electromagnetic field, not least
because of the requirements of gauge invariance.
The difficulty with quantum mechanical perturbation theory for the interaction
of atomic/molecular systems with radiation is this: the spectrum of the unperturbed
atom/molecule consists of a continuum corresponding to the half-axis [Σ, ∞) for
some Σ ≤ 0, and discrete energy levels E 0 , E 1 , . . . below the continuum, that is
E 0 ≤ E 1 ≤ · · · < Σ [66–68]. The spectrum of the free electromagnetic field Hamiltonian consists of a simple eigenvalue at 0, corresponding to the vacuum state Ψ 0 ,
and absolutely continuous spectrum on the half-axis [0, ∞). This means that when
coupling between particles and radiation is admitted, all the discrete energy levels of the atomic system including E 0 become thresholds of continuous spectra; a
quantum theory of matter and the electromagnetic field therefore requires the perturbation theory of continuous spectra. The quantum mechanical theory of electrons
17 The Lorentz Theory of the electron for example [77].
B. Sutcliffe and R.G. Woolley
for the answer (for example, loss of permutation symmetry, loss of parity—in the
case of chirality) and have no clear connection to Löwdin’s’s Coulomb Hamiltonian.
An alternative account that is based on the Coulomb Hamiltonian may however
be possible in the light of the fact that each part of it in the division made in (1.26)
has a completely continuous spectrum. As noted in Sect. 1.3.4 the formal eigenvectors of ˆ
H elec from the ground state up can exhibit extensive degeneracy. It might be
that ‘broken symmetry’ solutions corresponding to molecular structure could result
from treating the two parts as asymptotic states in a scattering or reaction process in
a manner analogous to that used in standard S-matrix theory. Such a state would be
characterised as a ‘resonance’ and would have to be long-lived to be describable as
a molecule. Only a true bound state, of infinite lifetime, such as a bound eigenstate
of the molecular Hamiltonian is really independent of how it was formed; resonant
states have histories that describe the environment of their preparation. The Potential Energy Surface would then appear only as an auxiliary concept through the involvement of the clamped-nuclei Hamiltonian in the construction of the states {ϕ m }
required for (1.36). That however remains subject to further investigation.
When Bohr introduced his quantum theory of the Rutherford atomic model of the
hydrogen atom he made a drastic change in the status of electrodynamics. Hitherto,
it had been understood 17 that charged particles affected, and were affected by, the
electromagnetic field, and that was the root cause of the failure of a dynamical classical atom (‘radiation damping’ is a strong coupling interaction). Bohr relegated the
electromagnetic field to a perturbation theoretic—weak coupling—role; the charges
interacted among themselves according to Coulomb’s law, to be treated as a strong
coupling situation, and would exist permanently in the stationary states selected by
the quantization conditions unless perturbed by an ‘external’ electromagnetic field
which produced ‘transitions’. That perturbation theory viewpoint was maintained
when quantum theory was applied to the atom and the electromagnetic field, and
largely survives to this day, to the extent that the electromagnetic field is frequently
regarded as a classical system. Such a spectroscopic viewpoint is not appropriate
in the present context; quantum electrodynamics teaches us that there is no strict
separation of charged particles and the (quantized) electromagnetic field, not least
because of the requirements of gauge invariance.
The difficulty with quantum mechanical perturbation theory for the interaction
of atomic/molecular systems with radiation is this: the spectrum of the unperturbed
atom/molecule consists of a continuum corresponding to the half-axis [Σ, ∞) for
some Σ ≤ 0, and discrete energy levels E 0 , E 1 , . . . below the continuum, that is
E 0 ≤ E 1 ≤ · · · < Σ [66–68]. The spectrum of the free electromagnetic field Hamiltonian consists of a simple eigenvalue at 0, corresponding to the vacuum state Ψ 0 ,
and absolutely continuous spectrum on the half-axis [0, ∞). This means that when
coupling between particles and radiation is admitted, all the discrete energy levels of the atomic system including E 0 become thresholds of continuous spectra; a
quantum theory of matter and the electromagnetic field therefore requires the perturbation theory of continuous spectra. The quantum mechanical theory of electrons
17 The Lorentz Theory of the electron for example [77].