sk nature of matter waves [2 of 3]

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WHAT IS THERE WAVING?If an electron is a wave

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Beginning of Matter Waves

When de Broglie first came up with the idea of matter waves, he was not able to pin point what actually waves. The idea came to him when he saw the probability of as an analogy between electrons and photons. Photon as a wave-particle was quite well established; but matter wave at that moment was more a mathematical construct than a reality since electron wave was not yet known. However, it happened that the idea turned out unexpectedly to be very helpful and so he carried on with it.

Photon wave

Electron wave


Wave Nature not Considered in Beginning

When de Broglie was working on his thesis, he was not overly concerned with the nature of the wave. At the beginning, he was only trying to find a theory to compromise their coexistence of both wave and particle properties in the photon. He regarded the coexistences as a curious kind of dualism that may be intrinsic in the nature of things:

“When in 1922-1923, I had my first ideas about wave mechanics, I was guided by the vision of constructing a true physical synthesis, resting upon precise concepts, of the coexistence of waves and particles. I never questioned then the nature of the physical reality of waves and particles.”*

When the particle wave concept became a celebrated theory, de Broglie began to feel the need to investigate into its physical reality.


Electromagnetic Nature of LightThe classical electromagnetic theory of Maxwell provided a physical basis to the nature of light. It is simply the oscillations of the electric and magnetic fields. Although no further explanation was given to the nature of the fields, the classical model of the photon envisages a wave propagating in the 𝒛𝒛 direction and the electromagnetic fields (𝐸𝐸 and 𝐵𝐵)* waving in the direction transverse to the propagation. They are simply the oscillations of the electric and magnetic fields, although no further explanation was given to the nature of the fields. Matter wave presents a more mysterious shroud over its nature.

B

E

P

Magnetic field Electric field

Poynting vector

Direction of Poynting vectorMagnetic field


Plane Matter WavesMatter wave presents a more mysterious shroud over its nature. At first, de Broglie thought that these waves were sinusoidal and plane in nature with their fronts perpendicular to the particle's direction of propagation, just like plane electromagnetic waves. However, Broglie later realized that a plane monochromatic wave is but an idealization which is not physically viable.


Pilot WavesAccording to de Broglie, all particles were accompanied by actual physical waves which acted like a pilot guiding the particle along its trajectories. The wave is physically real and occupies a certain region in space while the particle is a material point having a certain position in the wave. He called them the pilot waves. He believed that these distinctive assignments to both wave and particle are in closest accord with classical concepts of waves and particles.


Probability Wavesde Broglie also incorporated the probability element advocated by Born into this interpretation in that the probability of finding the particle is proportional to the intensity of the wave at the point. In the classical picture, when the particle wave incidents on a boundary between two media, it splits into a reflected wave and a refracted wave. The probabilities of the particle in these two opposite waves are determined by the amplitude of these waves. Thus the difficulty of having one particle partially reflected and partially refracted is lifted.

Incident wave Reflected wave

Refracted wave

%

%


Probability Wave too fast

However, such waves at times will be travelling at speed greater than that of light. This is taboo in the theory of relativity. At the same time, prediction of particle energy in bichromatic waves basing upon this hypothesis did not agree with experiment. As a result, de Broglie had to give up the interpretation.


Mathematical Analysis

de Broglie also tried to break down a wave into complex waves represented by Fourier integrals - forming a wave by the superposition of a number of component waves.In essence, the wave was a physical wave of very weak amplitude whose essential role was to guide the motion of the particle. This interpretation was untenable and was subsequently discarded as well.

𝑓𝑓 𝑥𝑥 =𝑎𝑎𝑜𝑜2

+ �𝑛𝑛=1

∞

𝑎𝑎𝑛𝑛 𝑐𝑐𝑐𝑐𝑐𝑐 𝑛𝑛𝑥𝑥 + �𝑛𝑛=1

∞

𝑏𝑏𝑛𝑛 𝑐𝑐𝑠𝑠𝑛𝑛 𝑛𝑛𝑥𝑥

Fourier analysis


Mathematical Wave

At a certain stage, de Broglie thought that matter was purely made of waves which were the only reality in nature. In order to explain such a reality, he came up with a vague theory of mathematical structure. According to him, a particle is a localized concentration of energy in the form of waves with extremely short wavelengths. In mathematical terms, a particle is represented by a point-singularity in the wave field. This kind of singularity was non-physical in nature. But no further light was shed on the term except some mathematical manipulations.


Wave function

In modern quantum mechanics, the reality of the de Broglie wave has undergone tremendous changes. The wave is no longer real. A “wave” isn't what is normally imagined as something that moves up and down and moves in one direction, like ripples in water. It's just a function that evolves with time and has a different value at different point in space.


Wave function 𝜓𝜓

The familiar wave is replaced by a mathematical function called the wave function 𝜓𝜓 (psi). This wave does not "exist" per se in physical space. It can be drawn (superimposed) on physical space, but that just means that it has a value at every point there. The absolute value of the function is the squared |𝜓𝜓(𝑥𝑥)|2 of the wave function. It gives the probability density of finding the particle in a given location. Here, it is the wave function is waving and what it waves is probability, not a physical entity.


Is the matter-wave an extended object?

