Chapter 29
Quantum Theory

The 20th century gave us a completely new, puzzling and unexpected way of looking at the physical world, as described by the theories of Quantum Mechanics and Relativity. An interesting difference between the two is that, while Relativity was born "fully dressed", i.e. as a complete and mature theory, Quantum Mechanics, had a very slow and gradual development, that is continuing into the present age.

While both theories produce a very counter-intuitive and "unimaginable" picture of the world, it is a fact that, in spite of its undisputable successes and predictive power, as of today it is not really clear what Quantum Theory really means, and the debate is still wide open about the nature of physical reality, as described by the quantum rules.

As an anticipation to what we will learn, here are some of the major innovations introduced by the quantum theory :

1.

at the atomic and sub-atomic level, physical processes and physical quantities are not continuous but quantized, i.e. they occur in discrete chunks and/or steps. (Note : you are discreet if you mind your own business, while you are discrete if you are made of small finite chunks)

In contrast, the whole body of Classical Physics is based on the idea of continuity. Calculus, which was "invented" by Newton to express his description of the world, can be thought of as the mathematical tool to describe continuous processes. But we have already encountered a discontinuous process: when light is produced, it is not accelerated from 0 up to c, but it instantenuously acquires its final speed.

2.

in the quantum picture, determinism has to be abandoned in favour of uncertainty and probabilism. As it has been said, Nature does not know what it will do next.

3.

one of the most intriguing puzzles of the quantum world is the dual nature (wave and particle) of both matter and radiation. How this can be is still an open question.

But let us proceed in order : it all started, even though nobody at the time realized it, with Max Planck, at the end of the 20th century. When studying radiation (Chapter 13), we had seen that a body at a given temperature emits electro-magnetic radiation which depends uniquely on its temperature. A systematic study would show that the emission spectrum has a distorted bell shape, with the peak of the curve shifting to higher and higher frequencies as the temperature increases (think of a heating a metal rod and observing the gradual shift of emitted radiation from infrared to red to blue light...). A cold body would emit preferentially in the radio-wave range, while a super hot one would produce UV and X-rays.

The problem was that the observed spectra of radiation could not be explained in terms of Classical Physics, based upon ideas of continuity. Trying to solve the problem, Planck realized that agreement with the data could be found by assuming that radiation can be emitted only in integer multiples of a minimum unit.

Useful comparison: think of how there is a smallest unit for any given currency, and payments can be made only in multiples of that unit. Obviously if the fundamental unit is small enough, one is not aware of the "quantization".

The smallness of the fundamental unit was certainly the case for the emission of radiation: to reconcile his model to the data, Planck found that, for a given frequency f, energy was emitted in finite packets of energy quanta, according to :

E = hf

where h is an exceedingly small constant, $h = 6.6\times 10^{-34}$ joule second.

Planck presented his conclusions on Dec 14, 1900, and we like to identify that date as the "birth of Quantum Mechanics" . Still, the event went largely unnoticed in the world of Physics.

The next episode in the saga sees again Einstein as a protagonist, and it involves the photoelectric effect. After the 1896 discovery of the electron, physicists performed several studies of the properties of this newly discovered constituent of matter. Among the new phenomena associated with it, it was seen that, shining light onto a metal surface, electrons could be pulled out of the material. So far, nothing to be too surprised about : light waves, it was well known, do carry energy, and it is not hard to imagine that, when this energy is transferred to an electron, it might give it enough of a kick to free it from the binding force that keeps it inside the material. It is like surf pounding on a dam, and getting some chunks of concrete off it...

The problem with the Photoelectric Effect was that it was not exhibiting a wave-like behaviour. As we have seen, the energy of a wave (including electro-magnetic waves) is proportional to the (square of) its amplitude; in the case of light, this is in turn related to the light's intensity. It was then logical to expect that the photo-electric effect could be enhanced by hitting the metal with light of higher and higher intensity. But this was not what was observed : the observations showed that, for a given metal, low frequency (e.g. red) light could not extract any electron no matter how intense the light source, while on the other side, if the frequency was high enough (e.g. blue), even a very dim light could pull out the electrons. Moreover, when electrons were extracted, their energy was not related to the intensity (i.e. the energy) of the incoming light, but again only to its frequency.

