You know nothing

It is common for physics grad students to remember the angular momentum operator . Many will even remember that it commutes with and we can therefore form a set of simultaneous eigenfunctions to describe a particle in angular momentum space. Some may even recall that the standard conventions yield the following eigenequations.

But do you know how to demonstrate that and are the eigenvalues for and ?

for Stache:

Particle Decay is a Random Phenomena: Part II

The formalism behind random decay is simple. Take to be the number of particles and the probability that a particle will decay in a given time. We can then write the differential euquation for the number of partices as follows:

.

Where represents the "lifetime" of the particle. Upon integrating we arrive at the familiar characteristic of exponential decay

Where this math gets interesting is when a particle can decay in multiple ways. Each additional decay channel should decrease the overall "lifetime" of the particle. The savvy observer notices that "lifetimes" add like resistors in parallel.

Of course determining why the different channels have different decay rates involves physics, namely:

.

See you next time when put the pion together with this equation and explore the physics of branching ratios.

Charged Pion Decay

Pions are the lightest hadrons and therefore the charged pion can only decay into a neutral pion (beta decay) and/or lighter leptons (electrons and muons). The decay must conserve charge and lepton number; since we need a final state with a charge carrying lepton we must also have the associated neutrino to conserve lepton number. Therefore the weak force is the only mechanism by which the decay may proceed. Thus we arrive at the first order diagram for charged pion decay is as follows (beta decay would manifest itself as a radiating from the blob):

The first vertex is indicated with a blob because the pion is a composite particle and the interaction of the constituent quarks is not fully understood. The outgoing particles are an anti-lepton and its associated neutrino (either a positron or a muon). From the diagram we see that the physics of the decay is independant of which lepton we choose, this independance is known as lepton universality. By examining pion decays and measuring the ratio of electron to muon decays we can test the concept of universality.

Čerenkov Radiation

This phenomena, aside from providing the eerie lighting in this picture from Idaho National Lab, is a central tool for many particle physics experiments.  Actually, it's so important, it won Chernkov the Noble prize in 1958.  We recall that it's analogous to a sonic boom, but for light, and we can explain it in purely in terms of classical electrodynamics.

The derivation is in full in Jackson, but I'll sketch it here.  We consider a charged particle moving through a medium, where all the distant interactions of the particle with the atoms in the material are represented by a macroscopic dielectric constant .   Taking the Fourier transforms of the E&M wave equations, we get the E and B fields as follows:

Then, after integrating in k and recognizing the modified Bessel function, we get three field components in terms of  , in each direction, where here 1 is parallel to the particle's velocity.  I'll skip straight to the limit where    (a and b are impact parameters, and lambda is  ), which gives:

We integrate over frequency to get energy over distance , which has Jackson explains is done rather elegantly by finding the energy flow through a cylinder radius a.  After this integration and applying our limit, we get an expression which is multiplied by:

If lambda has a positive real part, this vanishes at large distance as the energy is deposited near the path.  If lambda is purely imaginary, the exponential is unity and the expression is independent of scattering distance a, so some of the energy escapes to infinity as radiation!  When is lambda imaginary?  When epsilon is real (there is little absorption) and  

 .  

Or, the speed of the particle is greater than the phase velocity of the E&M fields in the material.  That's it!  

We've now seen that with the right circumstances, namely the speed of the particle is greater than the phase velocity in the material, some of the energy of the particle escapes as "Cherenkov" radiation.  You can go on to calculate the angle of emission and the photon yields, but I'll leave that to another post in which I'll describe the practical applications of this effect.

Source: Jackson's Classical Electrodynamics

Better Know a Particle: The Pion

Yukawa proposed a theory of strong interactions involving a "meson mediator particle" in 1935. The following year the physics community thought they had discovered the particle but in fact they have found the mu meson. Eventually in 1947 experiments conducted in the Pyrenees and the Andes mountain ranges discovered this elusive particle which have come to know as the pion.

