Tag Archives: Einstein

Mathematics and the periodic table

It isn’t surprising that math is involved in the periodic table. Decades before the existence of atoms was shown for sure (Einstein in 1905 on Brownian motion — https://physicsworld.com/a/einsteins-random-walk/) Mendeleev arranged the known elements in a table according to their chemical properties. Math is great at studying and describing structure, and the periodic table is full of it. 

What is surprising, is how periodic table structure arises from math that ostensibly has absolutely nothing to do with chemistry.  Here are 3 examples.

The first occurred exactly 60 years ago to the month in grad school.  The instructor was taking a class of budding chemists through the solution of the Schrodinger equation for the hydrogen atom. 

Recursion relations are no stranger to the differential equations course, where you learn to (tediously) find them for a polynomial series solution for the differential equation at hand. I never really understood them, but I could use them (like far too much math that I took back then).

So it wasn’t a shock when the QM instructor back then got to them in the course of solving the hydrogen atom (with it’s radially symmetric potential). First the equation had to be expressed in spherical coordinates (r, theta and phi) which made the Laplacian look rather fierce. Then the equation was split into 3, each involving one of r, theta or phi. The easiest to solve was the one involving phi which involved only a complex exponential. But periodic nature of the solution made the magnetic quantum number fall out. Pretty good, but nothing earthshaking.

Recursion relations made their appearance with the solution of the radial and the theta equations. So it was plug and chug time with series solutions and recursion relations so things wouldn’t blow up (or as Dr. Gouterman put it, the electron has to be somewhere, so the wavefunction must be zero at infinity). MEGO (My Eyes Glazed Over) until all of a sudden there were the main quantum number (n) and the azimuthal quantum number (l) coming directly out of the recursions.

When I first realized what was going on, it really hit me. I can still see the room and the people in it (just as people can remember exactly where they were and what they were doing when they heard about 9/11 or (for the oldsters among you) when Kennedy was shot — I was cutting a physiology class in med school). The realization that what I had considered mathematical diddle, in some way was giving us the quantum numbers and the periodic table, and the shape of orbitals, was a glimpse of incredible and unseen power. For me it was like seeing the face of God.

The second and third examples occurred this year as I was going through Tony Zee’s book “Group Theory in a Nutshell for Physicists”

The second example occurs with the rotation group in 3 dimensions, which is a 3 x 3 invertible matrix, such that multiplying it by its transpose gives the identity, and such that is determinant is +1.  It is called SO(3)

Then he tensors 2 rotation matrices together to get a 9 x 9 matrix.  Zee than looks for the irreducible matrices of which it is composed and finds that there is a 3×3, a 1×1 and a 5×5.  The 5×5 matrix is both traceless and symmetric.  Note that 5 = 2(2) + 1.  If you tensor 3 of them together you get (among other things 3(2) + 1)   = 7;   a 7 x 7 matrix.

If you’re a chemist this is beginning to look like the famous 2 L + 1 formula for the number of the number of magnetic quantum numbers given an orbital quantum number of L.   The application of a magnetic field to an atom causes the orbital momentum L to split in 2L + 1 magnetic eigenvalues.    And you get this from the dimension of a particular irreducible representation from a group.  Incredible.  How did abstract math know this.  

The third example also occurs a bit farther along in Zee’s book, starting with the basis vectors (Jx, Jy, Jz) of the Lie algebra of the rotation group SO(3).   These are then combined to form J+ and J-, which raise and lower the eigenvalues of Jz.  A fairly long way from chemistry you might think.  

All state vectors in quantum mechanics have absolute value +1 in Hilbert space, this means the eigenvectors must be normalized to one using complex constants.  Simply by assuming that the number of eigenvalues is finite, there must be a highest one (call it j) . This leads to a recursion relation for the normalization constants, and you wind up with the fact that they are all complex integers.  You get the simple equation s = 2j where s is a positive integer.  The 2j + 1 formula arises again, but that isn’t what is so marvelous. 

j doesn’t have to be an integer.  It could be 1/2, purely by the math.  The 1/2 gives 2 (1/2) + 1 e.g two numbers.  These turn out to be the spin quantum numbers for the electron.  Something completely out of left field, and yet purely mathematical in origin. It wasn’t introduced until 1924 by Pauli — long after the math had been worked out.  

Incredible.  

A mathematical kludge and its repair

If you are in a train going x miles an hour and throw a paper airplane forward at x feet per second (or x * 3600/5280 miles per hour, relative to someone outside the train sees the plane move a bit faster than x miles an hour.  Well that’s the whole idea of the Galilean transformation.  Except that they don’t really see velocities adding that way for really fast velocities (close to the speed of light).

Relativity says that there are no privileged sites of observation and that no matter how fast two observer frames are moving relative to each other light will zing past both at the same speed (3 x 10^8 meters/second, 186,000 miles/second).

All of Newton’s mechanics and force laws obeys the Galilean transformation (e.g. velocities add).  Maxwell conceived a series of 4 laws linking electricity and magnetism together, which predicted new phenomena (such as radio waves, and the fact that light was basically a form of wave traveling through space).