The quandary was resolved by Einstein : recalling Planck's results, he made the hypothesis that light carries energy in well defined packets, each packet carrying an amount E = hf. If we also assume that to pull an electron out of the metal it takes an amount of energy above a certain minimum value, the peculiarity of the photoelectric effect can be explained : if the energy of an individual quantum of red light is not enough to pull out an electron, increasing their number will not have any effect (except for the extremely unlikely occurrence of two or more quanta hitting the same electron within a short time interval). On the other side, even a single blue photon could do the trick. The energy of ejected electrons couls also be explained in terms of Einstein hypothesis:

let's call W₀ the minimum amount of energy required to free an electron from within a metal (one can think of a mechanical analogy : if an object is inside a hole, I need to give it a minimum amount of kinetic energy to kick it out of the hole, this amount being the difference in gravitational potential energy between the bottom and the top of the hole). Energy conservation than tells us that the maximum kinetic energy an extracted electron can have is given by:

E^max_kin = hf - W₀,

which does not depend upon the light's intensity but on its frequency. Light therefore, concluded Einstein, behaves like a particle.

Obviously, this interpretation could not be easily accepted, since it was very well established that light was a wave. If questioned, Einstein would have answered that he did not know how to reconcile the two pictures, but nevertheless that was how nature behaved. Later experiments confirmed the picture in full, even though nobody (including Einstein) can picture how something can be a wave and a particle at the same time.

Einstein, circa 1955 : fifty years of conscious brooding have brought me no nearer to the question of "what are light quanta?". Nowadays every clod thinks he knows it, but he is mistaken

An equally compelling demonstration of the particle nature of light is given by the scattering of light off electrons: in 1923, A. Compton, experimenting with X-rays, showed that the energy acquired by electrons when hit by X-rays was accompanied by a decrease in frequency of the scattered X-ray, according to

E^kin_electron = hf_in - hf_out

which is what one expects from energy conservation if one treats the X-ray/electron interaction as a a standard "billiard ball" elastic collision. This phenomenon (known now as the Compton effect) also allows to verify that photons, even though massless, do carry momentum. From the relativistic expressions for energy and momentum, one has

$E = \gamma mc^{2}$ and $p = \gamma m v$

so that

E/p = c²/v

Even though this last expression refers to massive objects, one can ask whether it also applies to massless ones. In the case of light, v=c, therefore:

$E/p = c \longrightarrow p = E/c = hf/c = h/\lambda$

that allows us to assign a momentum even to massless objects. By using this result, experiments would show that, as one expects from elastic collisions, both energy and momentum are conserved in collisions between particles and radiation

The significance of the particle/wave duality of electromagnetic radiation is even more puzzling when one realizes that one aspect or the other are brought into evidence depending on the experiment we do: it is as if light knew what we are trying to measure.....

Here is a classic experiment : an ordinary beam of light contains extremely large numbers of photons (question : how many photons per second are emitted by a 60 W light bulb? see Example 1 page 893), and we know that if we send such a beam through a double-slit arrangement we will observe interference patterns.

But it is also possible to create a light beam of such low intensity, that only a few photons per second are emitted. If we send such a low intensity light beam through the double slit, it is reasonable to assume that each photon will go either through one slit or the other, but not both. Still, if we run the experiment long enough, and we record, e.g. with a photographic film, the global distribution of photons on a screen at some distance from the slits, again we would observe an interference pattern! How can a photon know not to go on a section of the screen where it would undergo destructive interference with "future" photons? or does a photon maybe interfere with itself? To a large extent, these questions remain unanswered as of today.

The interpretation of the double slit experiment with light is puzzling enough, but this is just the beginning.... As physicists were compelled to accept this un-explainable dual nature of light (and electro-magnetic radiation in general), a French prince-turned-physicist, Louis de Broglie, came up with an even more outrageous proposition

given that radiation is both wave and particle, wouldn't it be natural to suspect that matter is also simultaneously particle and wave ?

This proposition by an unknown Physics Graduate Student, based on purely theoretical grounds, was very revolutionary, but Einstein liked it, and that was enough for it to be taken seriously. De Broglie proposal was to extend to ordinary matter the relation between momentum and wavelength we have found for radiation

$p = h/\lambda$

and to state that any object of momentum p = mv is at the same time also a wave of wavelength given by the expression above.

What does it mean that a material object (an electron, a proton, a cell, a football, an elephant) is a wave, or, at least, has a wavelength/frequency associated to it ? We (or at least I) don't know, but it is a fact that the wave property of matter can be detected under the proper conditions. Not too long after de Broglie proposals, experiments showed that beam of electrons, scattering off the regular lattice of a crystal do show interference patterns similar to those prieviously observed with X-rays.

Remembering some of the facts we have learnt when studying wave optics, we can do a few numerical examples to see when and how the wave nature of matter can be detected.....

