In 1924, Bohr, Kramers and Slater tried to introduce into microphysics conservation principles that hold only on the average. This attempt was abandoned in the light of the Compton-Simon experiment (and its later refinements). Since that time, except for a moment of doubt in 1936, it has been thought that the classical conservation laws (for energy and momentum) hold in quantum theory for each individual interaction, in a way that yields the classical exchange-and-balance of momentum (or energy) familiar from the laws of elastic collisions. It has been thought, that is, that in each individual “collision” what one part of the total system loses in linear momentum (say) another part gains, so as to maintain the same total amount afterwards as before. To those familiar with discussions of the interpretation of quantum theory, however, it will be apparent that the very concepts needed to express this idea of an elastic collision are generally not admitted in the theory. For one needs the idea that both before and after collision the total system has a well-defined value for the conserved quantity and, moreover, that the various component parts of the system have well-defined values for those quantities out of which the conserved one is composed. But in the case of spin, for example, it is customary to say that although total spin may be conserved in certain interactions one cannot attribute values to the separate components of spin (in various directions) because the associated operators do not commute. (Thus the whole sum is not even the sum of its parts.) Moreover, one does not generally refer to the values of quantities except in eigenstates. Hence, in general, one would not refer to the value of the conserved quantity before and after interaction, except for interactions initiated in eigenstates.
In 1924, Bohr, Kramers and Slater tried to introduce into microphysics conservation principles that hold only on the average. This attempt was abandoned in the light of the Compton-Simon experiment (and its later refinements). Since that time, except for a moment of doubt in 1936, it has been thought that the classical conservation laws (for energy and momentum) hold in quantum theory for each individual interaction, in a way that yields the classical exchange-and-balance of momentum (or energy) familiar from the laws of elastic collisions. It has been thought, that is, that in each individual “collision” what one part of the total system loses in linear momentum (say) another part gains, so as to maintain the same total amount afterwards as before. To those familiar with discussions of the interpretation of quantum theory, however, it will be apparent that the very concepts needed to express this idea of an elastic collision are generally not admitted in the theory. For one needs the idea that both before and after collision the total system has a well-defined value for the conserved quantity and, moreover, that the various component parts of the system have well-defined values for those quantities out of which the conserved one is composed. But in the case of spin, for example, it is customary to say that although total spin may be conserved in certain interactions one cannot attribute values to the separate components of spin (in various directions) because the associated operators do not commute. (Thus the whole sum is not even the sum of its parts.) Moreover, one does not generally refer to the values of quantities except in eigenstates. Hence, in general, one would not refer to the value of the conserved quantity before and after interaction, except for interactions initiated in eigenstates.
Bohr et al. once believed that both the conservation of energy and the conservation of momentum in a microscopic process are statistically conserved. And then Bothe et al. verified that the laws of the conservations of total energy and total momentum of an electron and an X-ray photon before and after a single collision are strictly valid by using the Compton scattering experiment, and such results also denied statistical conservation. There are expressions about strict conservations of total energy and total momentum of the system in Compton scattering experiment without quantum mechanical concepts in the introductions of some quantum mechanics books. Different from this, the aim of this paper is attempt to express those with quantum mechanical concepts. There is no expression about the strict conservation and the statistical conservation of physical quantities in quantum mechanics books. Our viewpoints are that the total energy and the total momentum of an isolated system are strict conservation by exchanging energy and momentum within the system in the evolution of the system state, and the momentum and kinetic energy (an interaction energy is not all belong to anyone subsystem) of a subsystem are statistical conservation. The “strict conservation” in journal articles is different from ours. A recent paper by the corresponding author of this paper and his coauthors (Int.J Theor. Phys. , 2020, 59 : 229) used the terms of strict conservation and statistical conservation (we offer more precise statements in this paper, and these statements have some differences with that of a conserved physical quantity) , and offered the expression ways of the strict conservation of total momentum for an isolated composite system and the statistical conservation of momentum for a subsystem in terms of the concepts of entangled states and eigenvalues. In this paper, we try to use these concepts and mixed states to express the strict conservations of total energy and total momentum of the system before and after a Compton scattering, and also explain the statistical conservations of energy (no interaction after scattering) and momentum of a subsystem. Similarly, we can express the strict conservations of some physical quantities of an isolated composite system and the statistical conservations of some physical quantities of a subsystem with quantum mechanical concepts.
When a theory is framed trying to explain a discrepant system of facts, it is a necessary process of thought to take some branch of the theory as more completely true than the rest, and to adjust the remaining parts in such a way that they will fit in with this base, though they may still conflict with one another. This has certainly been true of the quantum theory; the speculations connected with it have as their base the law of the conservation of energy. Now a critical examination of fundamentals does not by any means justify this faith. It is of course a fact of observation that in the gross energy is conserved, but this only means an averaged energy; and as pure dynamics has failed to explain many atomic phenomena there seems no reason to maintain the exact conservation of energy, which is only one of the consequences of the dynamical equations. Indeed it is hardly too much to say that had the photoelectric effect been discovered a century ago, it is probable that no one would ever have suggested that the status of the first law of thermodynamics was in any way different from that of the second. On the other hand, Bohr's theory, and especially Sommerfeld's extension of it, have given great encouragement to the belief that in dynamics lay the way to the complete truth, so that in consequence of the triumphs of that theory there has been little thought in other directions. Another impediment is that our whole ideas are saturated with the principles of energy, so that denying it leaves hardly any foundation from which to start.
