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## EPGY Physics: Summer Institute

### Singapore June 2007

Hello, Gary. Maybe you have already forgotten the question I posed to you a few weeks ago. It is ok..

Here I have another question to ask:
We all know that the radiation from a radioactive material has random in nature. Despite the fact that we can precisely determine the half-life of that material, we still do not know how many particles (alpha and beta) or how much EM wave (gamma ray) will be emitted in the next instant of time. It is because radiation is completely random and unpredictable..
My question is: is the probability of emitting one alpha or beta particle from a certain substance can be estimated by solving schrodinger's equation? Is the random nature of radiation similar to the random nature of a particle in quantum mechanics?
Yes, you can get the probabilities from the Schrodinger equation. And these probabilites directly relate to the half life of some substance (the higher the probability, the shorter the half life).
Basically you make a model of the potential of a nucleus and then calculate the probability for an alpha particle (say) to tunnel through that barrier. Of course in practice it works a little backwards. From the observed half-life, you know the probability and from that infer the potential barrier. Once you deal with large nuclei (usually the ones that yield alpha particles) it is too difficult to write down anything terribly specific about the potential from first principles.
Hello, Gary. It seems that I cannot help not to miss you:P
I have an interesting question to pose to you here, as following:
If the atom model we have studied is correct, then there should be vacuum between the fast-moving electrons and the nuclei (in between there is absolutely nothing). From the diameter of whole atoms and that of the nuclei, we can obtain that the vacuum occupies over 90% of the space in an atom. If everything is made up of atoms, for example human beings, as a result we should be transparent...or not that solid like what we are now. Can quantum mechanism explain this?
I am *finally* getting around to answer your question. I have been trying to catch up and am almost there.
Let's see there are two tracks I want to discuss in answering this

1) Recall that we need to be careful in discussing quantum object's trajectories, especially electrons in atoms. Though it is true that if you *measured* the electrons and nuclei that their volume would be very very VERY small compared to the volume of the atom. (Atom has a radius of about 10^-10 m and nucleus about 10^-15 m (and the electron much smaller than that)) we can not say necessarily that an electron is in one place and not the other (unless measured). [Though this is based upon most interpretations of qm, the Bohm interpretation would say there is really all that space there]. Ok but now to the meat of the question.

2) The reason things are opaque has to do with the interaction of light and matter. This interaction is often very tricky to work out but the basic idea is that light couples to electric charges and there is plenty of charge around an atom. A light wave will interact with an electron rather readily. It will "shake" and electron and that electron will re-radiate light. (I need to be careful as this is a semi-classical description). If you think about EM waves they are extended in space then they will envelop an atom and interact readily. But the main optical properties of materials is explained in its bulk properties.

A contrast can help explain this a little (and has to do a bit with cosmology). In the early universe the energy and pressure of the universe were so high that atoms could not form (if an electron bound with a proton, something would come by quickly and kick it away). This phase of the universe is known as the plasma phase (a gas of free electrically charged particles). In this phase, light could not propagate far since there was so much charge around- the universe was opaque. One the average energy of the universe went below 13.6 eV, (about 370,000 after the big bang) Hydrogen atoms could form and remain stable (particles did not have much energy to kick the electrons out). At this point, light could travel much further because, on the whole, there was not much electrical charge (now you had a gas of neutral H and He). The universe became transparent at this point and light could travel long distances. This light that was freed is the Cosmic Microwave background observed today (after it has cooled down to 2.7 K). So it is the density of charges around that relates to how light propagates (or doesn't) through material. [Glass is an interesting substance in this regard]

### Singapore Dec 2004

• 12/21/2004: Lerh Feng Low asks:
• I hope you're enjoying the beaches of Hawaii :) I've one small question, something I ought to have thought of, but I didn't: Stephen Hawking described String theory as "A theory of physics in which particles are described as waves of strings; unites quantum mechanics and general relativiy (The Universe in a Nutshell)". If it does unite those two theories, why isn't it hailed as the theory of quantum gravity? Thank you!
Well, I am back in my messy office, no more sunsets....sigh. Yeah, Hawking fell into the trap that those at the end of the 19th Century fell in (he later corrected himself). He mentioned many years back that physics is coming to an end...once a Theory of Everything (TOE) was developed, which he thought was right around the corner. Well that is rubbish. Even with such a theory, there is still plenty of physics to do and discover.

Back to your question...The thing is is that string theory (with one caveat I will mention in a bit) has not predicted anything about the nature of the universe or the elementary particles we observe. It is a mathematically consistent theory (with many aspects still to be explored) however it does not match up with observations. It is an interesting story in how physicists perception of this problem is changing in the past several years (I will post or attach an interesting description by Susskind, yes him again, on this matter).