Even though incredibly successful, Maxwell’s laws led to an equation (the wave equation) which didn’t obey the Galilean transformation.  This led Lorentz to modify it so the wave equation did obey Galileo.  If you’ve got some mathematical background an excellent exposition of all this is to be found in “The Geometry of Spacetime” by James J. Callahan pp. 22 – 27.

The Lorentz transformation is basically a kludge which makes things work out.  But he had no understanding of why it worked (or what it meant).  The equations produced by the Lorentz transformation are ugly.

Here are the variables involved.

t’ is time in the new frame, t in the old, x’ is position in the new frame x in the old. v is the constant velocity at which the two observation frames are moving relative to each other. c is the common symbol for the velocity of light.

Here are the two equations

t’ =  ( t – vz/c^2 )/ sqrt (1 – v^2/c^2)

x’ = ( z – vt ) /  sqrt (1 – v^2/c^2)

Enter Einstein — he derived them purely by thought.  I recommend Appendix 1 in Einstein’s book “Relativity”.  Amazingly you do not need tensors or even calculus to understand his derivation — just high school algebra (and not much of that — no trigonometry etc. etc.)  You will have the pleasure of watching the power of a great mind at work.

One caveat.  The first few equations won’t make much sense if you hit the appendix without having read the rest of the book (as I did).

Light travels at c miles/hour, so multiplying c by time gives you where it is after t seconds.  In equations x = ct.  This is also true for another reference frame x’ = ct’.

This implies that both x – ct =  0 and x’ – ct’ = 0

Then Einstein claims that these two equations imply that

(x – ct) = lambda * (x’ – ct’) ; lambda is some nonzero number.

Say what?  Is he really saying  0 = lambda * 0.

This is mathematical fantasy.  Lambda could be anything and the statement lacks mathematical content.

Yes, but . . .

It does not lack physical content, which is where the rest of the book comes in.

This is because the two frames (x, t) and (x’ , t’) are said to be in ‘standard configuration which is a complicated state of affairs. We’ll see why y, y’, z, z’ are left out shortly

The assumptions of the standard configuration are as follows:

  • An observer in frame of reference K defines events with coordinates t, x
  • Another frame K’ moves with velocity v relative to K, with an observer in this moving frame K’ defining events using coordinates t’, x’
  • The coordinate axes in each frame of reference are parallel
  • The relative motion is along the coincident xx’ axes (y = y’ and z = z’ for all time, only x’ changes, explaining why they are left out)
  • At time t = t’ =0, the origins of both coordinate systems are the same.

Another assumption is that at time t = t’ = 0 a light pulse is emitted by K at the origin (x = x’ = 0)

The only possible events in K and K’ are observations of the light pulse. Since the velocity of light (c) is independent of the coordinate system, K’ will see the pulse at time t’ and x’ axis location ct’, NOT x’-axis location ct’ – vt’ (which is what Galileo would say). So whenever K sees the pulse at time t and on worldline (ct, t), K’ will see the pulse SOMEWHERE on worldline (ct’, t’).

The way to express this mathematically is by (3) (x – ct) = lambda * (x’ – ct’)

This may seem trivial, but I spent a lot of time puzzling over equation (3)

Now get Einstein’s book and watch him derive the complicated looking Lorentz transformations using simple math and complex reasoning.

Catching God’s dice being thrown

Einstein famously said “Quantum theory yields much, but it hardly brings us close to the Old One’s secrets. I, in any case, am convinced He does not play dice with the universe.”  Astronomers have caught the dice being thrown (at least as far as the origin of life is concerned).

This post will contain a lot more background than most, as I expect some readers won’t have much scientific background.  The technically inclined can read the article on which this is based — http://www.pnas.org/content/115/28/7166

To cut to the chase — astronomers have found water, a simple sugar, and a compound containing carbon, hydrogen, oxygen and nitrogen around newly forming stars and planets.  You need no more than these 4 atoms to build the bases making up the DNA of our genes, all our sugars and carbohydrates, and 18 of the 20 amino acids that make up our proteins. Throw in sulfur and you have all 20 amino acids.  Add phosphorus and you have DNA and its cousin RNA (neither has been found around newly forming stars so far).

These are the ingredients of life itself. Here’s a quote from the article — “What I can definitively say is that the ingredients needed to make biogenic molecules like DNA and RNA are found around every forming protostar. They are there at an early stage, incorporating into bodies at least as large as comets, which we know are the building blocks of terrestrial planets. Whether these molecules survive or are delivered at the late stage of planet formation, that’s the part of it we don’t know very well.”

So each newly formed star and planetary system is a throw of God’s/Nature’s/Soulless physics’ dice for the creation of life.

As of 1 July 2018, there are 3,797 confirmed planets around 2,841 stars, with 632 having more than one (Wikipedia).  And that’s just in the stars close enough to us to study.  Our galaxy, the milky way, contains 400,000,000,000.

Current estimates have some 100,000,000,000 galaxies in the universe.  https://www.space.com/25303-how-many-galaxies-are-in-the-universe.html.  That’s a lot tosses for life to arise.

Suppose that some day life is found on one such planet.  Does this invalidate Genesis, the Koran?  Assume that they are the word of God somehow transmitted to man.  If the knowledge we have about astronomy (above), biology etc. etc. were imparted to Jesus, Mohammed, Abraham, Moses — it never would have been believed.  The creator had to start with something plausible.