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Cosmic Microwave Background

May 30, 2015 1 comment

The Cosmic Microwave Background (CMB) radiation is a very faint but observable form of radiation that is coming to us (and to other places too) from all directions. (By ‘radiation’ here is meant photons of light, or electromagnetic waves, from a wide range of possible frequencies or energies.) In today’s standard model of cosmology, this radiation is believed to emanate from about a time 200,000 to 400,000 years after the Big Bang – a timeframe known as ‘last scattering’ because that was when superheavy collisions between photons of light and other particles (electrons, protons, neutrons, etc) eased off to a degree that photons can ‘escape’ into the expanding space. At the time of last scattering, this radiation was very hot, around {3000}^{\circ}K (in the Kelvin scale). And in time, as space expanded, radiation cooled to its currently observed value of 2.726^\circ K.

One of the amazing facts about this radiation is that it almost perfectly matches Planck’s radiation formula (discovered in 1900) for a black body:

\displaystyle I(\nu) = \frac{8\pi h}{c^3} \cdot \frac{\nu^3}{e^{h\nu/kT} - 1}.

In this formula, \nu is the frequency variable (a positive real number that gives the number of cycles per second of a wave) and I(\nu) is the energy density as a function of frequency \nu.

The other variables are: T is the temperature of the black body which is assumed to be in equilibrium (so the temperature is uniformly constant throughout the body of radiation), c is the speed of light in vacuum, h is Planck’s constant, and k is Boltzmann’s constant from statistical mechanics.

If you plot the graph of this energy density function (against \nu) you get a curve that looks like a skewed ‘normal distribution’. Here are some examples I hijacked from the internet:

Various Planck radiation density graphs depending on temperature T.

Here you see various plots of Planck’s function for different temperatures T. The horizontal axis labels the frequency \nu, and the vertical gives the energy density I(\nu) per frequency. (Please ignore the rising black dotted curve.)

You’ll notice that the graphs have a maximum peak point. And that the lower the temperature, the smaller the frequency where the maximum occurs. Well, that’s what happened as the CMB radiation cooled from a long time ago till today: as the temperature T cooled (decreased) so did the frequency where the peak occurs.

To those of us who know calculus, we can actually compute what frequency \nu gives maximum density and give our answer in terms of the temperature T. All we do is compute the derivative of I(\nu) with respect to \nu and set it to zero and solve the resulting equation for \nu. You will get an equation whose solution isn’t so trivial to solve, so we’ll need some software or a calculator to approximate it. Anyway, I worked it out (and you can check my answer) and obtained the following:

\displaystyle\nu_{\max} = 2.82 \frac{kT}{h}.

(The equal sign here is an approximation!)

The \nu_{\max} is the frequency that gives maximum density and as you can see it is a straight linear function of temperature. The greater the temperature, the proportionately greater the max frequency. The colder the temperature gets the smaller the max-frequency \nu_{\max} gets, and from Planck’s energy equation E_{\max} = h\nu_{\max}, so also does the energy of the radiation drop.

Now plug in the observed value for the temperature of the background radiation, which is T = 2.726 (degrees Kelvin), and working it out we get (approximately)

\displaystyle\nu_{\max} = 160.2 \text{ GHz}

This frequency lies inside the microwave band which is why we call it the microwave radiation! (Even though it does also radiate in other higher and lower frequencies too but at much less intensity!)

Far back in time, when photons were released from their collision `trap’ (and the temperature of the radiation was much hotter) this max frequency was not in the microwave band.

Homework Question: what was the max-frequency \nu_{\max} at the time of last scattering? What frequency band does it belong to? In the visible range? Infrared? Ultraviolet? Higher still? (I’m dying to know! 😉 )

(It isn’t hard as it can be figured from the data above.)

Anyway, I thought working these out was fun.

The CMB radiation was first discovered by Penzias and Wilson in 1965. According to their measurements and calculations (and polite disposal of the pigeons nesting in their antenna!), they measured the temperature as being 3.5^\circ K plus or minus 1 Kelvin. (So the actual value that was confirmed later, namely 2.726, fits within their range.) The frequency of radiation that they detected, however, was not the maximum yielding one but was (as they had it in the title of their paper) 1080 Mc/s — which is ‘mega cycles per second’ and is the same as MHz (megahertz). The wavelength value corresponding to this is 7.35 cm. To do the conversion between wavelength \lambda and frequency \nu for electromagnetic waves we use the simple formula

\displaystyle\lambda\nu = c

where c is the speed of light (in vacuum).

And that’s the end of our little story for today!

Cheers, Sam Postscript.

The sacred physical constants:

Planck’s constant h = 6.6254 \times 10^{-27} \text{ erg sec}

Boltzmann’s constant k = 1.38049 \times 10^{-16} \text{ erg/K}

Speed of light c = 2.9979 \times 10^{10} \text{ cm/sec}.  

