Month of Equations: What Does Gauss' Law in Magnetism Really Mean?
October 2: Gauss' Law in Magnetism
Meaning of Equation: "Magnetic monopoles do not exist in nature"
But how do we infer this one line result from the above equation. Let us find out.
The inverted triangle in the above equation is known as the del operator. It reads as ( i ∂/∂x + j ∂/∂y + k ∂/∂z ) where i, j and k are unit vectors along x, y and z direction respectively. This operator acts on some function, in this case the magnetic field. If we take the dot product of this operator with a vector, it is known as divergence. Getting complicated? Let us make it easier.
Divergence, as the same suggests, answers the question: Does any point in space act as a source or sink of some quantity? Suppose you take two reference points A and B along the course of a river separated by some distance. Suppose x units of water is crossing point A. If the same amount of water crosses B after travelling some distance, then there is no divergence in the flow of water (here our vector function is the water flow). If lesser water crosses B than A, then the field (water flow) has negative divergence, i.e. it has a sink and if more water flows through point B, then there is positive divergence, i.e. there is a source in the field. All we want is the net flow, meaning there can be any number of sources or sinks between the points A and B, but if the total flow through B remains same as that of A, then also net divergence is zero. Simple enough to understand!
Now this equation says that divergence of magnetic field B is zero. From our above understanding of divergence, this means there is no source or sink of magnetic field anywhere in the universe. But we know that magnetic field has a source, the magnet. Here is the point! This equation tells that the net divergence of B is zero. So there are equal number of sources and sinks of magnetic field in the universe. Here source is the north pole of the magnet from where field lines originate and the sink is the south pole of the magnet where the field lines end. So even if you cut the magnet into two pieces, there is a new magnet with new north pole and new south pole to keep the net divergence zero. So magnetic monopoles (isolated north or south pole) do not exist in nature. If they exist, the R.H.S. of above equation will have to change. This equation is one of the 4 Maxwell's equations of electrodynamics.
'Month of Equations': What Does 'The Schrodinger Wave Equation' Really Mean?
October 5 - The Schrodinger Wave Equation
The Schrodinger Wave Equation (SWE) was a landmark in the history of mankind, for it lay the basic framework of one of the most successful theory a.k.a quantum mechanics. Understanding of SWE is very amusing. You already know what this equation actually means.
Meaning of Equation:
The total energy of a particle is the sum of its kinetic and potential energy.
What???? Are you kidding me! Is that what this equation tells?
Yes! If you look closely, this is what the Schrodinger wave equation means. The left part of the equation adds kinetic energy(h^2/2m ∇^2) and potential energy (V) and sums it equal to the total energy E. SWE is the most fundamental equation of quantum mechanics. All the information about a quantum system is already embedded in this equation just like everything about a mass m is contained in Newton's second law. The difference is that in classical mechanics, the fundamental quantity is position and in quantum mechanics, it is wavefunction.
Consider throwing a particle in a box, you can only tell where the particle will probably be. You can visualize it using an analogy. Suppose your friend is in his bedroom. Knowing the nature of your friend, you can tell that there is 80% probability that he is playing video games, 15% probability that he is sleeping and 5% probability that he is studying. So there is a spread of probability (probability distribution) but when you open the door, he is, say, sleeping. The distribution collapses into one value.
This is what this ψ(x) tells us. ψ(x) is the wave function of the particle. Since by being probabilistic, a particle exhibits a wave like nature in the box, ψ(x) is hence known as the wave function. Itself it doesn't represent anything physical but its square represents the probability of finding a particle in the particular state.
The image above shows a typical wavefunction of a particle in a box. One feature of the wavefunction is that it must be zero at the boundary of the box. This means the probability of finding the particle at the boundary is 0. Look carefully, this is exactly what every graph shows.
Another important implication of SWE, after energy conservation is quantization of energy. Consider the same problem of particle in a box as taken above. When you solve SWE for this particle, the energy levels that we get for a particle are quantized. This means the particle can only take certain energy values, unlike in classical mechanics. See Particle in a box.
So the Shrodinger wave equation tells the evolution of the wavefunction of a particle. If you know the wavefunction, you can derive all other quantities of the system.
Month of Equations: What Does 'Time Dilation' Really Mean?
October 1: Velocity Time Dilation
We begin our Month of Equations with the famous equation of Time Dilation. The mathematical equation of velocity time dilation is given above. Now let us try to understand what it really means.
Meaning of Equation
The faster you travel through space, the slower you travel through time
To understand this, let us go back in the year 1905 and meet a patent clerk at Bern, Switzerland who is trying to find a solution to a question no one was asking. So, Einstein had this question: Suppose I have a master clock in a town that sends signals to synchronize all the other clocks in the town, at the speed of light. The clocks will synchronize. No problem so far. But what happens if we try to send the signal to a clock on a moving train? Light will have to speed up or slow down to catch up with the train. But according to Maxwell's equations of electrodynamics, light always travels at the same speed. So either Newton is right (in saying time is absolute, same for everyone) or Maxwell is. It can't be both. Einstein wanted to know who was right between the two.
Einstein then came up with another brilliant thought experiment. Imagine yourself standing on a platform of a station. Two lighting bolts strike in front of you as shown above. The bolts were 300 m apart. For you, on the platform, they were simultaneous. But if a person views the 2 bolts by standing in the middle of a 300 m long moving train, they would not be simultaneous because he would be moving towards one bolt and away from the other. So then who is right? The person on the platform who says the bolts struck at the same time or the person on the train who says the bolts were not simultaneous? It turns out that- both are correct. Absolute simultaneity is not possible. Thus, time is not absolute. It means Newton is the one who gets it between the eyes.
