Friday, January 13, 2012
Charge, Parity and Time Reversal (CPT) Symmetry
 From our everyday experience, it is easy to conclude that nature obeys the laws of physics with absolute consistency. However, several experiments have revealed certain cases where these laws are not the same for all particles and their antiparticles. The concept of a symmetry, in physics, means that the laws will be the same for certain types of matter. Essentially, there are three different kinds of known symmetries that exist in the universe: charge (C), parity (P), and time reversal (T). The violations of these symmetries can cause nature to behave differently. If C symmetry is violated, then the laws of physics are not the same for particles and their antiparticles. P symmetry violation implies that the laws of physics are different for particles and their mirror images (meaning the ones that spin in the opposite direction). The violation of symmetry T indicates that if you go back in time, the laws governing the particles change.
 There were two American physicists by the names of Tsunng-Dao Lee and Chen Ning Yang suggested that the weak interaction violates P symmetry.  This was proven by an experiment which was conducted with radioactive atoms of colbalt-60 that were lined up and introduced a magnetic field to insure that they are spinning in the same direction. In addition, it was also found that the weak force also does not obey symmetry C. Oddly enough, the weak force did appear to obey the combined CP symmetry. Therefore the laws of physics would be the same for a particle and it’s antiparticle with opposite spin.
Surprise, surprise! There was a slight error in the previous experiment that was just mentioned. A few years later, it was discovered that the weak force actually violates CP symmetry. Another experiment was conducted by two physicists named Cronin and Fitch. They studied the decay of neutral kaons, which are mesons that are composed of either one down quark (or antiquark) and a strange antiquark (or quark). These particles have two decay modes where one will decay much faster than the other, even though they all have identical masses. The particles with the longer lifetimes will decay into three pions (denoted with the symbol π0), however the kaon ‘species’ with the shorter lifetimes will only decay into two pions. They had a 57 foot beamline, where they only expected to see the particles with slower decay rate at the end of the beam tube. In astonishment, one out of every 500 decays where from the kaons species that had a shorter lifetime. The main conflict with seeing the short-lived mesons at the end of the beam tube is because they are traveling relavistic speeds and therefore ignoring the time dilatationthat they are supposed to undergo. Thus, the experiment has shown that the weak force causes a small CP violation that can be seen in kaon decay.

Charge, Parity and Time Reversal (CPT) Symmetry

From our everyday experience, it is easy to conclude that nature obeys the laws of physics with absolute consistency. However, several experiments have revealed certain cases where these laws are not the same for all particles and their antiparticles. The concept of a symmetry, in physics, means that the laws will be the same for certain types of matter. Essentially, there are three different kinds of known symmetries that exist in the universe: charge (C), parity (P), and time reversal (T). The violations of these symmetries can cause nature to behave differently. If C symmetry is violated, then the laws of physics are not the same for particles and their antiparticles. P symmetry violation implies that the laws of physics are different for particles and their mirror images (meaning the ones that spin in the opposite direction). The violation of symmetry T indicates that if you go back in time, the laws governing the particles change.

There were two American physicists by the names of Tsunng-Dao Lee and Chen Ning Yang suggested that the weak interaction violates P symmetry. This was proven by an experiment which was conducted with radioactive atoms of colbalt-60 that were lined up and introduced a magnetic field to insure that they are spinning in the same direction. In addition, it was also found that the weak force also does not obey symmetry C. Oddly enough, the weak force did appear to obey the combined CP symmetry. Therefore the laws of physics would be the same for a particle and it’s antiparticle with opposite spin.

Surprise, surprise! There was a slight error in the previous experiment that was just mentioned. A few years later, it was discovered that the weak force actually violates CP symmetry. Another experiment was conducted by two physicists named Cronin and Fitch. They studied the decay of neutral kaons, which are mesons that are composed of either one down quark (or antiquark) and a strange antiquark (or quark). These particles have two decay modes where one will decay much faster than the other, even though they all have identical masses. The particles with the longer lifetimes will decay into three pions (denoted with the symbol π0), however the kaon ‘species’ with the shorter lifetimes will only decay into two pions. They had a 57 foot beamline, where they only expected to see the particles with slower decay rate at the end of the beam tube. In astonishment, one out of every 500 decays where from the kaons species that had a shorter lifetime. The main conflict with seeing the short-lived mesons at the end of the beam tube is because they are traveling relavistic speeds and therefore ignoring the time dilatationthat they are supposed to undergo. Thus, the experiment has shown that the weak force causes a small CP violation that can be seen in kaon decay.