Einstein summation convention

March 25, 2015 1 comment

Suppose you have a list of n numbers A_1, A_2, A_3, \dots, A_n .

Their sum A_1 + A_2 + A_3 + \dots + A_n is often shorthanded using the Sigma notation like this

\displaystyle\sum_{k=1}^n A_k

which is read “sum of A_k from k=1 to k=n.” This letter k that varies from 1 to n is called an ‘index’.

Vectors. You can think of a vector as an ordered list of numbers A = (A_1, A_2, A_3, \dots, A_n) .

If you have two vectors A = (A_1, A_2, A_3, \dots, A_n) and B = (B_1, B_2, B_3, \dots, B_n) their dot product is defined by multiplying their respective coordinates and adding the result:

\displaystyle A\bullet B = A_1B_1 + A_2B_2 + A_3B_3 + \dots + A_nB_n.

Using our summation notation, we can abbreviate this to

\displaystyle A\bullet B = \sum_{k=1}^n A_k B_k.

While working thru his general theory of relativity, Einstein noticed that whenever he was adding things like this, the same index k was repeated! (You can see the k appearing once in A and also in B.) So he thought, well in that case maybe we don’t need a Sigma notation! So remove it! The fact that we have a repeating index in a product expression would mean that a Sigma summation is implicitly understood. (Just don’t forget! And don’t eat fatty foods that can help you forget!)

With this idea, the Einstein summation convention would have us write the above dot product of vectors simply as

\displaystyle A\bullet B = A_k B_k

In his theory’s notation, it’s understood that the index k here would vary from 1 to 4, for the four dimensional space he was working with. That’s Einstein’s index notation where 1, 2, 3, are the indices for space coordinates (i.e., A_1, A_2, A_3 ), and k=4 for time (e.g., A_4 ). One could also write space-time coordinates using the vector (x_1, x_2, x_3, t) where t is for time. 

(Some authors have k go from 0 to 3 instead, with k=0 corresponding to time and the others to space coordinates.)

I used ‘k’ because it’s not gonna scare anyone, but Einstein actually uses Greek letters like \mu, \nu, \dots instead of the k. The convention is that Greek index letters range over all 4 space-time coordinates, and Latin indices (like k, j, m,etc)  for the space coordinates only. So if we use \mu instead of k the dot product of the two vectors would be

\displaystyle A\bullet B = A_\mu B_\mu.

So if we write A_\mu B_\mu it means we understand that we’re summing these over the 4 indices of space-time. And if we write  A_k B_k it means that we’re summing these over the 3 indices of space only. More specifically,

\displaystyle A_\mu B_\mu = A_1B_1 + A_2B_2 + A_3B_3 + A_4B_4

and

\displaystyle A_k B_k = A_1B_1 + A_2B_2 + A_3B_3.

There is one thing that I left out of this because I didn’t want to complicate the introduction and thereby scare readers! (I already may have! Shucks!) And that is, when you take the dot product of two vectors in Relativity, their indices are supposed to be such that one index is a subscript (‘at the bottom’) and the other repeating index is a superscript (‘at the top’).  So instead of writing our dot product as A_\mu B_\mu it is written as

\displaystyle A\bullet B = A_\mu B^\mu.

(This gets us into covariant vectors, ones written with subscripts, and contravariant vectors, ones written with superscripts. But that is another topic!)

How about we promote ourselves to Tensors? Fear not, let’s just treat it as a game with symbols! Well, tensors are just like vectors except that they can involved more than one index. For example, a vector such as in the above was written A_\mu , so it involves one index \mu . What if you have two indices? Well in that case we have a matrix which we can write M_{\mu \nu} . (Here, the two indices are sitting side by side like good friends and aren’t being multiplied! There’s an imaginary comma that’s supposed to separate them but it’s not conventional to insert a comma.)

The most important tensor in Relativity Theory is what is called the metric tensor written \large g_{\mu\nu} . It describes the distance structure (metric = distance) on a curved space-time. So much of the rest of the geometry of space, like its curvature, how to differentiate vector fields, curved motion of light and particles, shortest path in curved space between two points, etc, comes from this metric tensor \large g_{\mu\nu} .

The Einstein ‘gravitational tensor’ is one such tensor and is written G_{\mu \nu} . Tensors like those are called rank 2 tensors because they involve two different indices. Another good example of a rank 2 tensor is the energy-momentum tensor often written as T_{\mu \nu} . This tensor encodes the energy and matter distribution in spaces that dictate its geometry — the geometry (and curvature) being encoded in the Einstein tensor G_{\mu \nu} . (If you’ve read this far, you’re really getting into Relativity! And I’m very proud of you!)

You could have a tensors with 3, 4 or more indices, and the indices could be mixed subscripts and superscripts, like for example D_{\mu \nu}^{\alpha \beta} and F_{\tau}^\gamma .

