**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 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 role 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. Physicists devised experiments to examine how the temperature of an object is related to the object's rate of radiating electromagnetic tadiation. One of these experiments measured the temperature of objects as it radiated electromagnetic energy and found that the radiated energy is proportional to the fourth power of the temperature. This empirical relationship is now known as the Stephan-Boltzman law, but at the time of its discovery no one could explain why the rate of radiation was proportional to the fourth power of the temperature, and say, why not the third power. It took the application of 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 = ( x

_{f}- x_{i}) / ( t_{f }- t_{i })

In situations where it becomes (practically) 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 examples 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 (social) field of knowledge, one of its requirements is that any one 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 general 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. In the analysis of a typically experiment we produce a graph of the distribution of the measured values and then apply statistical theory of mathematics to determine, from this distribution, the average value of the measurement, etc. The distribution of the measured values describes all the statistical information about the expeiment. A value that corresponds to the tallest part of the distibution graph, for example, is the most likely value of the measured variable.

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 shape of the distribution will have be a "bell curve." The width of this curve gives an indication of the precision of the measurement. The narrower the width and thus the sharper the curve, the less variations are present from the average value. So, a distribution with a narrow width presents a highly precise measurement.

Early quantum theory evolved
from the formulations of Planck and *de*Broglie. 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 *t he wavefunction*

ii. There is an equation, called **Schrodinger's wave equation****,
th**at 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. In addition, it turns out that the notion of probability in QM is completely different from the one we use in classical physics especially in the context of measurements. To see this difference, let us use the example that is referred to in the excellent introductory book on quantum weirdness: "Quantum Enigma," by Bruce Rosenblum and Fred Kuttner. In this book these authors make a comparison between the probability of locating an electron in an atom with the probability of locating a pea in a standard shell game (a pea is under one of three possible half coconut shells). They argue that when we say that the pea has a probability of 1/3 to be under a given shell in effect we are making a statement about our own personal knowldge of the physical relity (the shell game operator knows for certain under which shell the pea is located!), but when QM states that the probability of finding an electron shared by atoms A and B to be 30% in atom A and 70% in atom B, it really says that the electron could be (and is) both in atom A and in atom B. But 30% of our measurements will find it in atom A and 70% of our measurements show that it is in atom B. These two notions of probability (classical versus QM) are very different!

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:
Monday, April 12, 2010
Malekis@union.edu**

To learn more about these issues read my web pages on Quantum Measurement posted at the site: