About Schrödinger's Cat

What for

The situation with the Schrödinger cat, probably, has an idea of ​​the majority of Khabrovites who are interested in physics. Therefore, I will not state it. The discussion is held around the interpretation of the state of the cat. Here are the alternatives:

1. The cat "And alive And dead." This is described in quantum mechanics as a superposition of the “alive” and “dead” states and, therefore, some interference effects are possible, similar to the case of light scattering on two slits.
2. The cat is "OR alive OR dead." This interpretation prohibits the above superposition and, therefore, prohibits interference effects.

My task is to present the point of view that follows, as it seems to me, from reading the book “Quantum Mechanics” by Feynman.

Where do the legs grow from

And legs grow out of the principle of superposition. It reads:



Let the system be able to $$$| s_1>$ in which the measurement of the observed s always gives the result $$$s_1$

and

let the system be able to $$$| s_2>$ in which the measurement of the observed s always gives the result $$$s_2$ ,

then the system can be prepared in a superposition state $$$c_1 | s_1> + c_2 | s_2>$ Where $$$| s_1 | ^ 2 + | s_2 | ^ 2 = 1$ . In this state, when measuring the observed s value $$$s_i$ will be observed with probability $$$| c_i | ^ 2$ . They say, differently, that the meaning $$$s_i$ will be observed with amplitude $$$c_i$ .



The principle of superposition for two states leads to the principle of superposition for an arbitrary number of admissible states of the system for the observed observable. But it is important for us only for two states - living and dead.



Note how to prepare the system in superposition - this is another question. The question is technical. And the principle says that you can prepare a superposition. But how to cook, he does not say anything about it.



Superposition leads to interference effects. And experimentally, it manifests itself only in interference. State interference is what distinguishes quantum mechanics from classical . Interference cannot always be observed. Indeed, the visual picture of interference can change so quickly that a visualizing device with a long reaction time will display an average picture, which lubricates or even eliminates the effect of interference. But this is a matter of technology. But in the absence of superposition, no technique will detect interference.



Most probably know the difference between the interference pattern and the non-interference pattern when scattering by two slits. Here are the pictures of this interference (each box is a separate picture):







The interference patterns of two circular coherent waves, depending on the wavelength and the distance between the sources.



What possible interference effects with the superposition of the living and the dead, I do not presume to describe and, moreover, visualize them.

Superposition Examples

Superposition in ordinary space

A free particle is described by a wave function - de Broglie wave in coordinate space:

$$

$$P (x) = exp (-ipx)$$

Here p is the momentum, which is a fixed quantity (parameter), and x is the coordinate is a variable that can take any value of the coordinate. Different pulses give different possible states. Therefore, a superposition of de Broglie waves corresponding to different momenta is possible. This can be a finite superposition, a countable superposition, a continuous superposition in which the sum goes over into the integral. We get a state that does not have a specific pulse value: when measuring a pulse, various values ​​can be obtained and this is not an experimental error.



How vast is the class of functions represented by such a superposition? Remembering the math, we find in the countable superpositions the Fourier series and in the continuous superposition the Fourier – Fourier integral is the expansion in p. Here it is the mysterious connection of abstract mathematics and concrete physics! Extensive research mathanalysis describes a class of Fourier-decomposable functions. But for physics it’s just a superposition of plane waves with different momenta.

Superposition in momentum space

For the sake of symmetry, one can similarly consider the de Broglie wave in momentum space - a particle with a fixed coordinate:

$$

$$X (p) = exp (-ipx)$$

Here x is the coordinate, which is a fixed quantity (parameter), and the momentum p is a variable that can have any value of the momentum. Different coordinates specify different possible states. This means that a superposition of de Broglie waves corresponding to different coordinates is possible. This can be a finite superposition, a countable superposition, a continuous superposition in which the sum goes over into the integral. We get a state that does not have a specific coordinate: when measuring the coordinate, different values ​​can be obtained and this is not an experimental error.

Superposition in the energy space

Stationary state - a state with a fixed energy. It is described by the wave function - de Broglie wave in the energy space:

$$

$$E (t) = exp (-iet)$$

Here e is the energy, which is a fixed quantity (parameter), and time t is a variable that can take any value of time. Different energies set different possible states. This means that a superposition of de Broglie waves corresponding to different energies is possible. This is a superposition of stationary states that can describe non-stationary states - the superposition coefficients can depend on time.