If you have tensors like this, with more than 1 or 2 indices, you can still form their dot products. For example for the tensors D and F, you can take any lower index of D (say you take \nu and set it equal to an upper index of F — and add! So we get a new tensor when we do this dot product! You get

\displaystyle D_{\mu \nu}^{\alpha \beta} F_{\tau}^\nu = D_{\mu 1}^{\alpha \beta} F_{\tau}^1 + D_{\mu 2}^{\alpha \beta} F_{\tau}^2 +D_{\mu 3}^{\alpha \beta} F_{\tau}^3 +D_{\mu 4}^{\alpha \beta} F_{\tau}^4

where it is understood that since the index \nu is repeated, you are summing over that index (from 1 to 4) (as I’ve written out on the right hand side). Notice that the indices that remain are \mu, \alpha, \beta, \tau . So this dot product gives rise to yet another tensor with these indices – let’s give the letter C:

\displaystyle C_{\mu \tau}^{\alpha \beta} = D_{\mu \nu}^{\alpha \beta} F_{\tau}^\nu.

This process where you pick two indices from tensors and add their products along that index is called ‘contraction‘ – even though it came out of doing a simple idea of dot product. Notice that in general when you contract tensors the result is not a number but is in fact another tensor. This process of contraction is very important in relativity and geometry, yet it’s based on a simple idea, extended to complicated objects like tensors. (In fact, you can call the original dot product of two vectors a contraction too, except it would be number in this case.)

Thank you!
\Sigma\alpha\mu

Einstein and Pantheism

January 3, 2015 Leave a comment

Albert Einstein’s views on religion and on the nature and existence of God has always generated interest, and they continue to. In this short note I point out that he expressed different views on whether he subscribed to “pantheism.” It is well-known that he said, for instance, that he does not believe in a personal God that one prays to, and that he rather believes in “Spinoza’s God” (in some “pantheistic” form). Here are two passages from Albert Einstein where he expressed contrary views on whether he is a “pantheist”.

I’m not an atheist and I don’t think I can call myself a pantheist. We are in the position of a little child entering a huge library filled with books in many different languages.…The child dimly suspects a mysterious order in the arrangement of the books but doesn’t know what it is. That, it seems to me, is the attitude of even the most intelligent human being toward God.” (Quoted in Encyclopedia Britannica article on Einstein.)

Here, Einstein says that he does not think he can call himself a pantheist. However, in his book Ideas and Opinions he said that his conception of God may be described as “pantheistic” (in the sense of Spinoza’s):

This firm belief, a belief bound up with a deep feeling, in a superior mind that reveals itself in the world of experience, represents my conception of God. In common parlance this may be described as “pantheistic” (Spinoza)” (Ideas and Opinions, section titled ‘On Scientific Truth’ – also quoted in Wikipedia with references)

It is true that Einsteins expressed his views on religion and God in various ways, but I thought that the fact that he appeared to identify and not identify with “pantheism” at various stages of his life is interesting. Setting aside that label, however, I think that his general conceptions of God in both these quotes — the mysterious order in the books, a universe already written in given languages, a superior mind that reveals itself in such ways — are fairly consistent, and also consistent with other sentiments he expressed elsewhere.

Matter and Antimatter don’t always annihilate

October 11, 2014 2 comments

It is often said that when matter and antimatter come into contact they’ll annihilate each other, usually with the release of powerful energy (photons).

Though in essence true, the statement is not exactly correct (and so can be misleading).

For example, if a proton comes into contact with a positron they will not annihilate. (If you recall, the positron is the antiparticle of the electron.) But if a positron comes into contact with an electron then, yes, they will annihilate (yielding a photon). (Maybe they will not instantaneously annihilate, since they could for the minutest moment combine to form positronium, a particle they form as they dance together around their center of mass – and then they annihilate into a photon.)

The annihilation would occur between particles that are conjugate to each other — that is, they have to be of the same type but “opposite.” So you could have a whole bunch of protons come into contact with antimatter particles of other non-protons and there will not be mutual annihilation between the proton and these other antiparticles.

Another example. The meson particles are represented in the quark model by a quark-antiquark pair. Like this: p\bar q . Here p and q could be any of the 6 known quarks u, d, c, b, t, s and the \bar q  stands for the antiquark of q . If we go by the loose logic that “matter and antimatter annihilate” then no mesons can exist since p  and \bar q  will instantly destroy one another.

For example, the pion particle \pi^+ has quark content u\bar d  consisting of an up-quark u and the anti-particle of the down quark. They don’t annihilate even though they’re together (in some mysterious fashion!) for a short while before it decays into other particles. For example, it is possible to have the decay reaction

\pi^+ \to \mu^+ + \nu_\mu

(which is not the same as annihilation into photons) of the pion into a muon and a neutrino.

