## The 21 cm line of hydrogen in Radio Astronomy

This has been a wonderful discovery back in the 1950s that gave Radio Astronomy a good push forward. It also helped in mapping out our Milky Way galaxy (which we really can’t see very well!).

It arose from a feature of quantum field theory, specifically from the hyperfine structure of hydrogen. (I’ll try to explain.)

You know that the hydrogen atom consists of a single proton at its central nucleus and a single electron moving around it somehow in certain specific quantized orbits. It cannot just circle around in any orbit.

That was one of Niels Bohr’s major contributions to our understanding of the atom. In fact this year we’re celebrating the 100th anniversary of his model of the atom (his major papers written in 1913). Some articles in the June issue of Nature magazine are in honor of Bohr’s work.

Normally the electron circles in the lowest orbit associated with the lowest energy state – usually called the ground state (the one with n = 1).

It is known that protons and electrons are particles that have “spin”. (That’s why they are sometimes also called ‘fermions’.) It’s as if they behave like spinning tops. (The Earth and Milky Way are spinning too!)

The spin can be in one direction (say ‘up’) or in the other direction (we label as ‘down’). (These labels of where ‘up’ and ‘down’ are depends on the coordinates we choose, but let’s now worry about that.)

When scientists looked at the spectrum of hydrogen more closely they saw that even while the electron can be in the same ground state – and with definite smallest energy – it can have slightly different energies that are very very close to one another. That’s what is meant by “hyperfine structure” — meaning that the usual energy levels of hydrogen are basically correct except that there are ever so slight deviations from the normal energy levels.

It was discovered by means of quantum field theory that this difference in ground state energies arise when the electron and proton switch between spinning in the same direction to spinning in opposite directions (or vice versa).

When they spin in the same direction the hydrogen atom has slightly more energy than when they are spinning in opposite direction.

And the difference between them?

The difference in these energies corresponds to an electromagnetic wave corresponding to about 21 cm wavelength. And that falls in the radio band of the electromagnetic spectrum.

So when the hydrogen atom shows an emission or absorption spectrum in that wavelength level it means that the electron and proton have switched between having parallel spins to having opposite spins. When the switch happens you see an electromagnetic ray either emitted or absorbed.

It does not happen too often, but when you have a huge number of hydrogen atoms — as you would in hydrogen clouds in our galaxy — it will invariably happen and can be measured.

Now it’s a really nice thing that our galaxy contains several hydrogen clouds. So by measuring the Doppler shift in the spectrum of hydrogen — at the 21 cm line! — you can measure the velocities of these clouds in relation to our location near the sun.

These velocity distributions are used together with other techniques to map out the hydrogen clouds in order to map out and locate the spiral arms they fall into.

That work (lots of hard work!) showed astronomers that our Milky Way does indeed have arms, just as we would see in some other galaxies, such as in the picture shown here of NGC 1232.

The one UNKNOWN about the structure of our Milky Way is that we don’t know whether it has 2 or 4 arms.

**References:**

[1] University Astronomy, by Pasachoff and Kutner.

[2] Astronomy (The Evolving Universe), by Michael Zeilik.

(These are excellent sources, by the way.)

## August Kekule’s Benzene Vision

The first time I heard of August Kekule’s dream/vision was from my dear mother! (My mom is a geologist who obviously had to know a lot of chemistry.) I am referring to Kekule’s vision while gazing at a fireplace which somehow prompted him onto the idea for the structure of the benzene molecule C_{6}H_{6}. And then I heard that the story is suspect maybe even a myth cooked up by unscientific minds. Now I have learned that Kekule himself recounted that story which was translated into English and published in the *Journal of Chemical Education (*Volume 35, No. 1, Jan. 1958, pp 21-23, translator: Theodor Benfey). Here is an excerpt from that paper relevant to the story where Kekule talks about his discovery.

I was sitting writing at my textbook but the work did not progress; my thoughts were elsewhere. I turned my chair to the fire and dozed. Again the atoms were gamboling before my eyes. This time the smaller groups kept modestly in the background. My mental eye, rendered more acute by repeated visions of the kind, could now distinguish larger structures of manifold conformation: long rows, sometimes more closely fitted together all twining and twisting in snake-like motion. But look! What was that? One of the snakes had seized hold of its own tail, and the form whirled mockingly before my eyes. As if by a flash of lightning I awoke; and this time also I spent the rest of the night in working out the consequences of the hypothesis. Let us learn to dream, gentlemen, then perhaps we shall find the truth.

And to those who don’t think The truth will be given. They’ll have it without effort.

But let us beware of publishing our dreams till they have been tested by the making understanding.