Superposition in spin space

A circularly polarized photon is represented as a superposition of two linear polarizations.

Superposition in the space of fundamental particles

Photon as a superposition

In the standard model, a photon is a superposition of bosons $$$B ^ 0$ and $$$W ^ 0$ .

Neutrino as a Superposition

Each neutrino with a certain mass is a superposition of the electron, muon and tau neutrinos. And, conversely, an electron neutrino, a muon neutrino, and a tau neutrino are a superposition of three neutrinos with specific masses.

Kaon as a superposition

The kaon situation is similar to the neutrino situation.

Superposition in the space of life

Here I pass.



The principle of superposition does not mean whether all states are exhausted by superpositions. For example, are there physical states that are not representable as a superposition of plane waves? - I dont know.



Quote from Feynman:

We have now described one of the greatest achievements of theoretical physics. It is not based on elegant mathematical tricks, similar to the general theory of relativity, nevertheless, the predictions obtained are just as important as, for example, the prediction of the positron. Of particular interest is the fact that we have brought the principle of superposition to its logical end. Bohm and his associates believed that the principles of quantum mechanics were not fundamental and, ultimately, could not explain the new phenomena. However, these principles work. This does not prove that they are true, but I am ready to bet that the principle of superposition will stand for centuries!

Questions

If we have a superposition, can we say that the system consists of superpositional components? Does white light consist of a rainbow? A photon consists of bosons? What does “consist” mean? You can expand the function in Fourier series in sinusoids, or you can expand in Fourier series in Legendre polynomials, in Chebyshev polynomials, etc. So what does it consist of? Is any mathematical superposition physically feasible? The sine wave itself can be expanded in Fourier series according to Legendre polynomials. So, there may exist a certain device that decomposes monochromatic light into Legendre polynomials. So we can say that monochromatic light consists of “Legendre waves”? Then we can introduce the concept of “Legendre photon”. And in some situations it will be easier to operate with a Legendre photon, rather than an ordinary sinusoidal photon. You can imagine a radio on the waves of Legendre ...



It seems that if we find a superposition of states that were previously considered to be the states of different, non-superposable systems, then these states must be considered states, of some kind of a new unified system.

Cat

We pass to the Schrödinger cat. He may be in a state of “alive” and may be in a state of “dead”. So, by the principle of superposition, he can also be in a superposition of the states “alive” and “dead”. Perhaps not in the current situation, but in some other, it is necessary. Is it so? And what possible interference effects can be observed in this case?

And let's summarize the cat: let's move on to the concept of “animal”. We know that it can be in the states of a lion, a man, ... So, according to the principle of superposition, their superposition is also possible. This is absurd. We go further and “animal” generalize to the concept of “material object”. Then you need to allow a superposition of any material objects. This is even more absurd. It seems like you need to clarify the concept of a system. Apparently, you need to take a specific system (a specific cat), and not an abstract (animal). But he suggested that Heisenberg consider the proton and neutron as different states of the nucleon system and received interesting consequences.

What Feynman Says

Let's turn to Feynman. He is considering an experiment on the scattering of neutrons by a crystal.



I summarize Feynman's text like this.



After neutron scattering on a crystal, there are two neutron systems at the output:

1. Elastically scattered neutrons
2. Inelastically scattered neutrons having an inverted spin compared to the original

There is no superposition between the first and second systems. They are physically distinguishable. In the first system, neutrons are superposed and a typical interference pattern is obtained a). The picture b) of the addition of intensities in the second system) is superimposed on it. The final figure is c).







There is a superposition in the second system, but no interference. The interference is destroyed by the randomness of the phase shift upon spin flipping . The fact that this shows the interference of two specially prepared neutron beams with different spin orientations, including those with oppositely directed spins. There is interference with sufficient coherence of the beams. Moreover, the effect is insane in terms of common sense. If neutrons in the second beam have a spin rotated 360 degrees compared to the spin of the first beam, then maximum interference is observed. It does not climb into any gates. A 360 degree rotation did not lead to its original state. It is impossible to explain. However, from a formal point of view, everything is clear here. A neutron is described by a spinor, not a scalar as a scalar meson, or by a vector as a photon. Quantum mechanics admits the amplitudes described by spinors - quantities that coincide with themselves only with a double revolution - a rotation of 720 degrees. A rotation by one revolution can be accompanied by a multiplication of the amplitude by a complex number with module 1. The physics of the spinor will not change - the means will not change. But when superimposing beams, interference effects are possible. This is where the indicated interference effect comes from.

