Quantum mechanics 2 : Mathematical Nature of Physics Continued...

A few days back I had posted an introductory article on Mathematical Nature of Physics where I addressed the importance of mathematical thinking in physics in some detail. I postponed the Historical Development section as I thought this bears much more relevance to the previous post. Here, as an extension of the previous article, I am going to elucidate on how do we actually perceive terms like MASS, CHARGE, SPIN, ENERGY, etc. in physics and how these have acquired a kind of false notion among students, leading to all the unnecessary confusion. The problem of understanding Quantum Mechanics starts right here with the lack of understanding of the mathematical nature of Physics.

Following questions were asked by a first year student a few months back –

(1)    I think that particle is related to mass i.e. a particle has to have some mass associated with it (is this thinking right?) 

(2)    When we focus on particle property of photon we see that photons behave as particles, so how is it that photons have zero rest mass? 

Once we start talking about PHOTONS (discrete) instead of RADIATION (generally visualized as continuous by many), we must bear in mind that we are now in the realms of Quantum Mechanics and the way we think and understand these physical entities in Quantum Mechanics is drastically different than our Classical ways. Therefore first it is necessary to learn to start thinking in the way required by Quantum Mechanics, not doing so results in confusions of which the above questions are classical example. So let me first elaborate how we look at things in Quantum Mechanics.

I'll start with the following question: What is an electron? The first thing that probably comes to our mind is a small spherical ball like thing (typical intuitive thinking), spinning about its axis, having some negative charge and some certain mass, revolving around a nice nucleus (if it is in an atom) and of course the last but not the least it is a PARTICLE.

(1)    The problem here is that no one has ever seen an electron so far, no one knows what it is actually made up of hence we cannot assume it like a ball.

(2) The quantities like charge, mass, spin, etc. which seem so familiar and real, are all actually mathematical just as I pointed out in the previous article about energy : ENERGY is an abstract mathematical quantity. The best way to visualize it is as a number which remains constant before and after a process. Similarly, momentum (mv) is also a number which remains constant if no force is acting on a body. These are completely mathematical objects, not real, which you cannot see or touch. These are just NUMBERS which appears as a result of theoretical calculations. THEY DO NOT EXIST IN REALITY.

Concept of Charge

If I put forth the following questions : what is a CHARGE? Where does it come from? What gives electron its charge? You will find yourself in dark. The fact is that nobody knows what a charge is or where does this charge come from. Then how do we define a charge? The following is the way answer to this question : we know that there are 4 fundamental forces in nature strong, weak, electromagnetic and gravity. Suppose I give you a pair of objects which interact electromagnetically with some force (could be repulsive or attractive). Now I give you another pair of objects interacting electromagnetically but with force different from the first pair. Now you know that there must be something within these objects in different proportions due to which the forces vary. If this thing is more in quantity the force is more, and if less then force is also less. Now we want to find an exact relation between this unknown thing and force. For this we do a lot of experiments with different objects containing this thing in different proportions, and gather a lot of data. Once we have the data, our job is to find some sort of relation between these forces and this unknown thing. This relation will be EMPERICAL as we are not deriving it from theory but constructing it to fit the data. Coulomb did exactly the same thing and he came up with the famous INVERSE SQUARE LAW. But we need to name that unknown thing on which the strength of these forces depend and somebody did so and he called it as CHARGE. Now we have a mathematical relation between the forces and the so called charge but you see while doing all this we never really knew what is actually causing this interaction. We just named it as CHARGE without really knowing what it is. It is just a name.

Electrons, protons and every other physics entity which can interact electromagnetically are called as CHARGED particles because they contain this mysterious charge. Amount of charge in a body indicate how strongly the body can undergo electromagnetic interaction. Photons do not show this interaction so we conclude that it should not have any CHARGE (whatever it is).

SPIN

The same goes with the concept of SPIN. When we solve Schrodinger equation for a hydrogen atom in three dimensions (x, y, z), we get three quantum number n, l and m. When we solve it (actually Dirac equation) for Hydrogen atom in four dimensional space (x, y, z, t) , we get a fourth quantum number which people, unfortunately, named as SPIN causing unnecessary confusion. It doesn’t mean electron is actually spinning about its axis (if electron has any axis in the first place as we don’t know anything about its shape). It is a mathematical property, result of some mathematical calculations. The other three quantum numbers also are completely mathematical, their values are decided by boundary conditions imposed on the system. I am going to show this explicitly in later articles.

MASS

Let us look at a much more familiar term: Mass.

In modern classical mechanics, the most general formulation of the law governing the motion of mechanical systems is the principle of least action also known as Hamilton’s principle. According to this law, every mechanical system is characterised by a mathematical function known as Lagrangian, denoted by the letter L. If the system occupies the position x1 at time t1 and x2 at time t2, then the system moves between these two points in such a s way that the integral 

 

takes the least possible value (minimum). This principle forms the starting point of everything in mechanics. The Newton’s laws appear as corollaries of this principle under different conditions. It can be shown using symmetry arguments that in the simple case of a free particle 

 

Here the constant α is required to be positive in order of the integral S to be minimum and its value is independent of the motion of the particle but is different for different particles. Hence we conclude that it must depend upon the amount of matter within the particle and we call it the MASS of the particle. Thus the positivity of MASS is decided by a mathematical principle and not because we can't visualise it.

It is just a number, with condition that it cannot be negative, which comes out during theoretical analysis.

If I ask you where does an electron get its mass from (what is it inside an electron that results in mass)? The answer is not that easy and so far no body knows about this. Whatever that thing is must be absent in a photon and hence photon does not have mass. Just like charge, we assign different numbers to MASSES of different bodies depending on how strongly they interact gravitationally. But we dont know what is it inside these object which makes them interact gravitationally. Whatever it is we call it MASS. We don’t know what it is or where does it come from.

The other quantities like potential energy, momentum, angular momentum, etc. emerge from this principle by further mathematical analysis under different conditions and symmetry arguments.

With this view of things, if now we ask what is the difference between an electron and a photon, what should our answer be? We answer it in the following way : First we compile a list of all the mathematical quantities which represent the properties of these two objects such as : charge, mass, spin, isospin, lepton number, baryon number, charm, bottom, up, down, top, strange, etc. All these are numbers used in physics to indicate the properties of these objects. Make one such list for electron specifying these numbers, make another such list for photon specifying all these numbers. And now compare the two. We will observe that the values of some quantities are different for electron and photon. For example, charge of electron is some number but for photon it is zero. Whatever differences we see here are the only differences we can talk about in case of photon and electron. So question of WAVES and PARTICLES does not arise. A Proton will differ fron an elecron and photon in the values of these numbers.

Moral of the story : View electrons, proton, photons, etc. as physical objects having some specific numbers for above set of mathematical properties (charge, mass, etc.) which tell us about their behaviour/interactions in different conditions (forces). Much of the current theoretical and experimental research in High Energy Physics is devoted to the study of sources of these properties.