The reason for the mountains was simply that high energy particle accelerators were not a common tool for physics yet. The Cosmotron was not powered up until 1953 and the Bevatron wouldn't see beam till the following year. So physicists had to hike up a mountain in the hopes of seeing a pion created by cosmic collisions in the atmosphere.

Jumping forward a few decades we now know plenty about the pion. There are three flavours:. With identical masses for the charged flavours of approximately 140 MeV/c/c and slightly lower mass for the neutral flavour. The charged pions are each other's antiparticles but the discussion of the neutral pion's antiparticle is best left to another post. Pions are bound states of the lightest quarks, and as mesons are a quark-antiquark pair ().

Finally pions are unstable particles. The neutral pion decays electromagnetically into two photons with a mean lifetime around . The decay of the charged pion is very interesting and follows the form where the neutrino has opposite lepton number of the muon. Not only is this decay a weak decay and thus on a much slower timescale (26ns) but naively one would expect to be a more common decay mode due to larger phase space. The topic of why the muon channel dominates is again a topic for another post.

Links to Experiments and Results about pions:

http://pen.phys.virginia.edu/

http://pienu.triumf.ca/

http://prl.aps.org/abstract/PRL/v103/i5/e051802

Particle Decay is a Random Phenomena: Part III

Now that we understand what it means for a process to be random let's delve into the mathematics. The most important distribution to understand is the Poisson distribution. The probability of exactly n Poisson occurrences during a time interval where were expected is:

To satisfy the Poisson conditions events must be occurring with an average rate and be independent of each other, decaying pions certainly qualify. There are some interesting and useful features of this distribution. For instance the mean ()and the variance () are equal to . That means the standard deviation of a Poisson distributed data set is .

Say we measure 1,000,000 pions decays. We now want to state our measurement of the branching ratio (). How do we determine the statistical error of our measured value ()? Since we know the decays obey Poisson statistics we simply use the features we learned above combined with standard statistical estimation rules:

http://en.wikipedia.org/wiki/Standard_error_(statistics)

The important thing to note is that the "one over square root of N" has nothing to do with Poisson statistics but rather is a feature of sampling error.

Particle Decay is a Random Phenomena: Part I

The standard model tells us that all particles of the same type are indistinguishable. Therefore when describing a system of multiple particles (of the same time) we must take great care to insure our particles are treated as indistinguishable. Accidentally treating particles as identical can cause serious miscalculations. For example (not physics but very illustrative):

For the following questions assume I have two children, child A and child B: the probability of any individual child being a boy is 1/2, the probability of the child being born on a Tuesday is 1/7, etc. For each question I will apply a different set of constraints. Only the constraints mentioned in the question affect that question.

Q1: What is the probability I have two boys?

A1: 1/4.

Q2: Child A is a boy. What is the probability I have two boys?

A2: 1/2.

Q3: At least one of my children is a boy. What is the probability I have two boys?

A3: 1/3.

Q4: At least one of my children is a boy born on Tuesday. What is the probability I have two boys?

A4: 13/27.

The reason for the counterintuitive answers to questions 3 and 4 is that we don't know which child (A or B) satisfies the constraint. Since it could be either the probability of overlap changes the result. The more (less) probable the overlap the closer the answer is to 1/3 (1/2).

But what does this have to do with particle decay?

Quantum Mechanics says there are no hidden variables. If the time a particle had been "alive" affected the probability it would decay in the future then there would have to be some hidden variable to fully describe the particle! If we could look at a particle and tell how long it had been alive then particles would not be indistinguishable!

Particle Decay is a Random Phenomena: Prelude

Before we begin, a small exercise:

The probability for a pion to decay in 26ns is .

Q: We make a pion. What is the probability that it decays during the next 26ns?

A: .

Q: We make a pion. It lives for 26ns. What is the probability that it decays during the next 26ns?

A: .

Q: We find a pion sitting on the sidewalk. What is the probability that it decays during the next 26ns?

A: .