So what can superposition?

In no textbook on quantum mechanics that I met, I have not found a criterion for the possibility of superposition . And only at Feynman did I meet a recipe when he examined scattering on two slits.

Feynman test

Never add the amplitudes of different, different final states. As soon as a photon was received by one of the photon counters near the gaps, we can always, if necessary, find out not disturbing more than the system, which of the alternatives (mutually exclusive events) was realized. Each alternative has its own probability, completely independent of the other. We repeat, do not add amplitudes for different final conditions (by “final” we mean the moment when we are interested in probability, that is, when the experiment is “finished”). But it is necessary to add the amplitudes for different indistinguishable alternatives in the course of the experiment itself, before the whole process ends. At the end of the process, you can, if you want, say that you “do not want to look at the photon”. This is your own business, but you still cannot add amplitudes. Nature does not know that you are looking at it, and it does not matter whether you are interested in its data or not. So we should not add amplitudes.

So, if there are physically indistinguishable ways to reach the point at which we are considering the possibility of interference, then the amplitudes of these paths add up and we have interference. If they are physically distinguishable, then the probabilities add up and, therefore, there is no interference. By path is meant movement not only in ordinary space . So if there are two modes of decay of a particle with one outcome, then they should superposition.



Let us call the Feynman maxims given above the Feynman superposition criterion.

So, Feynman says that only physically indistinguishable trajectories can superposition .

Saying Dirac

... each photon interferes only with itself. Interference between two different photons never occurs.

Presumably this applies to any object. Therefore, the cat can interfere only with itself. A dead cat and a live cat are extremely different cats. And is it possible to call a corpse a dead cat? This is the corpse of a cat, but not a cat.

Doubt

What is considered a superposition under a system? If we consider states with different values ​​of the momentum of an electron, then these are undoubtedly different states of one system, called an electron. If we consider different energy states of a hydrogen atom, then this is also one system - a hydrogen atom. But Heisenberg proposed to consider the proton and neutron as different states of the nucleon. Then, what is the possible superposition of a proton and a neutron? But then why is it impossible to superposition the electron and positron? They say that this contradicts the law of conservation of electric charge. Then why is the superposition of different energy states not contrary to the law of conservation of energy? Does the photon take energy away? Then the charge can be carried away by a nascent particle. One can honestly declare (Kempfer, Lipkin) that superpositions with different electric charges were not observed in nature, although such a superposition does not contradict any laws.



On the physical distinguishability of the trajectories of the system. What are distinguishing marks? Space points? - Not. Time points? Charges: mass, electric, lepton, baryon? Spin? Only internal characteristics? Feynman says that these are marks in the external environment that can be detected. When neutrons are scattered on a crystal, a neutron with an inverted spin leaves a mark in the crystal — a nucleus with an inverted spin. Any inelastic scattering leaves a mark (energy, spin ...) in the scattering medium, but elastic not. So, when passing through the slits, only elastic scattering photons interfere.



With regard to distinguishability, one more thing can be said. Our knowledge is inaccurate, and what today is considered physically indistinguishable may be distinguishable tomorrow. This happened with the concepts of right, left. If we consider the right and left as pure conventions, then this convention should not be included in the fundamental formulas. But it turned out that for a weak interaction the concepts of “right”, “left” are by no means a convention: the right and left states are distinguished by weak interactions. And in the Lagrangian of weak interaction, separately included “right” and “left” members. Those. without answering the question “why is the right different from the left?”, nevertheless, they successfully answered the question “how does this happen?” This, however, is not new. Even Newton, to the reproaches that he did not explain the nature of gravitation, but simply gave the formula of the law of gravitation, answered something like this: yes, I do not know the nature of gravity and do not put forward any hypotheses on this subject, but I know how the law of gravitation is described and this is something. A similar approach resulted in a certain philosophy: some physicists bluntly state that the matter of physics is to find out “how?” And not “why?”. Well, in fact, what to answer the question “Why are Maxwell's equations valid?”. Nobody knows that.



