In the framework of quantum theory, we present one theorem and three corollaries regarding the direct connection between constants of motion of a physical system and degeneracies of its energy eigenvalues. It is shown that this connection emerges when there exist quantum operators which commute with the Hamiltonian, but not with each other. Further it is shown that if the commutator of these operators is a nonvanishing constant number then (a) all the eigenvalues of the system are degenerate, and (b) the degree of degeneracy is infinite. A number of examples are discussed including the parity degeneracy of the hydrogen atom and the infinite degeneracy of the Landau levels of a charged particle in a constant magnetic field.
In the framework of quantum theory, we present one theorem and three corollaries regarding the direct connection between constants of motion of a physical system and degeneracies of its energy eigenvalues. It is shown that this connection emerges when there exist quantum operators which commute with the Hamiltonian, but not with each other. Further it is shown that if the commutator of these operators is a nonvanishing constant number then (a) all the eigenvalues of the system are degenerate, and (b) the degree of degeneracy is infinite. A number of examples are discussed including the parity degeneracy of the hydrogen atom and the infinite degeneracy of the Landau levels of a charged particle in a constant magnetic field.
Nonrelativistic Hamiltonians with large, even infinite, ground-state degeneracy are studied by connecting the degeneracy to the property of a Dirac operator. We then identify a special class of Hamiltonians, for which the full space of degenerate ground states in any spatial dimension can be exhibited explicitly. The two-dimensional version of the latter coincides with the Pauli Hamiltonian, and recently-discussed models leading to higher-dimensional Landau levels are obtained as special cases of the higher-dimensional version of this Hamiltonian. But, in our framework, it is only the asymptotic behavior of the background 'potential' that matters for the ground-state degeneracy. We work out in detail the ground states of the three-dimensional model in the presence of a uniform magnetic field and such potential. In the latter case one can see degenerate stacking of all 2d Landau levels along the magnetic field axis.
Abstract In order to extend the use of group theoretical arguments to the problem of accidental degeneracy in quantum mechanics, a new type of constant of the motion, known as a conditional constant of the motion, is introduced. Such a quantity, instead of commuting with the Hamiltonian H for the system, satisfies the more general relation HA = A†H, where A† denotes the hermitian conjugate (adjoint) of the conditional constant of the motion A. This expression reduces, if A is hermitian, to the usual definition of a constant of the motion. Otherwise it defines a new type of invariance, and it is this which will be referred to as conditional invariance. A discussion of the difficulties arising from the lack of hermiticity of A, which is of course essential to its definition, is given. In particular it is shown, under fairly general conditions, that the process of introducing a variable parameter in the Hamiltonian enabling it to have simultaneous eigenfunctions with A, gives rise to an eigenvalue equation in this parameter with respect to which A may be chosen to be hermitian. Conditional invariance is contrasted with both dynamical and geometric invariance. It is found to be sometimes replaceable by either of the latter forms of invariance and for such, explicit conditions are given. Some applications of conditional invariance are discussed. These include a study of the crossing of potential energy curves, a new model of symmetry breaking, a possible means of calculating the exact number of bound states for certain potentials and conditions for the existence of bound states near to the continuum.
By analogy with the mathematical definition of degenerate systems, it is suggested that systems which are interrupted before they have traversed a full period, or which are in an external field which varies greatly in the time of one period, be called physically degenerate. It is further suggested, in harmony with the ideas of Ehrenfest and Tolman, that such systems show weak quantization, the quantization becoming less and less complete as the degeneracy increases. In the case of partially degenerate systems, the values of the related action variable and of the energy are grouped more or less closely about the mean values which they would all have if the systems were completely quantized. As examples, these ideas are applied to several problems, such as quantization in a small or rapidly changing magnetic field (including Glaser's results for the change of diamagnetic susceptibility of some gases with pressure, experiments by Breit and Ellett and by Wood and Ellett on depolarization, and the experiment of Gerlach and Stern in which atoms are projected into a strong field), absorption in the higher lines of the principal series of an alkali (including a discussion of quantization in hyperbolic and parabolic obits), and the continuous x-ray spectrum. It is then pointed out that this assumption demands the existence of quantum forces which have not been previously considered in quantum theory and whose action is to stabilize the stationary states, bringing the action variables and energy rapidly nearer and nearer to the proper values as time goes on. The suggestion is made that these forces are responsible for the large change of energy and action variables during a quantum transition. From this point of view the adiabatic theorem appears in a simple light; when the external parameters vary too rapidly, the quantization becomes poor, the quantum forces become strong and put the atom into a quantized state again. In any atom but hydrogen, quantum forces must be continuously acting to oppose the interchange of energy and momentum between electrons and to keep each electron quantized. These forces thus may prove to be important in the solution of problems of the quantum dynamics of the constitution of atoms, though they must be considered in connection with the dynamics of the oscillators.
Certain Hamiltonians, based on two coupled quantum mechanical spins, exhibit degenerate eigenvalues despite having no obvious non-Abelian symmetries.Operators acting to permute the degenerate states do not have a simple form when expressed as polynomials of the generators of rotations for the respective spins.As observed in [3], one such Hamiltonian helps explain resonances in the spin relaxation rate of optically pumped Rb 2 , as a function of applied magnetic field.We give an explanation of why the degeneracies exist, based on properties of the commutator and anti-commutator of the Hamiltonian and its image under magnetic field reversal.
Certain Hamiltonians based on two coupled quantum mechanical spins exhibit degenerate eigenvalues despite having no obvious non-abelian symmetries. Operators acting to permute the degenerate states do not have a simple form when expressed as polynomials of the generators of rotations for the respective spins. As observed in [1], one such Hamiltonian helps explain resonances in the spin relaxation rate of optically pumped Rb_2, as a function of applied magnetic field. We give an explanation of why the degeneracies exist, based on properties of the commutator and anti-commutator of the Hamiltonian and its image under magnetic field reversal.
The recent two papers have successively proposed the concepts and more precise expressions of strict conservation or statistical conservation of physical quantities, which are different from the expression of a conserved quantity in quantum mechanics. The Hamiltonian operator of the system discussed in this paper are independent of time, so it is commutative with the operators of an angular momentum itself, its component and its square, so that all these quantities are conserved quantities. In an isolated composite system, one of the angular momentum components of the total system are strictly conserved, while the angular momentum components of the subsystems can only be conserved statistically, both of which can be expressed in terms of eigenvalues and eigenfunctions. Generally, angular momentum itself does not have eigenvalues and eigenfunctions, so it cannot be expressed as strict conservation in terms of these concepts. Although there are eigenvalues and eigenfunctions for the square of angular momentum, they cannot express strict conservation either. And they can only be expressed as statistical conservation.