Objective-collapse theory

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https://en.wikipedia.org/wiki/De_Broglie%E2%80%93Bohm_theory

Objective-collapse theories, also known as models of spontaneous wave function collapse or dynamical reduction models, were formulated as a response to the measurement problem in quantum mechanics, to explain why and how quantum measurements always give definite outcomes, not a superposition of them as predicted by the Schrödinger equation, and more generally how the classical world emerges from quantum theory. The fundamental idea is that the unitary evolution of the wave function describing the state of a quantum system is approximate. It works well for microscopic systems, but progressively loses its validity when the mass / complexity of the system increases.

In collapse theories, the Schrödinger equation is supplemented with additional nonlinear and stochastic terms (spontaneous collapses) which localize the wave function in space. The resulting dynamics is such that for microscopic isolated systems the new terms have a negligible effect; therefore, the usual quantum properties are recovered, apart from very tiny deviations. Such deviations can potentially be detected in dedicated experiments, and efforts are increasing worldwide towards testing them.

An inbuilt amplification mechanism makes sure that for macroscopic systems consisting of many particles, the collapse becomes stronger than the quantum dynamics. Then their wave function is always well localized in space, so well localized that it behaves, for all practical purposes, like a point moving in space according to Newton’s laws.

In this sense, collapse models provide a unified description of microscopic and macroscopic systems, avoiding the conceptual problems associated to measurements in quantum theory.

The most well-known examples of such theories are:

• Ghirardi–Rimini–Weber (GRW) model
• Continuous spontaneous localization (CSL) model
• Diósi–Penrose (DP) model

Collapse theories stand in opposition to many-worlds interpretation theories, in that they hold that a process of wave function collapse curtails the branching of the wave function and removes unobserved behaviour.

History of collapse theories

The genesis of collapse models dates back to the 1970s. In Italy, the group of L. Fonda, G.C. Ghirardi and A. Rimini was studying how to derive the exponential decay law in decay processes, within quantum theory. In their model, an essential feature was that, during the decay, particles undergo spontaneous collapses in space, an idea that was later carried over to characterize the GRW model. Meanwhile, P. Pearle in the USA was developing nonlinear and stochastic equations, to model the collapse of the wave function in a dynamical way; this formalism was later used for the CSL model. However, these models lacked the character of “universality” of the dynamics, i.e. its applicability to an arbitrary physical system (at least at the non-relativistic level), a necessary condition for any model to become a viable option.

The breakthrough came in 1986, when Ghirardi, Rimini and Weber published the paper with the meaningful title “Unified dynamics for microscopic and macroscopic systems”, where they presented what is now known as the GRW model, after the initials of the authors. The model contains all the ingredients a collapse model should have:

• The Schrödinger dynamics is modified by adding nonlinear stochastic terms, whose effect is to randomly localize the wave function in space.
• For microscopic systems, the new terms are mostly negligible.
• For macroscopic object, the new dynamics keeps the wave function well localized in space, thus ensuring classicality.
• In particular, at the end of measurements, there are always definite outcomes, distributed according to the Born rule.
• Deviations from quantum predictions are compatible with current experimental data.  

In 1990 the efforts for the GRW group on one side, and of P. Pearle on the other side, were brought together in formulating the Continuous Spontaneous Localization (CSL) model, where the Schrödinger dynamics and the random collapse are described within one stochastic differential equation, which is capable of describing also systems of identical particles, a feature which was missing in the GRW model.

In the late 1980s and 1990s, Diosi and Penrose independently formulated the idea that the wave function collapse is related to gravity. The dynamical equation is structurally similar to the CSL equation.

In the context of collapse models, it is worthwhile to mention the theory of quantum state diffusion.