(Source: aps.org)

Thursday, August 25, 2011
The Stern-Gerlach Experiment
In 1922 at the University of Frankfurt in Frankfurt, Germany, Otto Stern and Walther Gerlach sent a beam of silver atoms through an inhomogeneous magnetic field in their experimental device. They were taking a look at the new concept of quantized spin angular momentum. If indeed the spin associated with particles could only take on two or some other countable number of states, then the atoms transmitted through the other end of their machine should come out as two (or more) concentrated beams. Meanwhile if the quantum theory was wrong, classical physics predicted that the profile of a single smeared-out beam would result on the detector screen, due to the magnetic field deflecting each randomly spin-oriented atom a different amount on a continuous, rather than discrete, scale.
As you can see above, the results of the Stern-Gerlach experiment confirmed the quantization of spin for elementary particles.
Spin and quantum states
A spin-1/2 particle actually corresponds to a qubit
|ψ> = c1|ψ↑> + c2|ψ↓>
a wavefunction representing a particle whose quantum state can be seen as the superposition (or linear combination) of two pure states, one for each kind of possible spin along a chosen axis (such as x, y or z). The silver atoms of Stern and Gerlach’s experiment fit in this description because they are made of spin-1/2 particles (electrons and quarks, which make up protons and neutrons).
Significantly, the constant coefficients c1 and c2 are complex and can’t be directly measured. But the squared moduli ||c1||2 and ||c2||2 of these coefficients represent the probability that a particle in state |ψ> will be observed as spin up or down at the detector.
||c1||2 + ||c2||2 = 1 : it is certain that the particle will be detected in one of the two spin states.
That means when we pass a large sample of particles in identical quantum states through a Stern-Gerlach (S-G) machine and detector, we are actually measuring the probabilities that the particle will adopt the spin up or spin down states along the particular axis of the S-G machine. This follows the relative-frequency interpretation of probability, where as the number of identical trials grows large the relative frequency of an event approaches the true probability that the event will occur in any one trial.
By moving the screen so that either the up or down beam is allowed to pass while the the other is stopped at the screen, we are “polarizing” the beam to a certain spin orientation along the S-G machine axis. We can then place one or more S-G machines with stops in front of that beam and reproduce all the experiments analogous to linear polarization of light.

The Stern-Gerlach Experiment

In 1922 at the University of Frankfurt in Frankfurt, Germany, Otto Stern and Walther Gerlach sent a beam of silver atoms through an inhomogeneous magnetic field in their experimental device. They were taking a look at the new concept of quantized spin angular momentum. If indeed the spin associated with particles could only take on two or some other countable number of states, then the atoms transmitted through the other end of their machine should come out as two (or more) concentrated beams. Meanwhile if the quantum theory was wrong, classical physics predicted that the profile of a single smeared-out beam would result on the detector screen, due to the magnetic field deflecting each randomly spin-oriented atom a different amount on a continuous, rather than discrete, scale.

As you can see above, the results of the Stern-Gerlach experiment confirmed the quantization of spin for elementary particles.

Spin and quantum states

A spin-1/2 particle actually corresponds to a qubit

|ψ> = c1> + c2>

a wavefunction representing a particle whose quantum state can be seen as the superposition (or linear combination) of two pure states, one for each kind of possible spin along a chosen axis (such as x, y or z). The silver atoms of Stern and Gerlach’s experiment fit in this description because they are made of spin-1/2 particles (electrons and quarks, which make up protons and neutrons).

Significantly, the constant coefficients c1 and c2 are complex and can’t be directly measured. But the squared moduli ||c1||2 and ||c2||2 of these coefficients represent the probability that a particle in state |ψ> will be observed as spin up or down at the detector.

||c1||2 + ||c2||2 = 1 : it is certain that the particle will be detected in one of the two spin states.

That means when we pass a large sample of particles in identical quantum states through a Stern-Gerlach (S-G) machine and detector, we are actually measuring the probabilities that the particle will adopt the spin up or spin down states along the particular axis of the S-G machine. This follows the relative-frequency interpretation of probability, where as the number of identical trials grows large the relative frequency of an event approaches the true probability that the event will occur in any one trial.