Countless spores of the inner life fill the universe, but only in a few rare beings do they find the soil for their development; in them the idea, whose origin is known to no men, comes to life in creative action. (J. Von Liebig)

I believe it is unnecessary to rule out or ridicule dreams, trances, visions in the pursuit of scientific truth. Because, after all, they still have to be tested and examined in our sober existence (as Kekule already alluded). I see them as extensions of thinking and contemplation, and surely there is nothing wrong with these.

## Einstein on theory, logic, reality

Long ago (late 1980s) I attended a lecture by Einstein biographer I. Bernard Cohen. (Cohen actually interviewed Einstein and published it in the Scientific American in the 1955 issue.)

In his lecture, Cohen described Einstein’s view of scientific discovery as a sort of ‘leap’ from experiences to theory. That theory is not logically deduced from experiences but that theory is “jumped at” — or “swooped” is the word Cohen used, I think — thru the imagination or intuition based on our experiences (which of course would/could include experiments). This reminds one of the known Einstein quote that “imagination is more important that knowledge.”

In his book Ideas and Opinions, Albert Einstein said:

“Pure logical thinking cannot yield us any knowledge of the empirical world; all knowledge of reality starts from experience and end in it. Propositions arrived at by purely logical means are completely empty as regards reality. Because Galileo saw this, and particularly because he drummed it into the scientific world, he is the father of modern physics—indeed, of modern science altogether.”

(See Part V of his “Ideas and Opinions” in the section entitled “On the Method of Theoretical Physics.”)

In a related passage from the same section, Einstein noted:

“If, then, it is true that the axiomatic foundation of theoretical physics cannot be extracted from experience but must be freely invented, may we ever hope to find the right way? Furthermore, does this right way exist anywhere other than in our illusions? May we hope to be guided safely by experience at all, if there exist theories (such as classical mechanics) which to a large extent do justice to experience, without comprehending the matter in a deep way?

To these questions, I answer with complete confidence, that, in my opinion, the right way exists, and that we are capable of finding it. Our experience hitherto justifies us in trusting that nature is the realization of the simplest that is mathematically conceivable. I am convinced that purely mathematical construction enables us to find those concepts and those lawlike connections between them that provide the key to the understanding of natural phenomena. Useful mathematical concepts may well be suggested by experience, but in no way can they be derived from it. Experience naturally remains the sole criterion of the usefulness of a mathematical construction for physics. But the actual creative principle lies in mathematics. Thus, in a certain sense, I take it to be true that pure thought can grasp the real, as the ancients had dreamed.”

Note his reference to theory as being ‘freely invented’ (and even ‘illusion’) which echo the role of intuition and imagination in the scientific development of theory (but which are probably not completely divorced from experience either!).

The last two quotes above incidentally can be found online in Standford’s Encyclopedia of Philosophy: Einstein’s Philosophy of Science

## The sphere in dimensions 4, 5, …

The volume of a sphere of radius R in 2 dimensions is just the area of a circle which is π R^{2}. (The symbol π is Pi which is 3.1415….)

The volume of a sphere of radius R in 3 dimensions is (4/3) π R^{3}.

The volume of a sphere of radius R in 4 dimensions is (1/2) π^{2} R^{4}.

Hold everything! How’d you get that? With a little calculus!

Ok, you see a sort of pattern here. The volume of a sphere in n dimensions has a nice form: it is some constant C(n) (involving π and some fractions) times the radius R raised to the dimension n:

V(n,R) = C(n) R^{n} —————– (1)

where I wrote V(n,R) for the volume of a sphere in n dimensions of radius R (as a function of these two variables).

Now, how do you get the constants C(n)?

With a bit of calculus and some ‘telescoping’ as we say in Math.

(In the cases n = 2 and n = 3, we can see that C(2) = π, and C(3) = 4π/3.)

First, let’s do the calculus. The volume of a sphere in n dimensions can be obtained by integrating the volume of a sphere in n-1 dimensions like this using the form (1):

V(n,R) = ∫ V(n-1,r) dz = ∫ C(n-1) r^{n-1} dz

where here r^{2} + z^{2} = R^{2} and your integral goes from -R to R (or you can take 2 times the integral from 0 to R). You solve the latter for r, plug it into the last integral, and compute it using the trig substitution z = R sin θ. When you do, you get

V(n,R) = 2C(n-1) R^{n} ∫ cos^{n}θ dθ.

The cosine integral here goes from 0 to π/2, and it can be expressed in terms of the Gamma function Γ, so it becomes

V(n,R) = 2C(n-1) R^{n} π^{1/2} Γ((n+1)/2) / Γ((n+2)/2).