Similarly, the situation “right-left” happened in the case of kaons. It would seem that there are two completely distinct types of kaons. One splits into two mesons. And the second is three. But Gell-Mann and Paice suggested that we are dealing with the decay of a single particle. And two decay modes arise due to the fact that this initial particle is a superposition of two other types of kaons.



All this is so. But, it is doubtful that someday the living and the dead cat will unite in one system and the difference between the living and the dead will become ephemeral.

What can't superposition?

Superposition of different particles

Let us imagine an experiment with two slits, when a plane wave of electrons passes through one slot and a plane wave of protons passes through the other. Let de Broglie wavelengths be constant and the same. Will there be interference? Formally considered, the proton wave is described in exactly the same way as the electron wave. And why not be interference? But in quantum theory, the fields of the wave will be the same only in the spatial part. And they will be different in charge and back. But let us remain within the framework of the usual examination of the experience on the cracks. Consider a beam of electrons and negative muons. There both charges and spins are the same. Will there be interference? The answer is Feynman's criterion. Once the particles are physically distinguishable, then there will be no interference. As in the experiment with identical particles, when the passage through the gap is checked, the check makes the particles distinguishable and the interference disappears, so in the experiment with different particles they are already physically distinguishable initially. There will be no interference. Conversely, if the interference seems to be the same particles, but from different sources is not observed, then the particles are distinguishable. Although this distinguishability (characteristic responsible for distinguishability), we have not yet found.

Superposition of the Dead and the Living

In principle, superposition speaks of the states of a particular quantum system. A dead cat and a living cat are completely different physical systems. Only physically indistinguishable alternatives can be superposed. And the dead and the living are physically very distinguishable. You could even say that there can be no more distinguishability than the dead from the living. Our ignorance of whether a cat is alive or dead does not arise as a result of superposition, but because of a lack of information, as in any classical probabilistic problem. And in the case of superposition, there is no talk of a lack of information, and, as the Copenhagen interpretation of quantum mechanics claims, it cannot be.



If the experiment shows the absence of a superposition of an elastically scattered neutron and an inelastically scattered neutron, then it is logical to say that a dead cat and a living cat cannot be superposed. They are fundamentally different from each other.

Superposition of living and living

A living system involves a continuous exchange of matter and energy with the environment. By this, she continuously marks - it becomes physically distinguishable. So, living things can not interfere. It cannot harden and remain identical. Living all the time is not identical to itself. This is a different system all the time.



So, using the Feynman criterion, we conclude that

1. A neutron with an inverted spin and not inverted during neutron scattering on a crystal does not superposition
2. Particles with different charges cannot superposition. A proton and a neutron are superposed only if you forget about electric charges - when the electromagnetic interaction is turned off. Really do not. And that means in reality only some traces of a possible superposition can be observed. For example, the proton-proton, neutron-neutron, and proton-neutron scattering cross sections will be as close as the electromagnetic interaction is weaker than the strong one.
3. Proton and electron cannot superposition
4. Molecules of different substances cannot superposition

Doubt

However, why is superposition of quark and lepton allowed in supersymmetry? Is it possible that the superposition is relative and the fact that in some conditions there is no superposition does not mean that it is impossible in other situations? Then the principle of superposition must be supplemented by a description of the situation of superposition. So in a situation of switching off the electromagnetic field, a superposition of the proton and neutron is possible.

conclusions

• Superposition is a matter of principle : whether or not superposition is determined by the laws of nature. Interference is a technical matter : if there is a superposition, then there is interference. Another thing is whether it is detected by devices - it depends on the speed of reaction of the registration device. But a necessary condition for interference is superposition.
• There is no superposition of the dead and the living . These are physically completely different systems, not the states of one system. No superposition - no interference. And so the Schrödinger cat is either alive or dead without any superposition of these conditions. It is impossible to cook such a superposition.
• There is no superposition of living and living . A living system at different points in time is a physically different system, and if an exact copy of the living one is received for interference, then it will immediately become inaccurate due to vital processes.

I set forth my point of view. And the supreme judge in physics is an experiment. He can refute any logic. Well, we look forward to an experiment on the interference of the dead and the living, living and the dead amoeba, for example.