Now if we consider quarkonium, i.e. a quark and its antiquark together, such as for instance \pi^0 = u\bar u or \eta = d\bar d , so that you have a quark and its own antiquark, then they do annihilate. But, before they do so they’re together in a bound system giving life to the \pi^0, \eta particles for a very very short while (typically around 10^{-23} seconds). They have a small chance to form a particle before they annihilate. It is indeed amazing to think how such Lilliputian time reactions are part of how the world is structured. Simply awesome! 😉

PS. The word “annihilate” usually has to do when photon energy particles are the result of the interaction, not simply as a result of when a particle decays into other particles.

Sources:

(1) Bruce Schumm, Deep Down Things. See Chapter 5, “Patterns in Nature,” of this wonderful book. 🙂

(2) David Griffiths, Introduction to Elementary Particles. See Chapter 2. This is an excellent textbook but much more advanced with lots of Mathematics!

Escher Math

http://upload.wikimedia.org/wikipedia/en/a/a3/Escher%27s_Relativity.jpg

Escher Relativity

You’ve all seen these Escher drawings that seem to make sense locally but from a global, larger scale, do not – or ones that are just downright strange. We’ll it’s still creative art and it’s fun looking at them. They make you think in ways you probably didn’t. That’s Art!

Now I’ve been thinking if you can have similar things in math (or even physics). How about Escher math or Escher algebra?

Here’s a simple one I came up with, and see if you can ‘figure’ it out! 😉

(5 + {4 – 7)2 + 5}3.

LOL! 🙂

How about Escher Logic!? Wonder what that would be like. Is it associative / commutative? Escher proof?

Okay, so now … what’s your Escher?

Have a great day!

 

 

Richard Feynman on Erwin Schrödinger

I thought it is interesting to see what the great Nobel Laureate physicist Richard Feynman said about Erwin Schrödinger’s attempts to discover the famous Schrödinger equation in quantum mechanics (see quote below). It has been my experience in reading physics that this sort of “heuristic” reasoning is part of doing physics. It is a very creative (sometimes not logical!) art with mathematics in attempting to understand the physical world. Dirac did it too when he obtained his Dirac equation for the electron by pretending he could take the square root of the Klein-Gordon operator (which is second order in time). Creativity is a very big part of physics.

 

“When Schrödinger first wrote it down [his equation],
he gave a kind of derivation based on some heuristic
arguments and some brilliant intuitive guesses. Some
of the arguments he used were even false, but that does
not matter; the only important thing is that the ultimate
equation gives a correct description of nature.”
                                    — Richard P. Feynman
(Source: The Feynman Lectures on Physics, Vol. III, Chapter 16, 1965.)
 
 
 

How to see Schrödinger’s Cat as half-alive

January 1, 2014 4 comments

https://samuelprime.files.wordpress.com/2014/01/5ef6c-schrodinger-cat.gif

Schrödinger’s Cat is a thought experiment used to illustrate the weirdness of quantum mechanics. Namely, unlike classical Newtonian mechanics where a particle can only be in one state at a given time, quantum theory says that it can be in two or more states ‘at the same time’. The latter is often referred to as ‘superposition’ of states.

If D refers to the state that the cat is dead, and if A is the state that the cat is alive, then classically we can only have either A or D at any given time — we cannot have both states or neither.

In quantum theory, however, we can not only have states A and D but can also have many more states, such a for example this combination or superposition state:

ψ = 0.8A + 0.6D.

Notice that the coefficient numbers 0.8 and 0.6 here have their squares adding up to exactly 1:

(0.8)2 + (0.6)2 = 0.64 + 0.36 = 1.

This is because these squares, (0.8)2 and (0.6)2, refer to the probabilities associated to states A and D, respectively. Interpretation: there is a 64% chance that this mixed state ψ will collapse to the state A (cat is alive) and a 36% chance it will collapse to D (cat is dead) — if one proceeds to measure or find out the status of the cat were it to be in a quantum mechanical state described by ψ.

Realistically, it is hard to comprehend that a cat can be described by such a state. Well, maybe it isn’t that hard if we had some information about the probability of decay of the radioactive substance inside the box.

http://upload.wikimedia.org/wikipedia/commons/2/29/Stern-Gerlach_experiment.PNG

Nevertheless, I thought to share another related experiment where one could better ‘see’ and appreciate superposition states like ψ above. The great physicist Richard Feynman did a great job illustrating this with his use of the Stern-Gerlach experiment (which I will tell you about). (See chapters 5 and 6 of Volume III of the Feynman Lectures on Physics.)

In this experiment we have a magnetic field with north/south poles as shown. Then you have a beam of spin-half particles coming out of a furnace heading toward the magnetic field. The result is that the beam of particles will split into two beams. What essentially happened is that the magnetic field made a ‘measurement’ of the spins and some of them turned into spin-up particles (the upper half of the split beam) and the others into spin-down (the bottom beam). So the incoming beam from the furnace is like the cat being in the superposition state ψ and magnetic field is the agent that determined — measured! — whether the cat is alive (upper beam) or dead (lower beam). (Often in physics books they use the Dirac notation |↑〉for spin-up state and |↓〉 for spin-down.)

