Probability amplitude, atquantum mechanicsIt is also calledQuantum amplitude, is a descriptionparticleQuantum behavior ofComplex function。For example, probability amplitude can describe the position of particles.When describing the position of particles, the probability amplitude is a wave function, expressed aspositionFunction of.Thiswave functionMust meetSchrodinger equation。
Chinese name
Probability amplitude
Foreign name
probability amplitude
Alias
Quantum amplitude
Nature
Complex function
Field
Mathematical Science
Related figures
De Broglie, Born, Schrodinger, Feynman
Definition
A complex function describing the quantum behavior of particles
De BroglieProposedwaveWhat is the physical meaning of?He himself once thought that the wave associated with particles was the "guide wave" guiding the movement of particles, and thus predicted the electronicDouble slit interferenceExperimental results.This wave propagates with phase velocity u=c ²/v and its group velocity is exactly the velocity u of particle motion.He did not give a clear answer to the nature of this wave, but said that it was virtual and immaterial.
quantum mechanicsOne of the founders ofSchrodingerIt was said in 1926 that electronicDe Broglie waveThe continuous distribution of electricity in space is described.In order to explain the fact that electrons are particles, he believes that electrons are composed of many waveswave packet。This statement was quickly rejected.Because.First, wave packets always diverge and disintegrate, which contradicts the stability of electrons;Second, it is difficult to explain the wave packet that the electron remains stable in the process of atomic scattering.
Currently recognizedDe Broglie waveThe essential explanation ofBorn(M. Born) in 1926.Before Born,EinsteinTalking about his own discussionphotonandelectromagnetic waveThe electromagnetic field is a kind of "ghost field".This field guides the movement of photons, while electromagnetic waves everywhereamplitudeThe square of determines the probability of the existence of a photon in the unit volume everywhere.Born developed Einstein's ideas.He retained the corpuscular nature of particles and believed that matter waves described the probability of particles being found everywhere.This means that de Broglie wave isProbability wave[1]。
Fig. 1 Jonson electron diffraction pattern
Born's concept of probability wave can be used as an electronic double slitdiffractionThe experimental results.The electron double slit diffraction pattern in Figure 1 (a) is exactly the same as the light double slit diffraction pattern, showing no particle property, nor any uncertain characteristics like probability.But that is the result of an experiment made with a large number of electrons (or photons).If the intensity of the incident electron beam is weakened so that one electron passes through the double slits in turn, the diffraction "pattern" will be as shown in the figures in Figure 2 with the accumulation of the number of electrons.Figure 2 (a) is an image formed by only one electron passing through the double slit.Figure 2 (b) is an image formed after several electrons are penetrated.Figure 2 (c) is an image formed after dozens of electrons are penetrated.These images show that the electron is indeed a particle, because the image is composed of points.They also show that the whereabouts of electrons are completely uncertain. Where an electron arrives is completely a probability event.With the increase of the total number of incident electrons, the diffraction pattern is shown in (d).These stripes completely submerge the probabilistic behavior of a single electron.This also shows that although the destination of a single electron is probabilistic, its probability still has certain laws under certain conditions (such as double slits).These are the core of Born's concept of probability wave.
The experimental results shown in Figure 2 clearly illustrateMatter waveIt's not a classic wave.Classical wave is a form of motion.stayDouble slit experimentNo matter how small the intensity of the human radiation is, the classic wave "should" show the diffraction fringes with continuous distribution of intensity on the screen behind the slit. Only the brightness is weak.But Figure 2 clearly shows that the main body of the matter wave is still a particle, and the motion of this kind of particle does not have a classical vibration form[1]。
Fig. 2 Matter wave is not a classical wave
The experimental results shown in Figure 2 also show that microscopic particles are not classical particles.In the double slit experiment, the diffraction pattern formed by a large number of electrons is a number of narrow stripes with approximately the same intensity.As shown in Figure 3 (a).If only one slit is opened and the other slit is closed, a single slit diffraction stripe will be formed.It is characterized by that there are almost only wide central bright lines with greater intensity (P ₁ and P ₂ in Figure 3 (b)).If seam 1 is opened first and seam 2 is closed at the same time, seam 2 will be changed after a section of H inch.At the same time, close slit 1, and the total diffraction pattern P ₁ ₂ formed by the results of this experiment will be the superposition of two single slit diffraction patterns[1]．
The intensity distribution is quite different from the double slit diffraction pattern when two slits are opened at the same time.