So time dilation is an outcome of the fact that light travels at the same speed in all reference frames. The flow of time in that frame adjusts itself to keep the speed of light constant. So moving clocks run slower than the stationary clocks and this is exactly what the equation tells us. If v, the speed of moving clock gets larger, Δt' gets larger, which means the time duration between two events gets longer and hence time slows down. So moving clocks run slower than the stationary clocks and this is exactly what the equation tells us.
'Month of Equations': What Does Kepler's Third Law Really Mean?
October 3: Kepler's Third Law
Meaning of Equation: "Farther a planet from its orbiting star, more time it takes to complete one revolution around it"
The equation of Kepler's third law is easy to comprehend. The time period T is directly proportional to 1.5th power of the semi major axis of the elliptical orbit of the planet. Here the semi major axis represents the distance between the centres of the star and the planet which can be roughly replaced by the distance between the star and the planet. Now since the dependence of T on a is governed by the power of 1.5 in the exponent, as a will increase, T will increase. Hence we can deduce that farther a planet, more time will it take to complete its orbit around its parent star.
Kepler derived this law empirically, by using the data of the mean period of a planet and its distance from the Sun. Upon finding this pattern, Kepler quoted:
"I first believed I was dreaming...But it is absolutely certain and exact that the ratio which exists between the period times of any two planets is precisely the ratio of the 3/2th power of the mean distance." translated from "Harmonies of the World" by Kepler (1619)
Month of Equations: What Does Saha's Equation Really Mean?
Meaning of the equation:
Saha's equation tells the degree of ionization of a gas in thermal equilibrium by relating it to the pressure and temperature of the gas.
Developed by an Indian astrophysicist Meghnad Saha in 1920, Saha's equation is a useful result of combining the quantum mechanics and statistical mechanics to explain the spectral classification of stars. This equation depicts the dependence of ionization of a gas on various physical parameters such that :
- Dependence on ionization energy : As the temperature of a gas is raised, the degree of ionization of a gas remains low until the ionization energy is greater than the gas temperature (evident from the exponential factor).
- Dependence on temperature: Afterwards, the degree of ionization i.e., the ratio of number density of ions to the number density of neutral atoms of a gas in thermal equilibrium increases abruptly with an increase in temperature and the gas becomes a plasma (composed of ions, electrons and few neutral atoms).
- Dependence upon number density of ions: When an atom becomes charged, it may recombine with an electron and become a neutral again. So, as the number of electrons increase, the ionization ratio decreases. In the simplest hydrogen plasma, the number of electrons are considered to be equal to the number of ions and hence, as the number density of ions increase in a plasma, the rate of neutralization of ions is enhanced too and the ionization ratio decreases.
Let us now try to understand the physical significance of this equation:
The stellar interiors and atmospheres, gaseous nebulae, and much of the interstellar hydrogen are plasmas. In fact nearly 99% of the universe is considered to be in a plasma state. It is indeed very surprising but we may actually say that luckily the Earth where we live is in the remaining 1% of the universe where plasmas do not occur naturally. However, as soon as we leave the Earth, we may again encounter plasma.
Saha's equation depicts that a gas attains a plasma state at extremely high temperatures and low number densities of charged particles. It is because of this reason that plasmas exist naturally in astronomical objects with temperature of millions of degrees and very low number densities of atoms around 1 per cm cube and due to their natural occurrence, plasma is considered to be the fourth state of matter.
'Month of Equations': What Does The Uncertainty Principle Really Mean?
October 9: The Uncertainty Principle
Meaning of Equation
You cannot simultaneously measure the position and momentum of a particle. The more precisely you know its position, less precisely you will know its momentum.
Heisenberg's uncertainty principle is one such principle that truly depicts the bizarre nature of the quantum world. It is inherent in all the systems exhibiting wave like nature and it arises in quantum mechanics due to matter waves associated with all the quantum objects. There are two main approaches to understand the uncertainty principle: Matrix mechanics and Wave mechanics. We'll discuss the former in this article.
In physics we define observable as something that can be measured. For example. position and momentum are observables in quantum mechanics. Each observable is written as an operator (self adjoint). It is no rocket science. You all have been dealing with operators since childhood. Addition, subtraction, multiplication, division etc. are all operators that act on certain numbers and produce the outcome. In calculus, d/dx is an operator, known as the differential operator, that acts on a function and produces some result. You see, operators themselves do not mean anything. We need something upon which these operators can act to produce a meaningful outcome.
Similarly in quantum mechanics, every observable is associated with an operator. The position operator, the momentum operator, the Hamiltonian (which gives the total energy of the system) etc. These operators act on a particular quantum state and produce the outcome. Now since we have turned quantum mechanics into matrix mechanics, let us see one more important mathematical property of such operators.
We all know what commutation is. Take two numbers x and y. Multiply them and you can see that xy = yx. Always, no exception. But this isn't true in the case of matrices. If A and B are two matrices then AB is not always equal to BA. Thus the commutator of two matrices, defined by [A,B] = AB - BA, isn't always zero as it is in the case of simple numbers. [A,B] actually quantifies how well the two observables described by these operators can be measured simultaneously
That's it. We now have all the tools to understand what the uncertainty principle really means. If x and p represent position and momentum operator, then [x,p] = ih/2π. It is not zero. Thus you cannot simultaneously determine the position and momentum of a particle. Both are incompatible.
An eigenstate of an observable represents the state of the wavefunction for a certain measurement value (the eigenvalue). For example, if a measurement of an observable A is performed, then the system is in a particular eigenstate Ψ of that observable. However, the particular eigenstate of the observable A need not be an eigenstate of another observable B: If so, then it does not have a unique associated measurement for it, as the system is not in an eigenstate of that observable.
Matrix mechanics is one of the most followed mathematical formulation of quantum mechanics. If you want to deeply study quantum mechanics, matrices are the key.