Often people get confused when thinking about particle decay. The reason for the confusion is that particles decay randomly and humans inherently do not understand randomness. Further confounding the problem is the human desire to treat everything else like a human. To this extent we say particles have a lifetime, which implies the particle has an age. But in fact the probability that a particle would have decayed in the previous second is the exact same as the probability that it will decay in the next second. The pion could be 1 million years old and the probability of it decaying in the next 26ns is still . However that is one unlikely pion to still be "alive".

Mathematically speaking: ; is the number of pions, and is the decay rate of pions.

Radiation Length

As part of the target length analysis thing I'm working on I use the radiation length of several different materials. I've been reading the Particle Data Group's review on the passage of particles through matter and I thought I'd toss out a short summary on radiation length.

High energy electrons (I'm not sure exactly what they classify as high energy) mostly lose energy through bremsstrahlung, while high energy photons mostly lose their energy by electron-positron pair production. The radiation length is a characteristic "length" for these processes in a material, although it is usually measure in g / cm^-2 . You get this by multiplying a length by the density of the material. The radiation length is defined for electrons as the length over which the electron loses all but 1/e of its energy, while for photons it is defined as 7/9 of the mean free path.

On a similar note, low energy electrons lose energy through various types of scattering as well as ionization. Since the ionization loss rises logarithmically in energy and the bremsstrahlung increases linearly, there is a critical energy where the energy lost to ionization and the energy lost to bremsstrahlung are equal. For most materials, this critical energy is in the tens of MeV, so it seems that radiation length can be a useful parameter for most of the energies we see at JLab.

Stern-Gerlach

Everyone remembers the Stern-Gerlach experiment; we usually recall it in the context of the discovery of quantized intrinsic spin.  However, I like very much the treatment of Sakurai in his Quantum Mechanics text, which uses Stern-Gerlach as the introduction to shock us into a quantum mechanical way of thinking.

The setup is familiar: Heat up silver atoms which escape through a hole in the oven.  Collimate the beam and send it though an inhomogeneous magnetic field; the force in the z direction on the atom is .  Since the atoms in the oven were randomly oriented, there's no preferred orientation of , so classically we expect to see a vertical splay of particles coming out of the magnetic field.  Of course, we instead see two distinct components, scattered up and down, which leads us to the idea of quantized spin angular momentum.

The fun part begins with sequential Stern-Gerlach setups, adding the ability to rotate the subsequent apparatus on its side to act in the x direction instead of the z. The picture below, stolen from Wikipedia which obviously it stole from Sakurai, shows the results.

The first result makes sense; we removed the minus z portion of the beam, so it doesn't recur.  The second, in which the plus z portion is now split 50-50 into plus and minus x portions, is interesting; maybe 50% of the split beam was z plus and x plus while the other 50% was z plus and x minus?  The third result is the doozy.  We get z minus out at the end, but didn't we remove it in the first splitting?  Apparently the x measurement in the middle destroys our previous information on the z direction of the spin!  We can't know both the spin in z and in x simultaneously; this is obviously not a spinning top.

Sakurai then calms our confusion with an analogy to polarized light, with the x and z spin directions above corresponding to zero and 45 degree polarized light.  In that context, this result would be right at home.  In E&M we write the 45 degree polarization in terms of a linear combination of the 0 degree coordinate vectors, x and y.  For spin, we must turn our thinking to an abstract spin space, where the base vectors Sz+ and Sz- take the place of base vectors x and y in physical space. The Sx spin directions can then be swapped into the idea of the 45 degree polarized light, so should be expressed in terms of our Sz+ and Sz- base vectors:



The missing piece is Sy, the last spin direction, which we correspond to circularly polarized light in this analogy.  Circularly polarized light is expressed as the same combination of base x and y vectors, but with the y portion 45 degrees out of phase.  This now brings imaginary numbers easily into play, as we express a light wave in exponential notation instead of cosines and pull out .  In our corresponding spin vector space we have now been lead to:



So, in a few paragraphs we've seen not only the weirdness of quantum mechanical phenomena, but we've constructed a complex vector space directly out of our observations in Stern-Gerlach and an analogy with our understanding of polarized light.

Source: Sakurai's Modern Quantum Mechanics.