Читать книгу Quantum Physics is NOT Weird - Paul J. van Leeuwen - Страница 38

Until 1925 a unified solid mathematically formal basis of quantum physics was missing. Quantum physics had become a haphazard collection of – albeit intelligent – assumptions. So, in that year, Werner Heisenberg (1901-1976) not being content with this state of affairs, came up with a new mathematical tool for quantum mechanics, matrix mathematics. The location and intensity of the spectral lines of glowing hydrogen gas could be calculated precisely with his matrix mechanics. However, Heisenberg’s matrix mathematics was for most physicists a rather exotic branch of mathematics. Moreover, it provided no insight, as classic mechanics does, into the underlying mechanisms, it produced only discrete outcomes.

Surprisingly, in matrix mathematics the outcome of A x B is not equal to B x A. This corresponds very well with the phenomenon that the outcome of a quantum experiment depends on the order of your measurements. Anyway, the results of Heisenberg’s matrix mechanics were in very good accordance with the observed discrete values for the wavelengths in the hydrogen spectrum. This made matrix mechanics a very valuable tool for doing quantum physics calculations. But it delivered no useful suggestion whatsoever for a possible interpretation of quantum phenomena.

Progress in quantum physics was now well underway. One year later, in 1926, Erwin Schrödinger (1887-1961) published what is now known as the Schrödinger equation for quantum mechanics. We will pay more attention to his mathematics later. Shortly thereafter, in 1927, Heisenberg formulated his famous uncertainty principle. It says that there is an inverse relationship between the accuracies with which the momentum p (mass times velocity) and the position of a particle can be measured. There is a fundamental limit to the accuracy of your measurements that is not dependent on your measuring instruments. In 1929, Heisenberg and Wolfgang Pauli published the foundations of relativistic quantum mechanics. With these new mathematical tools, a solid theoretical basis was finally laid. Physics now even possessed two different mathematical tools to make quantum mechanical predictions.

For the sake of completeness, you’ll find below the Schrödinger equation for a single particle. The explanation of the meaning of the symbols is beyond the scope of this book and is by no means necessary for understanding the conclusions that we will draw from the observations and experiments in quantum physics.

Schrödinger sought, and found, his famous equation because he was already proficient in wave mechanics and because he realized that the electron wave model of De Broglie for a hydrogen atom with one single electron could not explain why that hydrogen-atom was not flat as a dime. It should be something like a tiny sphere. Schrödinger’s discovery came in a flash of inspiration. It was not the result of a patient mathematical analysis. He was racking his brains searching intensely for a mathematical equation describing a 3-dimensional standing electron wave extending in three directions around the nucleus of the atom. But the solution was revealed to him in a flash of insight while spending a relaxed holiday with an extramarital love.

For the purpose of this book, you don’t have to understand the Schrödinger equation, let alone apply it. However, it is important to understand that the wave function calculated with the Schrödinger equation represents the wavelike development over time of the location, the momentum, and the energy of the particle, but not so that we can know those properties in an exact way. This has to do with the fact that a wave has no distinct location. It spreads out in time and space. You cannot even say that these properties exist, in a material sense, before measurement.

To avoid confusion of terms, I will define here the following strongly related, but different in meaning, quantum wave concepts as they are used in this book:

 Quantum wave or state wave: The non-material wave behaving conform the solution of the Schrödinger equation.

 Wave function or state function: The mathematical expression that is the solution of the Schrödinger equation describing the quantum wave.

The uncertainty principle, or uncertainty relation, of Heisenberg is shown below.

Δp.Δx ≥ h/4π

The uncertainty in the location x is Δx, the uncertainty in the momentum p is Δp. Their multiplied value has an unsurpassable lower limit specified by Planck’s constant h divided by 4π.

This uncertainty relation can be made somewhat plausible by considering the diffraction of light waves going through either a smaller or wider hole or slit. Figure 5.7 shows what happens when we shine a beam of light, photons, through a wide and through a small opening. With a wide opening, most photons will continue in an almost straight line, the light waves passing through are only slightly disturbed by the edges of the opening. When you shine a beam of light at the opening, the result will be a rather sharply defined spot of light.

A small opening – small when compared with the wavelength – will act as a point wave source (Huygens) from where the light waves will extend spherical. The result is then a fuzzy spot with decreasing intensity towards the edges like the bold curve in figure 5.6.

Figure 5.7: Wide opening, less diffraction. Small opening, strong diffraction

When we would observe the projected light pattern a little bit more closely, we would see an interference pattern, the so-called Fraunhofer diffraction ^[5].

Diffraction is a known phenomenon that plays an important role with photo cameras. The smaller the aperture – the diameter of the diaphragm – the sharper objects located in a wide range of distances relative to the camera will be displayed. But when the camera diaphragm is made too small, the image will blur again by wave diffraction. This is not a camera defect but the result of the wave behavior of light and, as you now understand, connected to Heisenberg’s uncertainty principle.

What is now the message of this? The answer is that as soon as you try to define the path of the photon using a small aperture, the photon will scatter in all directions. The reason for this scattering of the photon is that it ‘is’ (also) a wave that is influenced in its course by the edges of the opening. With a large opening, that influence is smaller.

You must have noticed the duality in the above descriptions. The photon behaving both as a wave and as a particle. Sometimes, that kind of language is almost unavoidable in talking about quantum physics. However, keep this in mind: the wave is NOT the particle, as we’ll see.