Most popular models

Three are the models, which are most widely discussed in the literature:

• Ghirardi–Rimini–Weber (GRW) model: It is assumed that each constituent of a physical system independently undergoes spontaneous collapses. The collapses are random in time, distributed according to a Poisson distribution; they are random in space and are more likely to occur where the wave function is larger. In between collapses, the wave function evolves according to the Schrödinger equation. For composite systems, the collapse on each constituent causes the collapse of the center of mass wave functions.
• Continuous spontaneous localization (CSL) model: The Schrödinger equation is supplemented with a nonlinear and stochastic diffusion process driven by a suitably chosen universal noise coupled to the mass-density of the system, which counteracts the quantum spread of the wave function. As for the GRW model, the larger the system, the stronger the collapse, thus explaining the quantum-to-classical transition as a progressive breakdown of quantum linearity, when the system’s mass increases. The CSL model is formulated in terms of identical particles.
• Diósi–Penrose (DP) model: Diósi and Penrose formulated the idea that gravity is responsible for the collapse of the wave function. Penrose argued that, in a quantum gravity scenario where a spatial superposition creates the superposition of two different spacetime curvatures, gravity does not tolerate such superpositions and spontaneously collapses them. He also provided a phenomenological formula for the collapse time. Independently and prior to Penrose, Diósi presented a dynamical model that collapses the wave function with the same time scale suggested by Penrose.

The Quantum Mechanics with Universal Position Localization (QMUPL) model should also be mentioned; an extension of the GRW model for identical particles formulated by Tumulka, which proves several important mathematical results regarding the collapse equations.

In all models listed so far, the noise responsible for the collapse is Markovian (memoryless): either a Poisson process in the discrete GRW model, or a white noise in the continuous models. The models can be generalized to include arbitrary (colored) noises, possibly with a frequency cutoff: the CSL model model has been extended to its colored version (cCSL), as well as the QMUPL model (cQMUPL). In these new models the collapse properties remain basically unaltered, but specific physical predictions can change significantly.

In collapse models the energy is not conserved, because the noise responsible for the collapse induces Brownian motion on each constituent of a physical system. Accordingly, the kinetic energy increases at a faint but constant rate. Such a feature can be modified, without altering the collapse properties, by including appropriate dissipative effects in the dynamics. This is achieved for the GRW, CSL and QMUPL models, obtaining their dissipative counterparts (dGRW, dCSL, dQMUPL). In these new models, the energy thermalizes to a finite value.

Lastly, the QMUPL model was further generalized to include both colored noise as well as dissipative effects (dcQMUPL model).

Tests of collapse models

Collapse models modify the Schrödinger equation; therefore, they make predictions, which differ from standard quantum mechanical predictions. Although the deviations are difficult to detect, there is a growing number of experiments searching for spontaneous collapse effects. They can be classified in two groups:

• Interferometric experiments. They are refined versions of the double-slit experiment, showing the wave nature of matter (and light). The modern versions are meant to increase the mass of the system, the time of flight, and/or the delocalization distance in order to create ever larger superpositions. The most prominent experiments of this kind are with atoms, molecules and phonons.
• Non-interferometric experiments. They are based on the fact that the collapse noise, besides collapsing the wave function, also induces a diffusion on top of particles’ motion, which acts always, also when the wave function is already localized. Experiments of this kind involve cold atoms, opto-mechanical systems, gravitational wave detectors, underground experiments.

Problems and criticisms to collapse theories

Violation of the principle of the conservation of energy. According to collapse theories, energy is not conserved, also for isolated particles. More precisely, in the GRW, CSL and DP models the kinetic energy increases at a constant rate, which is small but non-zero. This is often presented as an unavoidable consequence of Heisenberg’s uncertainty principle: the collapse in position causes a larger uncertainty in momentum. This explanation is fundamentally wrong. Actually, in collapse theories the collapse in position determines also a localization in momentum: the wave function is driven to an almost minimum uncertainty state both in position as well as in momentum, compatibly with Heisenberg’s principle.

The reason why the energy increases according to collapse theories, is that the collapse noise diffuses the particle, thus accelerating it. This is the same situation as in classical Brownian motion. And as for classical Brownian motion, this increase can be stopped by adding dissipative effects. Dissipative versions of the QMUPL, GRW and CSL model exist, where the collapse properties are left unaltered with respect to the original models, while the energy thermalizes to a finite value (therefore it can even decrease, depending on its initial value).

Still, also in the dissipative model the energy is not strictly conserved. A resolution to this situation might come by considering also the noise a dynamical variable with its own energy, which is exchanged with the quantum system in such a way that the total system+noise energy is conserved.