In a way, you can now see the initial beam emanating from the furnace as being in a superposition state.

Ok, so the superposition state of the initial beam has now collapsed the state of each particle into two specific states: spin-up state (upper beam) and the spin-down state (lower beam). Does this mean that these states are no longer superposition states?

Yes and No! They are no longer in superposition if the split beams enter another magnetic field that points in the same direction as the original one. If you pass the upper beam into a second identical magnetic field, it will remain an upper beam — and the same with the lower beam. The magnetic field ‘made a decision’ and it’s going to stick with it! 🙂

That is why we call these states (upper and lower beams) ‘eigenstates’ of the original magnetic field. They are no longer mixed superposition states — the cat is either dead or alive as far as this field is concerned and not in any ‘in between fuzzy’ states.

Ok, that addresses the “Yes” part of the answer. Now for the “No” part.

Let’s suppose we have a different magnetic field, one just like the original one but perpendicular in direction to it. (So it’s like you’ve rotated the original field by 90 degrees; you can rotate by a different angle as well.)

In this case if you pass the original upper beam (that was split by the first magnetic field) into the second perpendicular field, this upper beam will split into two beams! So with respect to the second field the upper beam is now in a superposition state!

Essential Principle: the notion of superposition (in quantum theory) is always with respect to a variable that is being measured. In our case, that variable is the magnetic field. (And here we have two magnetic fields, hence we have two different variables — non-commuting variables as we say in quantum theory.)

Therefore, what was an eigenstate (collapsed state) for the first field (the upper beam) is no longer an eigenstate (no longer ‘collapsed’) for the second (perpendicular) field. (So if a wavefunction is collapsed, it can collapse again and again!)

The Schrodinger Cat experiment could possibly be better understood not as one single experiment, but as a stream of many many boxes with cats in them. This view might better relate to the fact that we have beam of particles each of which is being ‘measured’ by the field to determine its spin status (as being up or down).

Best wishes, Sam

Reference. R. Feynman, R. Leighton, M. Sands, The Feynman Lectures on Physics, Vol. III (Quantum Mechanics), chapters 5 and 6.

Postscript. It occurred to me to add a uniquely quantum mechanical feature that is contrary to classical physics thinking.   The Stern-Gerlach experiment is a good place to see this feature.

We noted that when the spin-half particles emerge from the furnace and into the magnetic field, they split into upper and lower beams. Classically, one might think that before entering the field the particles already had their spins either up or down before a measurement takes place (i.e., before entering the magnetic field) — just as one might say that the earth has a certain velocity as it moves around the sun before we measure it. Quantum theory does not see it that way. In the predominant Copenhagen Interpretation of quantum theory, one cannot say that the particle spins were already a mix of up or down spins before entering the field. Reason we cannot say this is that if we had rotated the field at an angle (say at right angles to the original), the beams would still split into two, but not the same two beams as before! So we cannot say that the particles were already in a mix of those that had spins in one direction or the other. That is one of the strange features of quantum theory, but wonderfully illustrated by Stern-Gerlack.

Mathematics. In vector space theory there is a neat way to illustrate this quantum phenomenon by means of `bases’. For example, the vectors (1,0) and (0,1) form a basis for 2D Euclidean space R2. So any vector (x,y) can be expressed (uniquely!) as a superposition of them; thus,

(x,y) = x(1,0) + y(0,1).

So this would be, to make an analogy, like how the beam of particles, described by (x,y), can be split into to beams — described by the basis vectors (1,0) and (0,1).

However, there are a myriad of other bases. For example, (2,1) and (1,1) also form a basis for R2. A general vector (x,y) can still be expressed in terms of a superposition of these two:

(x,y) = a(2,1) + b(1,1)

for some constants a and b (which are easy to solve in terms of x, y). So this other basis could, by analogy, represent a magnetic field that is at an angle with respect to the original — and its associated beams (2,1) and (1,1) (it’s eigenstates!) would be different because of their different directions. As a matter of fact, we can see here that these eigenstates (collapsed states), represented by (2,1) and (1,1), are actual (uncollapsed) superpositions of the former two, namely (1,0) and (0,1). And vice versa!

Analogy: let’s suppose the vector (5,3) represents the particle states coming out of the furnace. Let’s think of the basis vectors (1,0) and (0,1) represent the spin-up and spin-down beams, respectively, as they enter the first magnetic field, and let the other basis vectors (2,1) and (1,1) represent the spin-up and spin-down beams when they enter the second rotated (maybe perpendicular) magnetic field. Then the particle state (5,3) is a superposition in each of these bases!

(5,3) = 5(1,0) + 3(0,1),

(5,3) = 2(2,1) + 1(1,1).