Fig. 3 Schematic diagram of electron double slit diffraction experiment
If they are classical particles, they have their own definite orbits when passing through the double slits, either through slit 1 or slit 2.If the particles passing through the slit l can also be diffracted, they will form a single slit diffraction pattern.Those particles passing through slit 2 will form another pictureSingle slit diffractionPattern.Whether two slits are opened at the same time or only one slit is opened in turn, the final diffraction stripe should be the superposition of two single slit diffraction patterns as shown in Figure 3 (b).The experimental results show that the actual microscopic particles do not behave like this.This shows that microscopic particles are not classical particles.When only one slit is opened, the actual particles form a single slit diffraction pattern.When two slits are opened at the same time, there are two possibilities for the movement of actual particles: either through slit 1 or through slit 2.If we still follow the classical particle assumption, in order to explain the double slit diffraction pattern, we must think that when passing through this slit, it seems that it "knows" that another slit is also open, so it acts according to the probability under the double slit condition.This statement is just a "anthropomorphic" imagination. In fact, it is impossible to know experimentally which slit a microscopic particle "exactly" passes through. We can only say that there are two possibilities when it passes through the double slit.Micro particles behave so strangely because of their undulation!But the objective facts are really like this![1]
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In order to quantitatively describe the state of microscopic particles, quantum mechanics introduceswave function, and denoted by Ψ.In general, the wave function is a function of space and time, and it is a complex function, that is, Ψ=Ψ (x, y, z, t). To generalize the relationship between Einstein's "ghost field" and the probability of the existence of photons, Born assumed that | Ψ | ²=ΨΨ * is the probability density of particles, that is, the probability of finding particles in the unit volume near the point (x, y, z) at time t.The wave function Ψ is therefore called the probability amplitude.yesDouble slit experimentFor example, if Ψ ₁ represents the probability amplitude distribution of particles near the bottom plate when single slit 1 is used, then | Ψ ₁ | ²=P ₁ is the probability distribution of particles on the bottom plate, which corresponds to the single slit diffraction pattern P ₁ (Fig. 3 (b)).The probability amplitude of single slit 2 is denoted by Ψ ₂, then | Ψ ₂ | ²=P ₂ represents the probability distribution of particles on the floor at this time, which corresponds to the single slit diffraction pattern P ₂. If two slits are opened at the same time, the classical probability theory gives that the probability distribution of particles on the floor should be[1]
four
But that's not the case!When two slits are open at the same time, there are two possibilities for the direction of each incident particle. They can pass through one of the slits "arbitrarily".At this time, it is not probability phase superposition, but probability amplitude superposition, that is, the corresponding probability distribution is
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The final result here will appear the cross term of Ψ ₁ and Ψ ₂.It is this cross term that gives the interference effect between two slits, making the diffraction patterns different under two conditions of two slits opening at the same time and two slits opening in turn.
The strange law of probability amplitude superposition isFeynman(R.P. Feynman)《Feynman Lecture on Physics》Is called "the first principle of quantum mechanics".He wrote: "If an event can be realized in several ways, the probability amplitude of the event is the sum of the probability amplitudes when each way is realized separately. Then there is interference."
It is of great significance in philosophy to introduce the concept of probability into physical theory.It means that under the given conditions, it is impossible to predict the results accurately, and only the probability of some possible results can be predicted.That is to say, the only positive result can not be given, and the conclusion can only be given by statistical methods.This theory is in direct contradiction with the strict causality of classical physics.Born said in 1926: "The motion of particles obeysProbability law, butprobabilityI still sufferLaw of causalityDominated. "Although this sentence keeps the law of causality effective in some way, the introduction of the concept of probability is still a very big change in the process of people's understanding of nature.Therefore, although all physicists agree that quantum mechanics is a very successful theory because its predicted results and experiments are extremely accurate, there is still a great debate about the philosophical basis of quantum mechanics.Copenhagen School, including Bohn, W. Heisenberg and other quantum mechanics masters, who insisted on the probabilistic or statistical interpretation of wave functions and believed that it showed the ultimate essence of nature.Feynman also wrote (1965): "At present, we are limited to calculating probability. We say 'present', but we strongly hope that it will always be this way to solve this puzzle. It is impossible for nature to act in this way."[1]
Others disagree with this conclusion, and Einstein is the main opponent.He said in 1927: "God is not playing dice with the universe." De Broglie's words (1957) were even more thought-provoking.He believes that uncertainty is the essence of physics, and such a claim "is not entirely tenable. When the understanding of physical reality reaches a deeper level in the future, we may make a new interpretation of the probability law and quantum mechanics, that is, they are the results of the completely determined numerical evolution of those variables that we have not yet discovered.The powerful method we are now beginning to use to break down atomic nuclei and generate new particles may one day reveal to us about this deeper level of knowledge that we do not yet know.It is very dangerous for the development of science to prevent the attempt to further explore the current view of quantum mechanics, and it also deviates from the lessons we have learned from the history of science.In fact, the history of science tells us that the acquired knowledge is often temporary. Beyond this knowledge, there must be broader new areas to explore. "Finally, we can also quote a paragraph from P.A.M. Dirac, the master of quantum mechanics, in 1972: "In my opinion, we do not have the basic laws of quantum mechanics. The laws still in use need to be significantly modified.When we make such drastic changes, of course, our concept of using statistical calculation to make physical explanations for the theory may be completely changed. "[1]