Relativistic collapse models. One of the biggest challenges in collapse theories is to make them compatible with relativistic requirements. The GRW, CSL and DP models are not. The biggest difficulty is how to combine the nonlocal character of the collapse, which is necessary in order to make it compatible with the experimentally verified violation of Bell inequalities, with the relativistic principle of locality. Models exist, that attempt to generalize in a relativistic sense the GRW and CSL models, but their status as relativistic theories is still unclear. The formulation of a proper Lorentz-covariant theory of continuous objective collapse is still a matter of research.

Tail problem. In all collapse theories, the wave function is never fully contained within one (small) region of space, because the Schrödinger term of the dynamics will always spread it outside. Therefore, wave functions always contain tails stretching out to infinity, although their “weight” is the smaller, the larger the system. Critics of collapse theories argue that it is not clear how to interpret these tails, since they amount to the system never being really fully localized in space.  Supporters of collapse theories mostly dismiss this criticism as a misunderstanding of the theory, as in the context of dynamical collapse theories, the absolute square of the wave function is interpreted as an actual matter density. In this case, the tails merely represent an immeasurably small amount of smeared-out matter, while from a macroscopic perspective, all particles appear to be point-like for all practical purposes.

Matter wave

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Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave. In most cases, however, the wavelength is too small to have a practical impact on day-to-day activities. Hence in our day-to-day lives with objects the size of tennis balls and people, matter waves are not relevant.

The concept that matter behaves like a wave was proposed by Louis de Broglie (/dəˈbrɔɪ/) in 1924. It is also referred to as the de Broglie hypothesis. Matter waves are referred to as de Broglie waves.

The de Broglie wavelength is the wavelength, λ, associated with a massive particle (i.e., a particle with mass, as opposed to a massless particle) and is related to its momentum, p, through the Planck constant, h:

${\displaystyle \lambda ={\frac {h}{p}}={\frac {h}{mv}}.}$

Wave-like behavior of matter was first experimentally demonstrated by George Paget Thomson's thin metal diffraction experiment, and independently in the Davisson–Germer experiment both using electrons, and it has also been confirmed for other elementary particles, neutral atoms and even molecules.

Historical context

At the end of the 19th century, light was thought to consist of waves of electromagnetic fields which propagated according to Maxwell's equations, while matter was thought to consist of localized particles (see history of wave and particle duality). In 1900, this division was exposed to doubt, when, investigating the theory of black-body radiation, Max Planck proposed that light is emitted in discrete quanta of energy. It was thoroughly challenged in 1905. Extending Planck's investigation in several ways, including its connection with the photoelectric effect, Albert Einstein proposed that light is also propagated and absorbed in quanta; now called photons. These quanta would have an energy given by the Planck–Einstein relation:

${\displaystyle E=h\nu }$

and a momentum

${\displaystyle p={\frac {E}{c}}={\frac {h}{\lambda }}}$

where ν (lowercase Greek letter nu) and λ (lowercase Greek letter lambda) denote the frequency and wavelength of the light, c the speed of light, and h the Planck constant.^[3] In the modern convention, frequency is symbolized by f as is done in the rest of this article. Einstein's postulate was confirmed experimentally by Robert Millikan and Arthur Compton over the next two decades.

de Broglie hypothesis

Propagation of de Broglie waves in 1d – real part of the complex amplitude is blue, imaginary part is green. The probability (shown as the color opacity) of finding the particle at a given point x is spread out like a waveform; there is no definite position of the particle. As the amplitude increases above zero the slope decreases, so the amplitude decreases again, and vice versa. The result is an alternating amplitude: a wave. Top: plane wave. Bottom: wave packet.

De Broglie, in his 1924 PhD thesis, proposed that just as light has both wave-like and particle-like properties, electrons also have wave-like properties. By rearranging the momentum equation stated in the above section, we find a relationship between the wavelength, λ, associated with an electron and its momentum, p, through the Planck constant, h:^[4]

${\displaystyle \lambda ={\frac {h}{p}}.}$

The relationship is now known to hold for all types of matter: all matter exhibits properties of both particles and waves.

When I conceived the first basic ideas of wave mechanics in 1923–1924, I was guided by the aim to perform a real physical synthesis, valid for all particles, of the coexistence of the wave and of the corpuscular aspects that Einstein had introduced for photons in his theory of light quanta in 1905.