So it is now quite conceivable that the initial mixed state of particles as they exit the furnace can in fact split in any number of ways as they enter any magnetic field! I.e., it’s not as though they were initially all either (1,0),(0,1) or (2,1),(1,1), but (5,3) could be a simultaneous combination of each — and in fact (5,3) can be combination (superposition) in an infinite number of bases.

Indeed, it now looks like this strange feature of quantum theory can be described naturally from a mathematical perspective! Vector Space bases furnish a great example!

**********************************************************

Principles of quantum theory

December 22, 2013 Leave a comment

One beautiful summer morning I spent a couple hours in a park reflecting on what I know about quantum mechanics and thought to sketch it out from memory. (A good brain exercise to recapture things you learned and admire.) This note is an edited summary of my handwritten draft (without too much math).

Being a big subject, I will stick to some basic ideas (or principles) of quantum theory that may be worth noting.

Two key concepts are that of a `state’ and that of an ‘observable’.

The former describes the state of the system under study. The observable is a thing we measure. So for example, an electron can be in the ground state of an atom – which means that it is in `orbital’ of lowest energy. Then we have other states that it can be in at higher energies.

The observable is a quantity distinct from a state and one that we measure. Such as for example measuring a particle’s energy, its mass, position, momentum, velocity, charge, spin, angular momentum, etc.

QM gives us principles / interpretations by how states and observables can be mathematically described and how they relate to one another. So here is the first principle.

Principle 1. The state of a system is described by a function (or vector) ψ. The probability density associated with it is given by |ψ|².

This vector is usually a mathematical function of space, time (sometime momentum) variables.

For example, f(x) = exp(-x^2) is one such example. You can also have wave examples such as g(x) = exp(-x^2) sin(x) which looks like a localized wave (a packet) that captures both being a particle (localized) and a wave (due to the wave nature of sin(x)). This wave nature of the function allows it to interfere constructively or destructively with other similar functions — so you can have interference! In actual QM these wavefunctions involve more variables that one x variable, but I used one variable to illustrate.

Principle 2. Each measurable quantity (called an ‘observable’) in an experiment is represented by a matrix A. (A Hermitian matrix or operator.)

For example, energy is represented by the Hamiltonian matrix H, which gives the energy of a system under study. The system could be the hydrogen atom. In many or most situations, the Hamiltonian is the sum of the kinetic energy plus the potential energy (H = K.E. + V).

For simplicity, I will treat a measurable quantity and its associated matrix on equal footing.

From matrix algebra, a matrix is a rectangular array of numbers – like, say, a square array of 3 by 3 numbers, like this one I grabbed from the net:

https://i0.wp.com/www.mathsisfun.com/algebra/images/matrix-3x3-det-a.gif

Turns out you can multiply such things and do some algebra with them.

Two basic facts about these matrices is:

(1) they generally do not have the commutative property (so AB and BA aren’t always equal), unlike real or complex numbers,

(2) each matrix A comes with `magic’ numbers associated to it called eigenvalues of A.

For example the matrix

http://upload.wikimedia.org/math/4/4/c/44c1569404535221ffac85abef04e4cd.png

(called diagonal because it has all zeros above and below the diagonal) has eigenvalues 1, 4, -3. (When a matrix is not diagonal we have a procedure for finding them. Study Matrix or Linear Algebra!)

Principle 3. The possible measurements of a quantity will be its eigenvalues.

For example, the possible energy levels of an electron in the hydrogen atom are eigenvalues of the associated Hamiltonian matrix!

Principle 4. When you measure a quantity A when the system is in the state ψ the system `collapses’ into an eigenstate f of the matrix A.

Therefore the system makes a transition from state ψ to state f (when A is measured).

So mathematically we write

Af = af

which means that f is an eignstate (or eigenvector) of A with eigenvalue a.

So if A represents energy then `a’ would be energy measurement when the system is in state f.

For a general state ψ we cannot say that Aψ = aψ. This eigenvalue equation is only true for eigenstates, not general states.

Principle 5. Each state ψ of the system can be expressed as a superposition sum of the eigenstates of the measurable quantity (or matrix) A.

So if f, g, h, … are the eigenstates of A, then any other state ψ of the system can be expressed as a superposition (or linear combination) of them:

ψ = bf + cg + dh + …

where b, c, d, … are (complex) numbers. Further, |c|^2 = probability ψ will `collapse’ into the eigenstate g when measurement of A is performed.

These principles illustrate the indeterministic nature of quantum theory, because when measurement of A is made, the system can collapse into any one of its many eigenstates (of the matrix A) with various probabilities. So even if you had the ‘exact same’ setup initially there is no guarantee that you would see your system state change into the same state each time. That’s non-causality! (Quite unlike Newtonian mechanics.)

Principle 6. (Follow-up to Principles 4 and 5.) When measurement of A in the state ψ is performed, the probability that the system will collapse into the eigenstate vector φ is the dot product of Aψ and φ.

The latter dot product is usually written using the Dirac notation as <φ|A|ψ>.  In the notation above, this would be same as |c|^2.