When people originally started to explore string theory as a TOE (1980s) the thought was that they would have this theory describing strings in a certain number of dimensions and the particles of the Standard Model (remember all the quarks and leptons we discussed?) would come right out of the theory, masses and all. Well after much gnashing of teeth and broken pencils they had many theories of strings yet none gave what we observe (like the electron is a spin 1/2 object with mass 0.511 MeV/c^2, electric charge -e, etc.). This is still pretty much the story (theories go by names such as E8 X E8, O(32), Type I, Type IIA, etc.) to this day. So in some sense there was a new problem, there were too many string theories! Which one was correct? None seemed to match observations, and if count the total number of such theories it is an immense number. Except in the 90s Ed Witten (a big name in physics) observed that these can be seen as related, that all the theories can be described as different sectors of what has become called M-Theory (adds a 11th dimension to unify them). Well this was a welcome addition but it still did not answer which representation corresponded to our universe.

So some termed this dilemma as going from a Theory if Everything to a Theory of Anything - one which it was nearly impossible to find which vacuum (or version if you will) corresponded to reality. Some physicists see this as a good thing and that string theorists ought to get off of their high horse in claiming that string theory will solve everything, including a cure the common cold. The new interpretation that is taking shape is that here is a sophisticated mathematical theory describing string (and other dimensional objects) interactions and that among the 10^100 or so possibilities, one is ours. We just have to find it. Not easy to do. Ok, enough about string theory. Oh the one thing string theory claims to have derived is the entropy of the horizon of a black hole (recall our discussion of such). (If you want to see a great new book on this topic, not as elementary as Greene's "The Elegant Universe" but more to the point probably good for those who have an undergraduate background in physics, "Out of This World: Colliding Universes, Branes, Strings, and Other Wild Ideas of Modern Physics " by Stephen Webb. I thoroughly enjoyed this book).

Ok back to your question (yet again). So basically string theory is not a TOE as it does not agree with experiment yet. There are some problems with string theory that those who believe that general relativity is a fundamental theory think cause it not to be a TOE. For one, when quantizing a string (making a quantum theory of strings) the background metric is taken to be flat, but why? GR tells us that near massive objects the curvature of spacetime is not flat. Another approach to Quantum gravity (QG) is to quantize spacetime itself. The current theory of this approach is called Loop Quantum Gravity. Thus there are two camps competing to devise a QG theory and they argue quite a bit why theirs is the better approach. However, neither can yet give details about our universe (except for BH entropy which both claim to derive). It will probably be a long time before such a theory will be universally accepted. (Though there are some current experiments that may be able to supply some evidence in the near term: some theories predict that gravity will deviate from 1/r^2 over short distances and several researchers are attempting to measure such deviations - no results yet).

Ok my fingers are tired, let me know if this is enough or if there are further questions.

• Hello Gary,
This is James from the Physics EPGY camp in Singapore from 29 November to 10 December. I was wondering about the effects on the frequency of EM waves due to the Doppler Shift. The doppler shift says that when a planet is moving away from say a star, the frequency of the waves decreases, and so it appears to be redshifted, whereas when it is moving towards, the frequency of the waves increase, and so it appears to be blueshifted.

Say a photon from a star reaches a planet that is receding in 'A', and reaches a planet that is approaching in 'B'. The photon in A appears to have a lower frequency, while that in B appears to have a higher frequency. If I remember correctly, from one of your lectures, the formula for finding out the energy of the photon is E=hf, where E is the energy of the photon, h is planck's constant and f is the frequency.

So in A, the photon will appear to have less energy than when it was released, whereas in B, the photon appears to have more energy than when it was released. So where does this energy come from and go to?

What am I missing out here? Is E=hf the full formula? Some websites say that E refers to the quantumn energy of a photon, which i don't understand.

I did some searching on the internet and only managed to find gravitational redshift which seems to seems to support this by virtue of the photon actually losing energy to become redshifted. I am not sure but problem seems relativistic in a sense as the energy is different as view by the planets and the sun.

Excellent question James! Give me a minute to give a good response.

Ok the simple answer is that energy is not a Lorentz invariant. What I mean by that is that, if you recall, the energy of an object is different in different IRFs. What is the same in both IRFs is the analog of the interval, in this case the magnitude of the energy-momentum 4-vector (qm). More specifically,

qmqm = (mc2)2 = E2 - (pc)2.

The term (mc2) is a Lorentz invariant yet E and p may change in different IRFs. (Obviously the momentum of a massive particle will vary in different IRFs, thus E must also vary).

For a photon the same applies, however the mass is zero. So what is a Lorentz invariant is

qmqm = 0 = E2 - (pc)2.
The energy of the photon need not be the same in different IRFs.

The quantum energy of a particle is hf just as you say. And, just as a massive particle's energy can change in different IRFs, so can a photon's. However, in one particular frame energy is conserved, it is just the amount of energy that is being conserved may be different in different frames.

This may not be an entirely adequate explanation, if not let me know and I will amplify.