— de Broglie^[5]

In 1926, Erwin Schrödinger published an equation describing how a matter wave should evolve – the matter wave analogue of Maxwell's equations — and used it to derive the energy spectrum of hydrogen.

Experimental confirmation

Demonstration of a matter wave in diffraction of electrons

Matter waves were first experimentally confirmed to occur in George Paget Thomson's cathode ray diffraction experiment^[2] and the Davisson-Germer experiment for electrons, and the de Broglie hypothesis has been confirmed for other elementary particles. Furthermore, neutral atoms and even molecules have been shown to be wave-like.

Electrons

Further information: Davisson–Germer experiment and Electron diffraction

In 1927 at Bell Labs, Clinton Davisson and Lester Germer fired slow-moving electrons at a crystalline nickel target. The angular dependence of the diffracted electron intensity was measured, and was determined to have the same diffraction pattern as those predicted by Bragg for x-rays. At the same time George Paget Thomson at the University of Aberdeen was independently firing electrons at very thin metal foils to demonstrate the same effect. Before the acceptance of the de Broglie hypothesis, diffraction was a property that was thought to be exhibited only by waves. Therefore, the presence of any diffraction effects by matter demonstrated the wave-like nature of matter. When the de Broglie wavelength was inserted into the Bragg condition, the observed diffraction pattern was predicted, thereby experimentally confirming the de Broglie hypothesis for electrons.

This was a pivotal result in the development of quantum mechanics. Just as the photoelectric effect demonstrated the particle nature of light, the Davisson–Germer experiment showed the wave-nature of matter, and completed the theory of wave–particle duality. For physicists this idea was important because it meant that not only could any particle exhibit wave characteristics, but that one could use wave equations to describe phenomena in matter if one used the de Broglie wavelength.

Neutral atoms

Experiments with Fresnel diffraction and an atomic mirror for specular reflection of neutral atoms confirm the application of the de Broglie hypothesis to atoms, i.e. the existence of atomic waves which undergo diffraction, interference and allow quantum reflection by the tails of the attractive potential. Advances in laser cooling have allowed cooling of neutral atoms down to nanokelvin temperatures. At these temperatures, the thermal de Broglie wavelengths come into the micrometre range. Using Bragg diffraction of atoms and a Ramsey interferometry technique, the de Broglie wavelength of cold sodium atoms was explicitly measured and found to be consistent with the temperature measured by a different method.

This effect has been used to demonstrate atomic holography, and it may allow the construction of an atom probe imaging system with nanometer resolution. The description of these phenomena is based on the wave properties of neutral atoms, confirming the de Broglie hypothesis.

The effect has also been used to explain the spatial version of the quantum Zeno effect, in which an otherwise unstable object may be stabilised by rapidly repeated observations.

Molecules

Recent experiments even confirm the relations for molecules and even macromolecules that otherwise might be supposed too large to undergo quantum mechanical effects. In 1999, a research team in Vienna demonstrated diffraction for molecules as large as fullerenes. The researchers calculated a De Broglie wavelength of the most probable C₆₀ velocity as 2.5 pm. More recent experiments prove the quantum nature of molecules made of 810 atoms and with a mass of 10,123 amu. As of 2019, this has been pushed to molecules of 25,000 amu.

Still one step further than Louis de Broglie go theories which in quantum mechanics eliminate the concept of a pointlike classical particle and explain the observed facts by means of wavepackets of matter waves alone.

de Broglie relations

The de Broglie equations relate the wavelength λ to the momentum p, and frequency f to the total energy E of a free particle:

${\displaystyle {\begin{aligned}&\lambda =h/p\\&f=E/h\end{aligned}}}$

where h is the Planck constant. The equations can also be written as

${\displaystyle {\begin{aligned}&\mathbf {p} =\hbar \mathbf {k} \\&E=\hbar \omega \\\end{aligned}}}$

or 

${\displaystyle {\begin{aligned}&\mathbf {p} =\hbar \mathbf {\beta } \\&E=\hbar \omega \\\end{aligned}}}$

where ħ = h/2π is the reduced Planck constant, k is the wave vector, β is the phase constant, and ω is the angular frequency.