The double slit experiment can be (and it has been) repeated for electron beams, with equally inexplicable results: even though one electron at a time is going through the slits, interfence patterns are observed. But one could add an extra twist :

one can put a detector to determine through which opening the electron is going through. If we do this, the electron "senses" that we are trying to reveal its mystery, and it refuses to collaborate: when our experiment includes some device capable of detecting the path of the electron, then the interference pattern does not appear, and electrons behave as particles.....

The only possible interpretation of the double slit experiment is that, even though each electron is bound to be going through only one slit, something else must be going through both. This something else, which somehow contains the wave-like nature of quantum objects, is what goes under the name of wave function.

What is the wave function? What is the meaning of the wave associated with particle-like objects? The best answer we have found so far is that the wave function describing the evolution of a particle is associated with the probability of finding the particle in any given location.

What Quantum Mechanics has taught us is that we cannot know for sure the outcome of any single event, but we can only estimate its probability. One possible way to interpret quantum phenomena is to assume that, when a given event occurs (e.g. an electron in a double slit experiment), all of the possible outcomes do take place, according to their relative probability. In the double slit experiment, the electron has a 50-50 probability of going through either slit, therefore the global result will be consistent with it doing so ....

Unless of course we decide to observe the event : in this case, we force the electron in one of its possible configurations, but as soon as we do that, we destroy the probability function.... It is as if the Lottery drawings were performed without the results being announced : in this situation,we could only compute the probability of becoming millionaire. But if we demand that the result be known, then only one of all the possible outcomes becomes reality.

In spite of this un-deterministic behaviour, the Quantum World has been shown to obey a set of well determined rules. The first (and still very valid, for non-relativistic phenomena) rigorous description of the quantum rules and the evolution of the wave-functions associated with atomic and sub-atomic particles was due to Erwin Schroedinger.

Schroedinger's Equation can be thought of as the quantum's equivalent of Newton's Law of Motion. It describes the behaviour and the evolution of the wave function in the presence (or the absence) of a given field of force. The equation is capable of describing phenomena like single and double slit experiments, the behaviour of electrons within the atom (see next lectures), weird phenomena like the tunneling effect, where a particle can find itself outside of a hole even though it has not enough energy to do so, etc., etc.

But Schroedinger's success was bittersweet, since , even though he found the correct equation, he never completely accepted its interpretation in term of probability waves as opposed to real waves. But not even his cat was able to solve the riddle....

Heisenberg's Uncertainty Principle

The centerpiece of the un-deterministic scene of Quantum Mechanics is given by Heisenberg's Uncertainty Principle, that quantifies the intrinsic limitations in the accurate knowledge of the quantum world. Originally the principle was motivated by a realization of the fact that the process of measuring an event unavoidably affects it, but by now we realize that the principle has an intrinsic validity, idependently of any observer's intervention.

The starting point, which is a direct consequence of wave-particle duality is the following: (see also the textbook for a derivation involving the diffraction of a wave crossing a narrow slit)

suppose we want to determine the position of an object; to do so we must "see it", i.e. we must hit it with a photon, and let the photon bounce back to our eyes. But, thinking of the photon as a particle, this is like the collision between biliard balls: by looking at the object we modify its velocity. Moreover, there is a standard rule of optics, that states that the accuracy with which we can determine the position of an object is limited by the wavelength of the radiation illuminating it. The better we want to determine the position of an object, the shorter is the wavelength we must use. But shorter wavelengths correspond to higher frequencies, therefore to higher energies. The more precisely we try to determine the position of an object, the more we alter its velocity....

Heisenberg was able to quantify these considerations, showing that the accuracy with which we can simultaneously determine the position of an object along any given axis and the projection of its momentum along the same axis is limited by the expression

$\Delta x\times\Delta p_{x} \geq h/2\pi$

Similarly, it can also be shown that an equivalent expression relates the energy of a particle and the time during which it can have a given value of energy. This is in line with the 4-dimensional vision of the world of Einstein's relativity, since, in relativity, the 4th component (i.e. the "time" component) of the momentum vector is in fact the energy. When applied to the 4th component, Heisenberg's principle reads

$\Delta E\times\Delta t \geq h/2\pi$

and again this result has ben veerified countless times in sub-atomic physics. As an example, "Particle Physics 101" tells us that unstable particles, i.e. particles that can exist only for a finite $\Delta t$ do not have a well defined value of mass, and the shorter their lifetime the larger is the uncertainty in mass. In the sub-atomic world, one also routinely witnesses violation of Conservation of Energy, but the violation is only allowed to occur over a limited time interval so as to be consistent with Heisenberg's principle $\Delta E\times\Delta t \geq h/2\pi$ .