Quantum Conservation Laws

Being given a stationary operator O it fulfills the Heisenberg equation (see Sections 2.3.5 and 2.4.6)  

Quantum Nanochemistry-Volume I Quantum Theory and Observability 

The first eminent example of this result is the Hamiltonian itself for a system as far it does not depend explicitly on time, i.e., through the potential energy, it fulfills the eonditions  

Going to analyze Ihe momentum operatorial behavior, it influences ihe translation of systems through the unitary translational operator, in the same way as introduced the time evolution operator  

Since the translational operator depends on momentum as well, it satisfies the specific functional commutation with coordinate (see Section 2.4.4), namely 

---

Calling ui = 1/A the characteristic frequency of the quantum CA, the general conservation law that holds for QCA-I systems may then be written as... 

According to the quantum-chemical law of conservation of orbital symmetry in products and reactants of the elementary reaction, all products of tetroxide decomposition should have the singlet orbits including dioxygen [4], The singlet dioxygen is formed as a result of R02 disproportionation, however, in a yield sufficiently less than unity [15]. 

All reactions of this scheme are in accordance with the quantum-chemical law of conservation of orbital symmetry [4]. 

Because of the large number of rotational levels in the upper and lower states, the overlap between the exciting laser line and the dopp-ler broadened absorption profile may be nonzero simultaneously for several transitions (u", / ) (v, f) with different vibrational quantum numbers v and rotational numbers J. This means, in other words, that the energy conservation law allows several upper levels to be populated by absorption of laser photons from different lower levels. 

If this is not an A but a C particle initially excited, then the quantum yields of fluorescence through both channels, r cc and q lr, can be found from the same formulas (3.706) and (3.708) but with the opposite initial conditions PA(0) 0. P( (()) = 1. The conservation law takes place in this case as well... 

As an extension of Noether s theorem to quantum mechanics, the hypervirial theorem [101] derives conservation laws from invariant transformations of the theory. Consider a unitary transformation of the Schrodinger equation, U(H — F)T = U(H — = 0, and assume the variational Hilbert space closed under a... 

A probing process would force a physical transition from box-Hilbert space states to an asymptotic (laboratory) physical space. The transitions from box states to asymptotic or laboratory states are produced by external sources forcing conservation laws as the case might be. If one wants to speak in terms of time evolution, such process cannot be given in terms of standard unitary quantum-mechanical time evolution. 

The conservation of angular momentum is a consequence of isotropy or spherical rotational symmetry of space (1.3.1). An alternative statement of a conservation law is in terms of a nonobservable, which in this case is an absolute direction in space. Whenever an absolute direction is observed, conservation no longer holds, and vice versa. The alignment of spin, that allows of no intermediate orientations, defines such a direction with respect to conservation of angular momentum. One infers that space is not rotationally symmetrical at the quantum level. 

Since the atomic SU(2) quantum phase, discussed in Section III.B, is defined by the angular momentum of the excited atomic state, the conservation law (62) can be used to determine the field counterpart of the exponential of the phase operator (41) and other operators referred to the SU(2) quantum phase [36,46], For example, it is easily seen that the operator... 

The phase information is transmitted from the quantum source (atom) to photons via the conservation laws. In fact, only three physical quantities are conserved in the process of radiation energy, linear momentum, and angular momentum [26]. All of them are represented by the bilinear forms in the photon operators. 

Since the wavelengths of visible and infrared photons are always large in comparison to the lattice constants, the scattering vector Q can take on only vanishingly small values. Therefore, Raman scattering detects phonons only at K 0, i.e. at the centre F of the first BZ (G = 0). In addition to the conservation laws, symmetry selection rules hold, which take into account the fact that Raman scattering is a two-quantum process. When g and u states are present, it is found for Raman processes that g g transitions are allowed and w g transitions are forbidden. Therefore, the lines in the Raman spectrum can be associated with the symmetry of the excited phonons. 

The equations achieved above via the conservation laws do not contains gradients directly. Still they allow to constraint dimensional quantities. Technically that appears as a consequence of presence of certain dimensional conserved quantities, such as pii (as we demonstrated its conservation is related to a gradient term of the equation of motion). The dimensionless combination on which the constraint is achieved contains such a conserved quantity. It originates from the initial conditions and that is why there is no analog of it in quantum theory where the atomic energy levels are determined only by the fundamental constants and not by any initial conditions. 

A very remote analogy is conservation laws in classical and quantum physics. Description of quantum mechanics in terms of classical mechanics is not well defined, which happens because of commutativity of classical values and noncommutativity of their quantum analogs. We should regularize it and as a result part of classical symmetries may be realized in such a way that some conservation laws cannot be measured at all (e.g., conservation of the angular momentum as a vector). That example turns our attention to problem of observations. 

Some effects may be a pure observational problem. We can illustrate it by comparing conservations in classical and quantum physics. We remark that we cannot check any conservation laws, but only their consequences. From the point of view of classical physics we expect that we can measure different components of angular momentum and check at some time whether they have the same values. From quantum physics we know that they would not have the same value and that we can directly check only conservation of one component of the angular momentum. Conservation of the angular momentum as a vector can be checked via some specific consequences, but not so directly. 

Imagine a well-isolated spaceship observed in a space-fixed coordinate system. Its energy is preserved, its center of mass moves along a straight line with constant velocity (the total, or center-of-mass, momentum vector is preserved), and it preserves the total angular momentum. The same is true for a molecule or atom, but the conservation laws have to be formulated in the language of quantum mechanics. 

---