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.
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.
Quantum mechanical uncertainty After reviewing the main characteristics of the classical description of light, let us discuss those aspects of the quantization of the electromagnetic field which are of relevance for the analysis of the phenomena we are interested in. It seems a reasonable place to start to make clear the fundamental difference between the classical and the quantum mechanical description of nature; we will come across this difference many times when discussing experiments, and it will often give us a headache. We have to deal with the physical meaning of what is called uncertainty. The starting point of the classical description is the conviction that natural processes have a “factual” character. This means that physical variables such as the position or momentum of a particle have, in each single case, a well defined (in general, time dependent) value. However, it will not always be possible to measure all the appropriate variables (for instance, the instantaneous electric field strength of a radiation field); furthermore under normal circumstances we are able to measure only with a finite precision. Hence the basic credo of classical physics should be given in the following form: we are justified in imagining a world with variables possessing well defined values which are not known precisely (or not known at all). In doing this we are not forming any conclusions that contradict our everyday experiences. This is the fundamental concept of classical statistics: we are satisfied with the use of probability distributions for the variables we are interested in, not from fundamental but purely practical reasons.
Introduction It is believed that the basic physical building blocks forming the world we live in may be categorized into particles of matter and carriers of force between matter. All known elementary constituents of matter and transmitters of force are quantized. For example, energy, momentum, and angular momentum take on discrete quantized values. The electron is an example of an elementary particle of matter, and the photon is an example of a transmitter of force. Neutrons, protons, and atoms are composite particles made up of elementary particles of matter and transmitters of force. These composite particles are also quantized. Because classical mechanics is unable to explain quantization, we must learn quantum mechanics in order to understand the microscopic properties of atoms – which, for example, make up solids such as crystalline semiconductors. Historically, the laws of quantum mechanics have been established by experiment. The most important early experiments involved light. Long before it was realized that light waves are quantized into particles called photons, key experiments on the wave properties of light were performed. For example, it was established that the color of visible light is associated with different wavelengths of light. Table 2.1 shows the range of wavelengths corresponding to different colors. The connection between optical and electrical phenomena was established by Maxwell in 1864. This extended the concept of light to include the complete electromagnetic spectrum. A great deal of effort was, and continues to be, spent gathering information on the behavior of light.
In the paradigmatic example of quantum measurements, whenever one measures a system which starts in a superposition of two states of a conserved quantity, it jumps to one of the two states, implying different final values for the quantity that should have been conserved. The standard law of conservation for quantum mechanics handles this jump by stating only that the total distribution of the conserved quantity over repeated measurements is unchanged, but states nothing about individual cases. Here however we show that one can go beyond this and have conservation in each individual instance. We made our arguments in the case of angular momentum of a particle on a circle, where many technicalities simplify, and bring arguments to show that this holds in full generality. Hence we argue that the conservation law in quantum mechanics should be rewritten, to go beyond its hitherto statistical formulation, to state that the total of a conserved quantity is unchanged in every individual measurement outcome. As a further crucial element, we show that conservation can be localised at the level of the system of interest and its relevant frame of reference, and is independent on any assumptions on the distribution of the conserved quantity over the entire universe.
e1921529118 (2021)], it was argued that while the standard definition of conservation laws in quantum mechanics, which is of a statistical character, is perfectly valid, it misses essential features of nature and it can and must be revisited to address the issue of conservation/nonconservation in individual cases. Specifically, in the above paper, an experiment was presented in which it can be proven that in some individual cases, energy is not conserved, despite being conserved statistically. It was felt however that this is worrisome and that something must be wrong if there are individual instances in which conservation does not hold, even though this is not required by the standard conservation law. Here, we revisit that experiment and show that although its results are correct, there is a way to circumvent them and ensure individual case conservation in that situation. The solution is however quite unusual, challenging one of the basic assumptions of quantum mechanics, namely that any quantum state can be prepared, and it involves a time-holistic, double nonconservation effect. Our results bring light on the role of the preparation stage of the initial state of a particle and on the interplay of conservation laws and frames of reference. We also conjecture that when such a full analysis of any conservation experiment is performed, conservation is obeyed in every individual case.
At the beginning of the century doubts were growing about the whole edifice of classical physics, including the laws of mechanics in their Lagrangian and Hamiltonian forms, of statistical thermodynamics, as formulated by Boltzmann and Gibbs, and of optics, namely Maxwell’s and Fresnel’s theory. Einstein proposed that electromagnetic radiation exists in discrete quanta whilst propagating in space as well as during the emission or absorption processes. Scattering by free electrons is a common interaction connected with the absorption of electromagnetic radiation. If the electromagnetic energy is large compared with the ionization energy, the atomic electron can be treated as free. The characteristic features of the scattered radiation have been explained by Compton, on the basis of an elastic collision of a photon with a free electron, where both energy and momentum are conserved. The concept of the matter wavelength holds for both relativistic and non-relativistic particles, provided that the appropriate form of momentum is used.
Elementary particles are found in two different situations: (i) bound to metastable states of matter, for which angular momentum is quantized, and (ii) free, for which, due to their high energy-momentum and leaving aside inner a.m. or spin, the h-quantization step is completely harmless. Perhaps Quantum Mechanics can be seen just as the simplest mathematical formalism where angular momentum (the magnitude of each of its three orthogonal projections) is by construction quantized: all possible values are taken from a discrete set. Indeed: (i) This idea finds support in very reasonable, completely classical physical arguments, if we place ourselves in the framework of Stochastic Electrodynamics (SED): there, all sustained periodic movement of a charge must satisfy a power balance that restricts the value of the average angular momentum, on each of its projections. (ii) It gives a natural explanation of the concept of "photon", as a constraint on the observable spectrum of energy-momentum exchanges between metastable physical states, in particular also for its discreteness. QM would be, in this picture, a semi-static theory, transparent to all the (micro)-dynamics taking place between apparently "discrete" events (transitions in the state of the system). For instance, (the magnitude of the projections of) quantum angular momentum would only reflect average values over a (classical) cyclic trajectory, a fact that we regard as almost obvious given the particularity of the corresponding addition rules in QM.