As in classical physics, also in quantum mechanics: physical processes are bound to obey conservation laws. However, there is an important difference. While in classical physics we can suppose that all variables have definite values (irrespective of whether we know them or not), quantum mechanics allows them to be intrinsically uncertain. In those cases conservation laws apply only to ensemble averages (expectation values), and hence are less informative. They show their full strength in measurement processes. Then the measured values satisfy the respective conservation laws in any individual experiment. Let me remind you of spin measurements on a system of two spin-½ particles, being in a state with zero total spin, which we used in Section 5.5 to illustrate the EPR paradox. The spin components, with respect to an arbitrarily chosen direction, of the two particles are indefinite; however, measuring them yields definite values (+½ or −½) that add up to zero, thus confirming the angular conservation law. In the following, we will consider some realistic experiments for which conservation laws provide a simple explanation. Energy and momentum conservation For an understanding of the interaction between matter and radiation, Einstein's photon concept (actually, he spoke of ‘light quanta’) proved to be very helpful. He assumed photons to be particle-like constituents of electromagnetic radiation, in particular light. They have the energy h ν, where ν is the frequency of the radiation field. (Since the electromagnetic spectrum extends over an extremely wide range, there is a huge number of different sorts of photons.) The crucial point is that photons are generated or annihilated only as a whole. This property gives rise to typical quantum effects.
Modified versions of the Schrödinger equation have been proposed in order to incorporate the description of measurement processes into the mathematical structure of quantum theory. Typically, these proposals introduce new physical constants, and imply small violations of momentum and energy conservation. These problematic features can be eliminated by assuming that wave function collapse is induced by the individual interactions that establish correlations between systems. The generation of a sufficient number of small, random shifts of amplitude between interacting and noninteracting branches of the wave function can bring about collapse on a scale consistent with our macroscopic experience. Two-particle interaction potential energies can be used as the basis for a collapse term added to the Schrödinger equation. The range of the interactions sets the distance scale of the collapse effects; the ratio of potential energies to the total relativistic energies of the particles determines the magnitude of the amplitude shifts, and the rate at which the interactions proceed fixes the timing parameters. Consistency with conservation laws in individual experiments is maintained because the collapse operator automatically takes into account the small, residual entanglement between the measured system and systems with which it has previously exchanged conserved quantities during interactions. Conservation is exact for momentum and orbital angular momentum, and it holds for energy within the accuracy allowed by the limited forms of energy describable in nonrelativistic theory.
When two fundamental particles collide, the total energy in the system remains the same. at is, if you sum the energies of each particle before the collision, you get the same result as you get by summing the energies of each particle aer the collision. We call this the principle of energy conservation. It is a sacrosanct law that no physicist is eager to part with. Similarly, if you sum the arrows corresponding to each particle’s momentum before the collision, you get the same result as you get by summing their momenta aer the collision. is is known as the principle of momentum conservation.
Classical statistical equilibrium appears to be based on a no-flow situation with a balance of pressure (including -dV/dx in the case of a potential present). It may be noted, however, that like quantum mechanics, a single particle has different probabilities of being at one x versus another. Furthermore, the Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein distributions may be obtained using reaction balance which is linked to conservation of energy i.e. g(f(ei))g(f(ej)) = g(f(el))g(f(ek)) where ei+ej=el+ei=e. Taking ln of the former expression and linking to the latter yields f(ei), the probability distribution. g=1 for the MB case, f(e)/(1-f(ei)) for the FD and f(e)/(1+f(ei)) for the BE. In classical mechanics, one has the conservation equation: p(x)*p(x)/2m + V(x) = E. In quantum bound systems one has: KE(ave)(x) + V(x) = E which appears to be very similar, although it is based on ensemble averages and there is a momentum distribution P(p/x) at each x. A question which arises is: What is the “conservation of energy” equation for a particular p of the P(p/x) distribution? In this note, we argue the “potential” term associated with a component p*p/2m is related to the part of the potential which contributes to the “creation” of p. In particular, we consider V(x)= Sum over k V(k) exp(ikx) where each k term renders a kick of momentum k. One may create a momentum p using k and p-k for each value of k. We further argue that: p*p/2m P(p/x) + contributing portion of V(x) = E P(p/x) ((1)) for all p. Thus, in quantum equilibrium, which involves a flow, there is a “constant E” type of equation i.e. ((1)), different in form but similar in idea, to that of classical statistical mechanical equilibrium. In previous notes, we referred to this as cycling, but in this note we wish to consider it in terms of dynamics. We further note that in both classical statistical mechanics and quantum mechanics, the energy-probability p equation makes use of the overall average energy which is bT for the classical case and E for the quantum. (b is constant)