Quantum Theory (Quantum Mechanics)
Introduction
Quantum Theory, also known as Quantum Mechanics (QM), is a theory that describes the micro-physics: i.e. the physics of atoms and molecules. This theory has to be modified in order for it to also describe the world of particle physics, i.e. the sub atomic physics. But in its modified forms, known as Quantum Electrodynamics and Quantum Chromodynamics, it also explains many aspects of particle physics. In general, QM it is one of the most successful theories of physics. Despite its success in formulating known phenomenon and predicting undiscovered ones, QM has remained a very controversial theory since its creation a century ago. Part of the controversy is because this theory does not make any assumptions about the physical world the way Classical Physics does. So, physicists who believe that classical physics is not just "a" model of reality, find it very difficult to believe in quantum theory. The other aspect of QM that creates controversy among the physics community is that it has (successfully) predicted physical phenomenon that are not only unexplainable by the classical physics, but they remain in direct conflict with many well regarded tenants of classical physics. To appreciate the first of these conflicts, we first need to examine some of the underlying assumptions of classical physics. Among these, we need to study the notion of measurables, i.e. aspects of the physical system that could be quantified and thus measured, and the role of statistics in the process of measurement.
Measurables (physical variables)
In classical physics study of a physical system begins with the identification (definition) of its measurables. These are aspects or properties of the system that can be measured and quantified. Once the measurables are identified, then the job of the experiment is to devise a scheme in order to measure these so called variables. The job of the theory, on the other hand, is to explain how these measurables are related to each other and to other previously measured variables. For example in the context of thermodynamics, a branch of physics that we examined earlier, temperature is a measurable. Another measurable is the irradiance, which is the energy of electromagnetic radiation per unit area per unit time. Experiments that measured the temperature of objects as it radiated electromagnetic energy found that the radiated energy is proportional to the fourth power of the temperature. This empirical relationship is known as the Stephan-Boltzman law. It took Planck's theory of radiation to relate this empirical relation to other well known theories of electromagnetic radiation. Another example is in Mechanics. Classical physics claims that the position and speed of an object are its measurables. These variables are then related to another variable, time, according to the expression:
speed = (final position - initial position) / (final time - initial time) , or in the short hand equation form:
v = ( xf - xi ) / ( tf - ti )
In situations where it becomes difficult to measure the value of these variables classical physics comes up with either a new measurable that can be measured easier, or one that can approximate the measurement. For example, in the case of motion of a collection of objects, say a flock of Starlings, classical physics introduces the concept of center of mass of the flock. So, even if one could not determine the position of individual birds flying in a flock, one could quantify the motion of the flock as a whole. Another such example that describs a "collection behavior" is the measurable: "wavelength". For water waves this is a measure of how water droplets (molecules) pass-on a disturbance, but it does not describe the motion of individual water molecules.
Some of other measurables in classical physics are: mass, force, acceleration, electric charge, electric current, magnetic moment, frequency, amplitude, index of refraction, etc.
Role of Statistics in Classical Physics
Because science is a communal field of knowledge, one of its requirements is that any person who follows exact procedures for a measurement must come up with the same results as any other person. Of course there will be slight variations even when the same original person makes the given measurement repeatedly. According to classical physics these variations are due to the notion that it is not possible to have 100% control on all aspects of a measurement. Despite this acceptance that there is "always room for error", the belief is that were we to have a 100% control, then there would be no variations in our measured values. Until then, we have to use statistics to determine the significance of variations in our measured values from trial to trial. Typically we produce a graph of the distribution of the measured values and we apply statistical theory of mathematics to determine, from this distribution, the average value of the measurement, etc.
One of the tenants of classical physics (in fact all of science) is that if the fluctuations in the measurement are completely random, then the distribution will have be a "bell curve". The width of this curve gives an indication of the precision of the measurement. The sharper the curve, the less variations are present from the average value, and therefore the more precise a measurement becomes.
Early quantum theory evolved from the formulations of Planck and deBroglie. This was called wave mechanics. Schrodinger's formulation put this theory into a very formal and mathematical formulation with its well defined postulates:
i. Given a physical system, there exists a function, called the wavefunction or state vector, that incorporates in it all information about all the measurables of this system
ii. There is an equation, called Schrodinger's wave equation, that fully describes the time development of the wavefunction.
iii. The value of any measurable is directly associated with a unique operator and can then be calculated using the wavefunction alone.
This theory includes very well defined procedures for calculating the system's wavefunction and its time development (evolution). This theory also imposes very strict and rigorous mathematical requirements that this wavefunction must satisfy. The mathematical restrictions follows the rules of Calculus and in this way are not different from those imposed in classical theories. The determination of the value of the measurable, on the other hand, follows procedures akin to those used in statistics. As a result, the predictions of this theory is, in a sense, "average", or if you like, statistical in its essence. So, although QM does not exclude the existence of definite (exact-valued) measurables, it does not provide us with any means to make an exact/definite determination of a measurable either! Furthermore, it is important to note that neither the wavefunction, nor the operator, the two tools necessary for determination of the measurement value, are in themselves measurables.
One of the most successful early predictions of quantum mechanics was on the emission of alpha particles from the nucleus. We'll cover this in detail in later parts of the course, but for now check out the applet on quantum tunneling developed by Professor Zoltan's team. It gives a great illustration of the interplay of wavefunction and measurables in addition to the phenomena of quantum tunneling itself.
Some New implications
a. Mixed States
A physical system, according to quantum theory, is described by a wavefunction which represents a state of that system. Since the value of a measurable of the system depends on the evaluation of the appropriate operator, the wavefunction itself could represent a mixture of states! That is to say, even though the result of a measurement is always definite, our presentation of what may result could be (correctly) indefinite.
One of the consequence of this is that we can no longer rely on locality (causality) as definitely as we have in classical physics. This disturbing prediction of QM was first verified in the laboratory in early 1980's. Today, it has been demonstrated even in advance undergraduate physics laboratory experiments. Still, many practicing physicists confuse the disturbing results of these experiments with "the weirdness of QM".
b. Heisenberg's Uncertainty Principle
Another disturbing feature of the quantum world is the very direct consequence of quantum formulation. In QM, it turns out, certain measurables which in classical physics were believed to be independent, now have a peculiar co-dependence. The nature of this CO-dependence is not that the two measurables cannot have independent values, but that the precision of any measurement on one of these will end up affecting the precision of the other's. An example of a co-dependent measurable pair is position and speed. So, were we to determine the position of an object in the in the quantum world very precisely, then we could not know the speed of the object very precisely. That is to say, a precise knowledge of the position necessitates an imprecise knowledge of the speed, and visa versa. This limitation, as it turns out, is a consequence of nature, not of the the experiment. As a result, it is wasteful to try to devise a way to measure both position and speed with high degree of precision. Again, all known experiments to date support this implication of quantum theory.
Questions on Quantum Mechanics
Last Modified: Wednesday, September 19, 2007 Malekis@union.edu
To learn more about these issues read my web pages on Quantum Measurement posted at the site:
Check out how thermodynamics relates to you lounging in front of a fire place!
Read my colleague, Chad Orzel's blog on his conversations with his Dog on Quantum Mechanics!