Heisenberg diffraction experiment: Create a pinpoint hole with a fine needle in an opaque sheet of paper or carton. Then make the room as dark as possible. Thereafter shine with a laser on the pinpoint hole. A laser projects coherent light, which means synchronous parallel traveling wave fronts. Observe the projection on a nearby wall of the laser light that passed the pinhole. What should you observe? What do you observe?

Heisenberg’s quantum matrix mechanics is, as already mentioned, mathematically complex and only gives you discrete outcomes, the predictions about the observed aspects of atomic phenomena. The Schrödinger equation is, at first sight, an entirely different beast. Its solution is a continuous wave. It made quantum physics calculations considerably easier for physicists proficient in wave mechanics. That is the reason why most physicists preferred the Schrödinger equation, much to the chagrin of Heisenberg. Contrary to Heisenberg’s matrix mechanics the solution of the Schrödinger equation – the state function – describes an in space and time continuous phenomenon, a wave. But it was not clear what the substance would be in which it was propagating. If it really was a substance.

With his equation Schrödinger had hoped to save the classic causality and continuity of the Newton universe. He initially believed that the solution of his equation, the state wave, represented the materially spatial "spread-out" charge and mass of the electron. Ultimately, however, it was shown that the Heisenberg matrix mechanics and the Schrödinger equation are mathematically equivalent and – more importantly – that the quantum wave is a physical wave but not a material wave. By the term ‘material’ I mean something that is in principle directly ‘tangible’, whereby ‘tangible’ also includes the use of material instruments. By the term ‘physical’ I mean something having a physical effect, which is not necessarily material.

Consider for a moment here the fact that atomic objects have both wave and particle properties, and that this duality finds a counterpart in those two seemingly different quantum theories of Heisenberg and Schrödinger. Heisenberg is about discrete states, the particle aspect, Schrödinger is about waves. However, they have been shown to be equivalent.

The controversy between Heisenberg and Schrödinger continued to simmer somewhat throughout their related lives and careers. Schrödinger finally had to abandon his opinion about the materiality of the state wave and thus his hope that it existed in the material universe. He ultimately left the quantum physics community and changed the direction of his interest to the study of living systems.

The uncertainty relationship gave rise to heated discussions and worrying sleepless through the night. Heisenberg initially thought that his uncertainty relation was the result of the inherent limitations of physical measuring instruments. But it turned out to be a fundamental aspect of nature and of quantum physics. His uncertainty relationship can even be derived mathematically from the Schrödinger equation. In this way these two gentlemen and their two theories are reunited in an uneasy duality.

In 1927 Max Born (1882-1970) proposed a (meta)physical interpretation of the quantum state wave representing the solution of the Schrödinger equation. To understand his interpretation, let’s first have a better look at the character of the solution of Schrödinger’s equation. The solution of the equation is a time- and location dependent complex state vector. The term complex means that the vector is defined by the combination of two terms of a different nature. The first term represents a real number, the second term is an imaginary ^[6] number. An imaginary number is a number that when squared gives you a negative result. It does not have a ‘tangible’ value.

A vector ^[7] is a mathematical object with two properties: length and direction. Force and speed, having both spacelike and numerical properties, are rather straightforward examples of vectors in physics. A vector in a diagram is usually represented by an arrow and in a text by a line or arrow above or below the symbol.

According to Born, the squared value of the length of the complex state vector, which is a real number, should be interpreted as the measure of the probability of finding the quantum object at a certain place and time. The length of the state vector at a certain location therefore directly represents the probability of finding the particle at that location.

It is interesting to try to see how Born may have come to his insight. His specialty was optics and then the question about light as a wave and light as a particle is an obvious concern. He must have been very familiar with semi-transparent mirrors (beamsplitters) that transmit half of the incident light and reflect the other half. He knew very well that Planck and Einstein had found that light is quantized in photons with a fixed amount of energy per photon.

Born must then have realized, combining these two facts, that the photons that reach the mirror must in their entirety either pass the mirror or be reflected. These photons are certainly not split in two halves. Such a split would mean a halving of their energy and therefore – according to Planck’s law – of their frequency and thus a doubling of wavelength. Which means a changing of the color of the light from, for example, violet to red. That is certainly not what is observed in beamsplitters. The probability for a photon of being reflected by an accurate beamsplitter is 50%. Born very probably mused on the meaning of the state vector of the Schrödinger wave function describing the process of a photon hitting a beamsplitter.

The value of the state vector describing the passage through the mirror therefore had to be connected, according to Born, with the probability to find the photon behind the beamsplitter. Because it concerns a probability, the outcome, whether it goes through the mirror or not, cannot be predicted for a single photon. Considering the huge numbers of photons that are sent through a beamsplitter in the average laboratory experiment, these numbers always average out, which accounts for the observed behavior of beamsplitters and other optical instruments.

The quantum uncertainty whether a single photon will pass through the semi-transparent mirror or not, is fundamental and is not the result of insufficient information concerning the photon and the mirror. Quantum probability is known by physicists as objective probability. Random generators using photons and beamsplitters – passing is 1, reflection is 0 – are generally accepted to produce fully reliable random binary numbers, bits.

So, according to Born, the objective probability that a particle is found at a certain time and place in a measurement is determined by the square of the amplitude – the length of the arrow – of the quantum wave function vector. The outcome is a natural number between 0 and 1 corresponding with a probability between 0% and 100%. Time and time again Born’s insight has proved to be excellently predictive in quantum experiments. Which is the way good scientific theories are confirmed. But why should an extra adjective – ‘objective’ – be assigned to quantum probability?