Next to the basic eigenvalues of A, there’s also it’s `average’ value or expectation value in a given state. That’s like taking the weighted average of tests in a class – with weights assigned to each eigenstate based on the superposition (as in the weights b, c, d, … in the above superposition for ψ). So we have:

Principle 7. The expected or average value of quantity A in the state described by ψ is <ψ|A|ψ>.

In our notation above where ψ = bf + cg + dh + …, this expected value is

<ψ|A|ψ> = |b|^2 times (eigenvalue of f)  + |c|^2 times (eigenvalue of g) + …

which you can see it being the weighed average of the possible outcomes of A, namely from the eigenvalues, each being weighted according to its corresponding probabilities |b|^2, |c|^2, … .

In other words if you carry out measurements of A many many times and calculate the average of the values you get, you get this value.

Principle 8. There are some key complementary measurement observables. (Classic example: Heisenberg relation QP – PQ = ih.)

This means that if you have two quantities P and Q that you could measure, if you measure P first and then Q, you will not get the same result as when you do Q first and then P. (In Newton’s mechanics, you could at least in theory measure P and Q simultaneously to any precision in any order.)

Example, position and momentum are complementary in this respect — which is what leads to the Heisenberg Uncertainty Principle, that you cannot measure both the position and momentum of a subatomic particle with arbitrary precision. I.e., there will be an inherent uncertainty in the measurements. Trying to be very accurate with one means you lose accuracy with they other all the more.

From Principle 8 you can show that if you have an eigenstate of the position observable Q, it will not be an eigenstate for P but will be a superposition for P.

So `collapsed’ states could still be superpositions! (Specifically, a collapsed state for Q will be a superposition, uncollapsed, state for P.)

That’s enough for now. There are of course other principles (and some of the above are interlinked), like the Schrodinger equation or the Dirac equation, which tell us what law the state ψ must obey, but I shall leave them out. The above should give an idea of the fundamental principles on which the theory is based.

Have fun,
Samuel Prime

The 21 cm line of hydrogen in Radio Astronomy

June 30, 2013 2 comments

 

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!).File:Galaxy NGC 1232.jpg

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

June 30, 2013 6 comments

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 C6H6. 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

 

https://i0.wp.com/www.soundswitheinstein.com/wp-content/uploads/2012/12/einstein-guitar.jpg

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, …

June 23, 2013 2 comments

The volume of a sphere of radius R in 2 dimensions is just the area of a circle which is π R2. (The symbol π is Pi which is 3.1415….)

The volume of a sphere of radius R in 3 dimensions is (4/3) π R3.

The volume of a sphere of radius R in 4 dimensions is (1/2) π2 R4.

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) Rn —————– (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) rn-1 dz

where here r2 + z2 = R2 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) Rn ∫ cosnθ 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) Rn π1/2 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) = πn/2 Rn / 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) π2 R5.

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?!!

Earth’s Magnetic Field

August 18, 2012 Leave a comment

 

I know just a little about the Earth’s magnetic field – also called the geomagnetic field. The following are from some notes I wrote a few years ago and came across lately (thought maybe worthwhile sharing in my own words). My notes were based on: Ency. Britannica; Wiki article on geomagnetic field; and ‘chapter 3′ of a physicist’s notes (whose name I’m missing).

1. Magnetic north and Earth’s true north aren’t the same!

2. In fact, it is the magnetic south pole that’s closer to the Earth’s north, by something like 11 degrees. (That is called ‘magnetic declination’, the angle difference from true north.) For precise navigation this 11 degrees could be taken into account.

3. The magnetic field lines (usually written as B in physics) start from magnetic north and end at magnetic south. (At least, that is the convention.) Magnetic fields affect only charged particles (like electrons and protons). These particles move along the field lines by spiraling around them (like a coiled wire).

(They go back and forth. The reason they spiral in doing so is explained by the magnetic force being F = q v x B, where q is the charge on the particle, v is its velocity, and B is the magnetic field. The force is always perpendicular to B and v, which is why they spiral.)

4. The Chinese appear to have been the first to discover the geomagnetic field in their effort to perfect their navigation technology. (About 1100′s AD or so.) Later Sir Edmond Halley (of Halley’s comet) mapped the magnetic field.

5. It was believed 100s of years ago that the geomagnetic field had extra-terrestrial origin. It was the brilliant mathematician Carl Friedrich Gauss (mid 1800s) who showed that the field actually had its origin from the Earth itself, and he gave a mathematical expression for it (using spherical harmonics).

6. The magnetic field of the earth can experience reversal — where magnetic north and south poles are interchanged. But this happens irregularly, from some 700,000 or so years to a few million years. (I think it’s still a mystery as to why this happened in the past.) These reversals are recorded in rocks that register the field’s direction in the past. And in turn, this has been valuable in determining the history and motion of plate tectonics – and discovery of the mid ocean ridges which affect continental motion.

