Chapter 2
A Comment on the Question of Degeneracies
in Quantum Mechanics
Michal Svrˇ cek
Abstract The problem of degeneracies, descending from the Born-Oppenheimer
(B-O) approximation serves as a “comeback backdoor” of the principle of complementarity, but on a much more subtle level. Quantum mechanics incorporates
both mechanical and field theory features, which results in the well-known particlewave aspects of complementarity. The degeneracy problem, however, prompts a
new type of “property-object” complementary phenomena. This leads to serious
consequences: Field theoretical methods, unlike mechanical ones, are incapable of
separating the internal and the external degrees of freedom with respect to the centre of gravity, but on the other hand adapt relativistically in a natural manner very
similar to the space-time formulas of Maxwell’s equations. The solutions of the
quantum field equations, relativistic in the mentioned specific sense, yield singularities at symmetric points that correspond to the well-known B-O degeneracies
giving the latter in actual fact a metaphysical attribute. However, Nature has in this
case a more sophisticated method or modus operandi to avoid degenerations and to
instigate symmetry violations.
In quantum mechanics, we often encounter degenerate states, which are authentic
and experimentally detectable. The most famous case of degeneracy removal is the
splitting of states under the influence of external electric or magnetic fields (Stark
and Zeeman effects). On the other hand, we also often come across virtual degenerate states that are the product of a simplified Hamiltonian, which we usually have to
choose due to the possibility of a realistic analytical solution when the total Hamiltonian does not directly provide such a solution. Since the step toward the answer
exploit the principle of superposition, the simplified Hamiltonian may lead to nonexistent fictional degenerations, which are eventually eliminated when taking the
total Hamiltonian into consideration. This removal is either resolved in perturbation
theory or in a non-perturbative approach based on multiconfigurational interaction.
Realistic degenerate states are mostly well defined and they do not therefore need
to be considered further here. In contrast, in the case of so-called virtual degenerate
M. Svrˇ cek (B)
Centre de Mécanique Ondulatoire Appliquée, CMOA Czech Branch, Carlsbad, Czech Republic
e-mail: m.sv@o2active.cz
M. Hotokka et al. (eds.), Advances in Quantum Methods and Applications in
Chemistry, Physics, and Biology, Progress in Theoretical Chemistry and Physics 27,
DOI 10.1007/978-3-319-01529-3_2,
© Springer International Publishing Switzerland 2013
41
A Comment on the Question of Degeneracies
in Quantum Mechanics
Michal Svrˇ cek
Abstract The problem of degeneracies, descending from the Born-Oppenheimer
(B-O) approximation serves as a “comeback backdoor” of the principle of complementarity, but on a much more subtle level. Quantum mechanics incorporates
both mechanical and field theory features, which results in the well-known particlewave aspects of complementarity. The degeneracy problem, however, prompts a
new type of “property-object” complementary phenomena. This leads to serious
consequences: Field theoretical methods, unlike mechanical ones, are incapable of
separating the internal and the external degrees of freedom with respect to the centre of gravity, but on the other hand adapt relativistically in a natural manner very
similar to the space-time formulas of Maxwell’s equations. The solutions of the
quantum field equations, relativistic in the mentioned specific sense, yield singularities at symmetric points that correspond to the well-known B-O degeneracies
giving the latter in actual fact a metaphysical attribute. However, Nature has in this
case a more sophisticated method or modus operandi to avoid degenerations and to
instigate symmetry violations.
In quantum mechanics, we often encounter degenerate states, which are authentic
and experimentally detectable. The most famous case of degeneracy removal is the
splitting of states under the influence of external electric or magnetic fields (Stark
and Zeeman effects). On the other hand, we also often come across virtual degenerate states that are the product of a simplified Hamiltonian, which we usually have to
choose due to the possibility of a realistic analytical solution when the total Hamiltonian does not directly provide such a solution. Since the step toward the answer
exploit the principle of superposition, the simplified Hamiltonian may lead to nonexistent fictional degenerations, which are eventually eliminated when taking the
total Hamiltonian into consideration. This removal is either resolved in perturbation
theory or in a non-perturbative approach based on multiconfigurational interaction.
Realistic degenerate states are mostly well defined and they do not therefore need
to be considered further here. In contrast, in the case of so-called virtual degenerate
M. Svrˇ cek (B)
Centre de Mécanique Ondulatoire Appliquée, CMOA Czech Branch, Carlsbad, Czech Republic
e-mail: m.sv@o2active.cz
M. Hotokka et al. (eds.), Advances in Quantum Methods and Applications in
Chemistry, Physics, and Biology, Progress in Theoretical Chemistry and Physics 27,
DOI 10.1007/978-3-319-01529-3_2,
© Springer International Publishing Switzerland 2013
41