In each pair, the second equation is also referred to as the Planck–Einstein relation, since it was also proposed by Planck and Einstein.

Special relativity

Using two formulas from special relativity, one for the relativistic mass energy and one for the relativistic momentum

${\displaystyle E=mc^{2}=\gamma m_{0}c^{2}}$

${\displaystyle {\vec {p}}=m{\vec {v}}=\gamma m_{0}{\vec {v}}}$

allows the equations to be written as

${\displaystyle {\begin{aligned}&\lambda =\,\,{\frac {h}{\gamma m_{0}v}}\,=\,{\frac {h}{m_{0}v}}\,\,\,\,{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\\&f={\frac {\gamma \,m_{0}c^{2}}{h}}={\frac {m_{0}c^{2}}{h}}{\bigg /}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\end{aligned}}}$

where ${\displaystyle m_{0}}$ denotes the particle's rest mass, ${\displaystyle v}$ its velocity, ${\displaystyle \gamma }$ the Lorentz factor, and ${\displaystyle c}$ the speed of light in a vacuum. See below for details of the derivation of the de Broglie relations. Group velocity (equal to the particle's speed) should not be confused with phase velocity (equal to the product of the particle's frequency and its wavelength). In the case of a non-dispersive medium, they happen to be equal, but otherwise they are not.

Group velocity

Albert Einstein first explained the wave–particle duality of light in 1905. Louis de Broglie hypothesized that any particle should also exhibit such a duality. The velocity of a particle, he concluded, should always equal the group velocity of the corresponding wave. The magnitude of the group velocity is equal to the particle's speed.

Both in relativistic and non-relativistic quantum physics, we can identify the group velocity of a particle's wave function with the particle velocity. Quantum mechanics has very accurately demonstrated this hypothesis, and the relation has been shown explicitly for particles as large as molecules.

De Broglie deduced that if the duality equations already known for light were the same for any particle, then his hypothesis would hold. This means that

${\displaystyle v_{g}={\frac {\partial \omega }{\partial k}}={\frac {\partial (E/\hbar )}{\partial (p/\hbar )}}={\frac {\partial E}{\partial p}}}$

where E is the total energy of the particle, p is its momentum, ħ is the reduced Planck constant. For a free non-relativistic particle it follows that

${\displaystyle {\begin{aligned}v_{g}&={\frac {\partial E}{\partial p}}={\frac {\partial }{\partial p}}\left({\frac {1}{2}}{\frac {p^{2}}{m}}\right)\\&={\frac {p}{m}}\\&=v\end{aligned}}}$

where m is the mass of the particle and v its velocity.

Also in special relativity we find that

${\displaystyle {\begin{aligned}v_{g}&={\frac {\partial E}{\partial p}}={\frac {\partial }{\partial p}}\left({\sqrt {p^{2}c^{2}+m_{0}^{2}c^{4}}}\right)\\&={\frac {pc^{2}}{\sqrt {p^{2}c^{2}+m_{0}^{2}c^{4}}}}\\&={\frac {pc^{2}}{E}}\end{aligned}}}$

where m₀ is the rest mass of the particle and c is the speed of light in a vacuum. But (see below), using that the phase velocity is v_p = E/p = c²/v, therefore

${\displaystyle {\begin{aligned}v_{g}&={\frac {pc^{2}}{E}}\\&={\frac {c^{2}}{v_{p}}}\\&=v\end{aligned}}}$

where v is the velocity of the particle regardless of wave behavior.

Phase velocity

In quantum mechanics, particles also behave as waves with complex phases. The phase velocity is equal to the product of the frequency multiplied by the wavelength.

By the de Broglie hypothesis, we see that

${\displaystyle v_{\mathrm {p} }={\frac {\omega }{k}}={\frac {E/\hbar }{p/\hbar }}={\frac {E}{p}}.}$

Using relativistic relations for energy and momentum, we have

${\displaystyle v_{\mathrm {p} }={\frac {E}{p}}={\frac {mc^{2}}{mv}}={\frac {\gamma m_{0}c^{2}}{\gamma m_{0}v}}={\frac {c^{2}}{v}}={\frac {c}{\beta }}}$

where E is the total energy of the particle (i.e. rest energy plus kinetic energy in the kinematic sense), p the momentum, ${\displaystyle \gamma }$ the Lorentz factor, c the speed of light, and β the speed as a fraction of c. The variable v can either be taken to be the speed of the particle or the group velocity of the corresponding matter wave. Since the particle speed ${\displaystyle v<c}$ for any particle that has mass (according to special relativity), the phase velocity of matter waves always exceeds c, i.e.