Some scientists tried to think of the electron as an extended object. An electron may be considered as a collection of millions of fragments instead of a single integrated particle. It spreads out as a hump and there is the powder of an electron at every point.

In such a picture there is no electron-particle. What one observes is only the fraction corresponds to the probability of finding the electron there. The denser are the powdery parts, the more likely is the electron found. The fractions behaves like an electron because they clump together the minute one tries to make an observation. So it is meaningless in asking what is it that is waving in the electron. An electron is an extended object. In the field of an atom, the orbital electrons extended smoothly like clouds round the nucleus.


Probability Density

Some other scientists would support the idea by saying that the product of the charge −𝑒𝑒 and the probability density |𝜓𝜓(𝑥𝑥)|2can be interpreted as a charge density.This is due to the motion of the electron in an atom. It moves so fast that the forces they exert on other charges are essentially equivalent to the forces exerted by a charge distribution prescribed by −𝑒𝑒|𝜓𝜓(𝑥𝑥)|2.


Is the matter-wave an extended object?

The idea of the smeared out electron is but a murky transition of a single particle to a collection of fractional particles. Though it is an intuitive attempt to explain the nature of the quantum wave, the idea of an electron as a smeared object or a charge distribution was met with much objections. Firstly because this form of the electron is different from the traditional form. Secondly the Charge density is only valid in the presence of large number of charged particles. An electron is an electron, not a collection of smaller particles.The renowned physicist Richard Feynman strongly protested: “the wave function of an electron in an atom does not, then, describe a smeared-out electron with a smooth charge density. The electron is either here, or there, or somewhere else, but wherever it is, it is a point charge”.


Wave Packets

Erwin Schrӧdinger (1877-1961) also worked on the idea that the de Broglie wave was formed by the superposition of several waves. His adeptness in mathematics enabled him to put his findings in complicated and abstract mathematical forms, among which the famous Schrӧdinger’s equation was one of the sublime examples. He came up with the notion that these waves worked well with the fictitious wave function 𝜓𝜓 which propagated in a fictitious space. Schroëdinger suggested that a particle was only a wave packet (Wellenpaket) of de Broglie waves. The wave packet assumed a well-defined locality in space and time. It is therefore an ideal candidate to represent highly localized matter. What is more, its group velocity coincided perfectly with the trajectory of the particle. This eliminated the dilemma that the individual waves may travel faster than the particle itself.

𝝍𝝍


Quantum Mechanics

Some physicists found the reality of the wave packet unacceptable. For one thing, such a group would be destroyed by dispersion during diffraction experiments, so that the particle would no longer be found in the scattered beams. A typical example is found in the refraction and reflection of a matter wave incident on a boundary between two media. It is extremely hard to accept that both the refracted and reflected wave group still represent the one and only original electron. For another, the wave group spreads out in time. It cannot therefore represent a particle in the aspect of stable existence.


End of de Broglie Wave

In quantum mechanics (QM), the de Broglie wave has become a mathematical construct. It is probability and not anything physical that is waving. So it can be said that the quest for the nature of de Broglie wave meets its end here. It is no longer of any physical meaning to ask the question: “What is it waving?” As a consequence, the original matter waves gradually lost much of their physical attributes and became grossly fictitious. The new wave idea turned out to be an abstract theory constructed over a purely mathematical substructure. The corpuscle itself becomes a term represented by symbols and abstract notions, representing a quantum world that is so contrary to conventional perception.


What is waving there?

de Broglie spent a lot of time much time after his formulation of matter-waves. His efforts went without much success and this dilemma stayed unsolved ever since. So in spite of all the successful experimental verifications of the existence of the de Broglie waves and its applications, the question remains unsolved from 1929 to the present time:

What exactly is in existence in a matter wave?

- a mathematical symbolism?- a particle?- a wave?


de Broglie’s wave

WHERE DID IT GO WRONG?


de Broglie equation

The relation between the wavelength and the momentum of a particle is the core essence of the ground breaking de Broglie equation.

In the equation, 𝜆𝜆 is called the de Broglie wavelength of the particle, 𝑝𝑝is its momentum, and ℎ is Planck’s constant:

𝜆𝜆 =ℎ𝑝𝑝

Wavelength of particle

Planck’s constant

Momentum of the particle



de Broglie’s Problem

However in spite of all the efforts in the rest of his life, de Broglie could not figure out what is there waving. Instead, it ends up in the hands of other physicists as probability wave without any physical substance. Probability became the fundamental essence of modern quantum mechanics. The earlier particle-wave picture is no longer seriously considered.This is why de Broglie was so upset with the later versions of quantum mechanics.


Where did it go wrong?In the earlier days of the discovery, electron was taken as the typical particle that exhibited the particle-wave phenomenon. So in our discussion, we concentrate on the case of the electron:


𝜆𝜆𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑜𝑜𝑛𝑛 =ℎ

𝑝𝑝𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑜𝑜𝑛𝑛

Wavelength of electron

Planck’s constant

Momentum of the electron

Electron𝒆𝒆−

sk nature of matter waves [2 of 3]

Science

wave nature

photon wave electron

wave propagating

wave mechanics

plane matter waves matter

particle wave concept

plane monochromatic

pilot waves