*©Fernando Caracena 2013*

## Waves, particles or both?

Planck broke through the looking glass and the way that physicists saw everything changed forever. Physics underwent a major paradigm shift. It was not an easy transition. Neils Bohr, one of the main architects of quantum mechanics said "If you aren't confused by quantum mechanics, you haven't really understood it." Also, "Anyone who is not shocked by quantum theory has not understood it." Another architect of quantum theory, Werner Heisenberg, said "I remember discussions with Bohr which went through many hours till very late at night and ended almost in despair…Can nature possibly be so absurd as it seemed to us in these atomic experiments?"

There is a series of lectures on quantum mechanics by Leonard Susskind from Stanford University in California that says most of what I would write here, perhaps more abstractly. I value my time, and his lectures are readily available, so I refer the reader to Susskind's lectures, which leaves me open to discuss other things about quantum mechanics, perhaps some philosophy.

Physics was proceeding along a path of development before quantum theory emerged that assumed that all things broken apart would finally lead to the smallest basic parts of it, which were its simplest possible parts. These simplest things would be particles of matter, which would be like point masses. Quantum phenomena, however, destroyed this entire line of thinking, and set physics on an entirely different path of development that destroyed all such simple pictures of the smallest bits of matter.

At the onset of quantum theory, not everything was known to be particles. There were known fields, such as that of gravity and electromagnetism. Fields fill space with values that vary continuously and smoothly from point to point, and which develop according to equations of motion. James Clerk Maxwell formulated the laws of electromagnetism, discovered by Michael Faraday, in terms of a few partial differential equations, which he used to predict the existence of electromagnet waves that happened to propagate at a speed equal to the speed of light. Further experimentation showed that light was really a portion of the spectrum of electromagnetic (EM) waves. Since the only difference between EM waves is their frequency, many physicists like to refer to the entire EM wave spectrum as "light."

The big problem in physics came when Max Planck discovered that light, which consists of waves, has also a particle aspect in that it can be emitted and absorbed only in discrete chunks of energy,the size of which is proportional to the frequency of a portion of the light spectrum

ε_{ν} = hν, (1a)

where ε_{ν} is the energy of a single excitation of that portion of the light spectrum having the frequency ν in Hz (cycles per second). The portion of electromagnetic waves having a frequency ν can exchange energy with other things in an integral number of units of ε_{ν}. As a result, the ν frequency portion of the EM spectrum can contain only an integer number of these excitations above the zero point energy level, which happens to be one half of this unit for each spectral component

E_{ν} = (n+½) ε_{ν} . (2)

The zero point energy,½ ε_{ν}, is a feature of quantum theory that perhaps is the source of dark energy that fills what appears to be the empty space of the physical vacuum. Some inventors claim to have been able to tap into this zero-point energy to build what amounts to a perpetual motion machine, but so far this has not been verified by scientists, although it remains a feature of a popular computer game.

Because of their discreteness, EM field quanta are called photons, where the name implies particles among others, such as electrons, neutrons, protons, positrons, etc. Associated with these field quanta are other particle-like properties, such as momentum and spin angular momentum. Lumped together in a single entity we cannot avoid picturing photons as little balls that travel very fast through the vacuum of space. In fact, an observed scattering of light by electrons or other charged particles, called Compton scattering, can be modeled mathematically as a relativistic billiard-ball collision.

In our intuition acquired from everyday experience, which is described by classical physics, we associate discreteness of physical properties with particles. From a classical perspective, we would tend to visualize particles as as points that carry all of the properties of those particles along a trajectory, which is a one dimensional curve embedded in three dimensions and time. With regard to EM waves, the minimal exchange with matter of the ν-spectral component is called a photon, which suggests a particle. However, EM theory treats the propagation of EM waves as continuous waves that fill portions of space. So we have two pictures of light competing for our attention: particles and waves. Neither picture is a complete description of light. Each by itself is inadequate to describe light phenomena. Quantum theory replaces the classical notions of our intuition with a theory that applies to matter at all scales. Everything is wave-like, but in the case of ordinary objects such as baseballs and bowling balls, the wave properties are so small that they are not noticeable, and are completely negligible. Quantum theory is a complete description of the wave-like behavior of matter at all scales. On the microscopic scale, however, it dominates in such a way that it results in the casualty of the loss of a clear, intuitive picture of what is really going on.

Richard Feynman, who among two others won the Nobel Prize for his work on Quantum Electrodynamics (QED) while talking about the double slit experiment in his famous lecture series, "The Character of Physical Law" mentioned that nobody understands quantum mechanics because it is beyond the range of ordinary experience. This particular lecture is a very good discussion of the quantum effects and the dual nature of quanta. By the way, his double slit thought experiment has been experimentally realized just recently.

*Momenta and other wave properties*

*Momenta and other wave properties*

The scalar product of the momentum four vector of a particle with itself is an invariant, which is just the square of its rest mass times the square of the speed of light. Note that from here on, the rest mass will be labeled, m, without any subscript. To show that the above statement is true, we just compute the square of the momentum four vector below νλand carry out the multiplication until we arrive at the result

**P** = (E/c, p), (3a)

**P**** •**** P** = (E/c)^{2} - **p**^{2}, (3b)

**P**** •**** P** = (γ mc)^{2 }- ( γmv)^{2}, (3c)

**P**** •**** P** = (γ mc)^{2 }[1 – (v/c)^{2}], (3d)

**P**** •**** P** = (γ mc)^{2} γ^{--2}, (3e)

**P**** •**** P** = (mc)^{2}. (3f)

The point of this exercise is to compute the relationship between energy and momentum for a particle of zero rest mass, such as the photon, which is

**P**** •**** P** =0,

or

(E/c)^{2} - **p**^{2} =0,

which results in the magnitude of the photon momentum

p = E/c . (4)

Now substitute for the photon energy using (1)

p = hν/c

now use the relation between the frequency, wavelength and phase velocity of electromagnetic waves

ν λ = c

p = h / λ. (5)

The relations between energy and momentum and the frequency and wavelength (1) and (5) can be redefined in terms of angular frequency (ω) and wave number (k) as follows:

ε_{ν} = ħ ω (6a)

p = ħ k, (6b)

where the angular frequency is

ω = 2π ν (7a)

and the wave number (magnitude of the wave vector) is

k=2π/ λ. (7b)

The above equations establish a correlation between the photon's wave properties and its particle properties.

In 1929 Luis de Broglie won the Nobel prize in physics for his theory that not only is there a wave particle duality for photons, but that this duality existed for all submicroscopic particles, such as electrons, protons, etc. As Richard Feynman would say, not only is light screwy, but electrons are screwy in the same way.

Prof Water Lewin of MIT presents a lecture of the Heisenberg's Uncertainty Principle in this video.

The next blog will discuss some of the early quantum theory.

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Wavicles--quantum quanderies