${\displaystyle v_{\mathrm {p} }>c,\,}$

and as we can see, it approaches c when the particle speed is in the relativistic range. The superluminal phase velocity does not violate special relativity, because phase propagation carries no energy. See the article on Dispersion (optics) for details.

Four-vectors

Using four-vectors, the De Broglie relations form a single equation:

${\displaystyle \mathbf {P} =\hbar \mathbf {K} }$

which is frame-independent.

Likewise, the relation between group/particle velocity and phase velocity is given in frame-independent form by:

${\displaystyle \mathbf {K} =\left({\frac {\omega _{o}}{c^{2}}}\right)\mathbf {U} }$

where

Four-momentum ${\displaystyle \mathbf {P} =\left({\frac {E}{c}},{\vec {\mathbf {p} }}\right)}$

Four-wavevector ${\displaystyle \mathbf {K} =\left({\frac {\omega }{c}},{\vec {\mathbf {k} }}\right)=\left({\frac {\omega }{c}},{\frac {\omega }{v_{p}}}\mathbf {\hat {n}} \right)}$

Four-velocity ${\displaystyle \mathbf {U} =\gamma (c,{\vec {\mathbf {u} }})=\gamma (c,v_{g}{\hat {\mathbf {n} }})}$

Interpretations

The physical reality underlying de Broglie waves is a subject of ongoing debate. Some theories treat either the particle or the wave aspect as its fundamental nature, seeking to explain the other as an emergent property. Some, such as the hidden variable theory, treat the wave and the particle as distinct entities. Yet others propose some intermediate entity that is neither quite wave nor quite particle but only appears as such when we measure one or the other property. The Copenhagen interpretation states that the nature of the underlying reality is unknowable and beyond the bounds of scientific inquiry.

Schrödinger's quantum mechanical waves are conceptually different from ordinary physical waves such as water or sound. Ordinary physical waves are characterized by undulating real-number 'displacements' of dimensioned physical variables at each point of ordinary physical space at each instant of time. Schrödinger's "waves" are characterized by the undulating value of a dimensionless complex number at each point of an abstract multi-dimensional space, for example of configuration space.

At the Fifth Solvay Conference in 1927, Max Born and Werner Heisenberg reported as follows:

If one wishes to calculate the probabilities of excitation and ionization of atoms [M. Born, Zur Quantenmechanik der Stossvorgange, Z. f. Phys., 37 (1926), 863; [Quantenmechanik der Stossvorgange], ibid., 38 (1926), 803] then one must introduce the coordinates of the atomic electrons as variables on an equal footing with those of the colliding electron. The waves then propagate no longer in three-dimensional space but in multi-dimensional configuration space. From this one sees that the quantum mechanical waves are indeed something quite different from the light waves of the classical theory.

At the same conference, Erwin Schrödinger reported likewise.

Under [the name 'wave mechanics',] at present two theories are being carried on, which are indeed closely related but not identical. The first, which follows on directly from the famous doctoral thesis by L. de Broglie, concerns waves in three-dimensional space. Because of the strictly relativistic treatment that is adopted in this version from the outset, we shall refer to it as the four-dimensional wave mechanics. The other theory is more remote from Mr de Broglie's original ideas, insofar as it is based on a wave-like process in the space of position coordinates (q-space) of an arbitrary mechanical system.[Long footnote about manuscript not copied here.] We shall therefore call it the multi-dimensional wave mechanics. Of course this use of the q-space is to be seen only as a mathematical tool, as it is often applied also in the old mechanics; ultimately, in this version also, the process to be described is one in space and time. In truth, however, a complete unification of the two conceptions has not yet been achieved. Anything over and above the motion of a single electron could be treated so far only in the multi-dimensional version; also, this is the one that provides the mathematical solution to the problems posed by the Heisenberg-Born matrix mechanics.