7. The motion of molten iron in the core of the Earth is generally credited for the creation of the geomagnetic field — this is called the geodynamo theory of Sir Bullard (about 1940s-50s). (The Earth’s crust has its contributions too, but they are fairly smaller.) Scientists study the inside of the Earth in part from how the geomagnetic field behaves and changes.

8. There are still some mysteries about geomagnetism. For example, why do the magnetic poles move about 10 km each year?

Higgs particle seems to exist

 

http://t3.gstatic.com/images?q=tbn:ANd9GcSLG3M0gZsrCfGEzxQkwVFIn2IDC4rfY2Tvn_I0rEQAuwAOg0BTVwIt looks like there will be a major announcement on Wednesday July 4th regarding the existence of the long postulated particle the Higgs boson of the Standard Model. Its existence had been conjectured in the mid 1960s. I’ve read several articles including some from prominent blogs maintained by top physicists in the area, and here is briefly what I gather from them:

1. The Higgs particle exists.

2. But some of its observed decay properties are at variance with those predicted by the Standard Model (the leading theory in Particle Physics) — and apparently significant. Particularly in the decay rates of the Higgs into two photons and into WW particles. The other decays seem to agree with theory.

3. The evidence for the Higgs particle is still indirect — in that they cannot see it directly, but they do see its ‘footprints’.

4. The evidence for its existence would appear to have risen to around a 4 sigma level of confidence (or maybe a little more?), which is about 99.99% certainty. In December 2011, the data back then allowed for a 3 sigma certainty, which is 99.7% confidence level. To be considered a discovery in Physics the criterion is to attain 5 sigma confidence — 99.9999%. (See the little table below for what these sigmas, or standard deviations, are.)

5. The Higgs mass is around 125 GeV. (Pretty heavy! About 133 times  the proton mass!)

6. Several independent experiments (at CERN and in the US) seem to be in general agreement.

So it looks like the Higgs exists but not quite exactly the one predicted. However, the experts will make the announcement on Wednesday. It is notable that they chose to announce on July 4th as well as having invited 5 key physicists, including Peter Higgs (shown in the photo here), to the event.

—————–
What do these sigma’s mean?  Basically, statistical degrees of certainty. Briefly,
1 sigma = 67% certainty;
2 sigma = 95% certainty;
3 sigma = 99.7% certainty (considered good evidence but not a discovery);
4 sigma = 99.99% certainty
5 sigma = 99.9999% certainty (considered ‘certain’ or a discovery).

 

Categories: News, Physics, Science

Errors in superluminal neutrino experiment

February 24, 2012 Leave a comment

Just as several physicists had suspected that something is wrong with the OPERA experiment (back in November 2011) that concluded that neutrinos have traveled faster than light, it now looks like their suspicions were correct.  Two major errors seem to have surfaced in the experiment at CERN — namely, two problems with the GPS synchronization system used and another problem that had to do with fiber optic cables used in transmitting the GPS signals.  This was reported by NATURE Magazine.  Einstein must be smiling in his grave.

The next issue to look forward to is whether the signals announced by CERN in December 2011 to be possible signs of the Higgs boson will in fact be confirmed to a 5 sigma level of certainty. (The data in December gave about a 3 sigma level of confidence.)  And how about supersymmetry (SUSY)? Still a no-show.

Categories: News, Physics, Science

Higgs boson announcement by LHC team

December 13, 2011 Leave a comment

The two most intriguing results from two independent experiments — ATLAS & CMS — at the Large Hadron Collider seem to be the decays of the Higgs into 2 photons, and the Higgs into W’s and Z’s & leptons. Their results are quite close enough to make it very interesting. But they are not conclusive proofs — 5 sigmas — to assure us of the Higgs’ existence. There may be other Higgs at work, beside only one (some predict 5 Higgs particles) — or the results could be attributable to some kind of new physics. We will need to wait into 2012 for the LHC to accumulate enough data to say for sure.  Today’s LHC announcement is not much different from what has been rumored: good & intriguing evidence but not conclusive.  I think we should prepare our minds for possible disappointments along the way, if that happens.

For authoritative blogs managed by particle physicists see: viXra Not Even WrongRésonaancesReference Frame.

Categories: News, Physics, Science

Physics literature inconsistent on dark energy

December 1, 2011 Leave a comment

In reading several articles on dark energy, I’ve encountered some that seem to suggest its ‘confirmation’ and others that suggest otherwise or uncertainty.  It’s status, as far as I’m concerned, is at the theoretical level.

There seems more agreement on dark matter, but there are physicists who do not subscribe to it. If supersymmetric (SUSY) particles do not show up at the Large Hadron Collider, then dark matter may be in trouble since many of its particles (called WIMPS) are conjectured to be of the SUSY kind. From what I’ve been reading, there’s growing doubt regarding supersymmetry today.

Categories: News, Physics, Science