Now compare this with the form for V in equation (1), you see that the R’s cancel and you have C(n) expressed in terms of C(n-1). After telescoping and simplifying you eventually get the volume of a sphere in n dimensions to be:

V(n,R) = π^{n/2} R^{n} / Γ((n+2)/2).

Now plug in n equals 4 dimensions and you have what we said above. (Note: Γ(3) = 2 — in fact, for positive integers N the Gamma function has simple values given by Γ(N) = (N-1)!, using the factorial notation.)

How about the volume of a sphere in 5 dimensions? It works out to

V(5,R) = (8/15) π^{2} R^{5}.

One last thing that’s neat: if you take the derivative of the Volume with respect to the radius R, you get its Surface Area! How crazy is that?!!

## Can a product of 4 consecutive odds be a perfect square?

Someone on twitter asked if a product of four consecutive odd positive numbers can be a perfect square.

My answer: No.

For example, 3 x 5 x 7 x 9 = 945 which is not a perfect square. (30^2 = 900 and 31^2 = 961.) Similarly, 7 x 9 x 11 x 13 = 9009, again is not a perfect square.

Here is my proof. Write the four consecutive positive odd numbers as:

2x+1, 2x+3, 2x+5, 2x+7

where x is a positive integer.

The middle two numbers 2x+3, 2x+5 cannot both be perfect squares since their difference is 2 — the difference between two positive consecutive perfect squares is at least 3. Let’s suppose it is 2x+3 that is not a perfect square. (The argument can still be adapted if it was 2x+5.) So 2x+3 is divisible by an odd prime p that has an odd power in its prime factorization. This prime p cannot divide the other three factors 2x+1, 2x+5, 2x+7, or else it would divide their differences from 2x+3, which are 2 or 4 (and p is odd). Therefore the prime p appears in the prime factorization of the product

(2x+1)(2x+3)(2x+5)(2x+7)

with only an odd power, hence this product cannot be a perfect square. QED

## Progress on the Twin Prime Conjecture

**The Chinese mathematician Yitang Zhang made great strides in one of the big problems in number theory called the Twin Prime Conjecture. He did not solve it but proved a related result that is considered quite remarkable — and he got his paper accepted for publication in the very prestigious math journal Annals of Mathematics.**

**If you wish you can read the sources below regarding the news on this, but I’d like to say things my way.**

**First, let’s explain some things. The ‘numbers‘ we’re talking about here are mainly positive whole numbers 1, 2, 3, 4, …, (which we call positive integers, or natural numbers). (So we’re not talking fractions here.)**

**A number is prime when it is not divisible by any other number beside 1 and itself. So for example, 5 is a prime, while 6 is not prime (we call it composite, since it is divisible by 2 and 3, beside 1 and 6).**

**So the sequence of prime numbers starts off and proceeds like this:**

**2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, …**

**and they go on and on forever. We know that there are infinitely many prime numbers (and the proof is not hard).**

**Now if you look closely at this list of primes, you notice something interesting. That there are ‘consecutive’ pairs of primes. Ones whose difference is 2. Like the pair 5, 7, and the pair 11, 13, and the pair 29, 31, etc etc. These are called twin primes.**

**The Twin Prime Conjecture says that there are infinitely many of these twin pairs of primes.**

**But unlike Euclid’s proof that there are infinitely many primes (which is easy), the proof of the Twin Prime conjecture is extremely hard or just unknown. (Or maybe it’s not true!) No one has been able (so far) to settle the question or find a proof of it.**

**Since that is too hard, mathematicians try to ask and maybe settle simpler, but related, questions that they may have a chance at answering. **

**For example, instead of looking at pairs of primes whose difference is 2, why not look at ones whose difference is some number L? **

**As a related conjecture, one can ask:**

**Q: Is there a number L such that there are infinitely many pairs of primes whose difference is no more than L?**

**When L = 2, this question reduces to the Twin Prime Conjecture. For L = 10, let’s say, we can have pairs like 7, 11, whose difference is 4 which is no more than 10. Another pair would be 17, 23, whose difference is 6 (again no more than 10). Etc.**

**Now what Professor Yitang Zhang proved is that the answer to question Q is YES! There is such a fixed difference L. He proved that if you take L to be 70,000,000 (70 million) then there are infinitely many pairs of primes whose difference is no more than 70 million.**

**(So in particular we don’t know if there infinitely many pairs of primes whose difference is not more than 100 (or 1000, or even a million). These would still be interesting but apparently quite hard to answer.)**

**Once again, we have very simple sounding questions (a babe can ask them!) that are very hard to prove, and indeed we do not even have proofs for them.**

**REFERENCES:**

**These two reference are short reports on the result.**

Nature Magazine — New Scientist

**The following reference is slightly longer and more detailed:
**