In 1955, Heisenberg reiterated this:

An important step forward was made by the work of Born [Z. Phys., 37: 863, 1926 and 38: 803, 1926] in the summer of 1926. In this work, the wave in configuration space was interpreted as a probability wave, in order to explain collision processes on Schrödinger's theory. This hypothesis contained two important new features in comparison with that of Bohr, Kramers and Slater. The first of these was the assertion that, in considering "probability waves", we are concerned with processes not in ordinary three-dimensional space, but in an abstract configuration space (a fact which is, unfortunately, sometimes overlooked even today); the second was the recognition that the probability wave is related to an individual process.

It is mentioned above that the "displaced quantity" of the Schrödinger wave has values that are dimensionless complex numbers. One may ask what is the physical meaning of those numbers. According to Heisenberg, rather than being of some ordinary physical quantity such as, for example, Maxwell's electric field intensity, or mass density, the Schrödinger-wave packet's "displaced quantity" is probability amplitude. He wrote that instead of using the term 'wave packet', it is preferable to speak of a probability packet. The probability amplitude supports calculation of probability of location or momentum of discrete particles. Heisenberg recites Duane's account of particle diffraction by probabilistic quantal translation momentum transfer, which allows, for example in Young's two-slit experiment, each diffracted particle probabilistically to pass discretely through a particular slit. Thus one does not need necessarily think of the matter wave, as it were, as 'composed of smeared matter'.

These ideas may be expressed in ordinary language as follows. In the account of ordinary physical waves, a 'point' refers to a position in ordinary physical space at an instant of time, at which there is specified a 'displacement' of some physical quantity. But in the account of quantum mechanics, a 'point' refers to a configuration of the system at an instant of time, every particle of the system being in a sense present in every 'point' of configuration space, each particle at such a 'point' being located possibly at a different position in ordinary physical space. There is no explicit definite indication that, at an instant, this particle is 'here' and that particle is 'there' in some separate 'location' in configuration space. This conceptual difference entails that, in contrast to de Broglie's pre-quantum mechanical wave description, the quantum mechanical probability packet description does not directly and explicitly express the Aristotelian idea, referred to by Newton, that causal efficacy propagates through ordinary space by contact, nor the Einsteinian idea that such propagation is no faster than light. In contrast, these ideas are so expressed in the classical wave account, through the Green's function, though it is inadequate for the observed quantal phenomena. The physical reasoning for this was first recognized by Einstein.

De Broglie's phase wave and periodic phenomenon

De Broglie's thesis started from the hypothesis, “that to each portion of energy with a proper mass m₀ one may associate a periodic phenomenon of the frequency ν₀, such that one finds: hν₀ = m₀c². The frequency ν₀ is to be measured, of course, in the rest frame of the energy packet. This hypothesis is the basis of our theory.”  (This frequency is also known as Compton frequency.)

De Broglie followed his initial hypothesis of a periodic phenomenon, with frequency ν₀ , associated with the energy packet. He used the special theory of relativity to find, in the frame of the observer of the electron energy packet that is moving with velocity ${\displaystyle v}$, that its frequency was apparently reduced to

${\displaystyle \nu _{1}=\nu _{0}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\,.}$

De Broglie reasoned that to a stationary observer this hypothetical intrinsic particle periodic phenomenon appears to be in phase with a wave of wavelength ${\displaystyle \lambda }$ and frequency ${\displaystyle f}$ that is propagating with phase velocity ${\displaystyle v_{\mathrm {p} }}$. De Broglie called this wave the “phase wave” («onde de phase» in French). This was his basic matter wave conception. He noted, as above, that ${\displaystyle v_{\mathrm {p} }>c}$, and the phase wave does not transfer energy.

While the concept of waves being associated with matter is correct, de Broglie did not leap directly to the final understanding of quantum mechanics with no missteps. There are conceptual problems with the approach that de Broglie took in his thesis that he was not able to resolve, despite trying a number of different fundamental hypotheses in different papers published while working on, and shortly after publishing, his thesis. These difficulties were resolved by Erwin Schrödinger, who developed the wave mechanics approach, starting from a somewhat different basic hypothesis.