By moving the screen so that either the up or down beam is allowed to pass while the the other is stopped at the screen, we are “polarizing” the beam to a certain spin orientation along the S-G machine axis. We can then place one or more S-G machines with stops in front of that beam and reproduce all the experiments analogous to linear polarization of light.

Saturday, August 20, 2011
The mention of “spin"  of a particle is one that immediately triggers an intuitive, simple  visualization in the minds of nearly everyone: perhaps, a ball rotating  with some angular velocity about an internal axis. This is certainly  what was visualized by some of those originally peering into the work of  the young Wolfgang Pauli (he of the Exclusion Principle fame), but it is a short-lived analogy: the original name of the phenomena was a “two-valued quantum degree of freedom”, and it behaves only as such.
In a simple sense, elementary particles do experience a sort of spin — that is, an intrinsic angular momentum about their own respective axes. However, the analogy of the spinning  top comes to a halt when another characteristic of spin comes into play  — it can only exist in one of two possible orientations at any given  time. This is not a quality that is exhibited by any spinning top. This  is why physicists deem spin a “non-classical” degree of freedom —  another way of stating that it is a phenomena not described by any  macroscopic physical analogy.
Another peculiar property of  particle spin is that it can either be 0, or any half-integer value  (usually implied to be a multiple of ħ, the reduced Planck constant).  Stephen Hawking made a clever analogy is made to demonstrate the  counter-intuitiveness of particle spin: imagine a deck of playing  cards. A spin-0 particle behaves like a point, and appears similar in  every orientation. A spin-1 particle behaves like an ace: a rotation of  360˚ brings it to its original orientation. A spin-2 particle behaves  like a face card: one half-rotation brings it to its original  orientation. The peculiar point is that a spin-1/2 particle, for  example, does not behave in this way: it takes two complete revolutions to return to its original position.

A final remark on the concept of spin and its importance: the Standard Model of particle physics is fundamentally interconnected with spin. The way we categorize elementary particles — into fermions (“matter” particles) and bosons (“force” particles) — is based on the fact that the former are spin-1/2 particles, and that the latter are of spin-1.

The  intrinsic symmetry and anti-symmetry present with bosons and fermions,  respectively, are what allow for (somewhat) stable particles to form —  protons, neutrons, etc. — and without this completely outlandish and  truly bizarre phenomenon, the known universe would not exist as we know  it today. Spin is but one example of how ill-equipped humans are to  intuitively comprehend the fundamental processes that govern our  universe, and an example of how striking it is that we remain unfazed  while making deeper-penetrating progress.

The mention of “spin" of a particle is one that immediately triggers an intuitive, simple visualization in the minds of nearly everyone: perhaps, a ball rotating with some angular velocity about an internal axis. This is certainly what was visualized by some of those originally peering into the work of the young Wolfgang Pauli (he of the Exclusion Principle fame), but it is a short-lived analogy: the original name of the phenomena was a “two-valued quantum degree of freedom”, and it behaves only as such.

In a simple sense, elementary particles do experience a sort of spin — that is, an intrinsic angular momentum about their own respective axes. However, the analogy of the spinning top comes to a halt when another characteristic of spin comes into play — it can only exist in one of two possible orientations at any given time. This is not a quality that is exhibited by any spinning top. This is why physicists deem spin a “non-classical” degree of freedom — another way of stating that it is a phenomena not described by any macroscopic physical analogy.

Another peculiar property of particle spin is that it can either be 0, or any half-integer value (usually implied to be a multiple of ħ, the reduced Planck constant). Stephen Hawking made a clever analogy is made to demonstrate the counter-intuitiveness of particle spin: imagine a deck of playing cards. A spin-0 particle behaves like a point, and appears similar in every orientation. A spin-1 particle behaves like an ace: a rotation of 360˚ brings it to its original orientation. A spin-2 particle behaves like a face card: one half-rotation brings it to its original orientation. The peculiar point is that a spin-1/2 particle, for example, does not behave in this way: it takes two complete revolutions to return to its original position.

A final remark on the concept of spin and its importance: the Standard Model of particle physics is fundamentally interconnected with spin. The way we categorize elementary particles — into fermions (“matter” particles) and bosons (“force” particles) — is based on the fact that the former are spin-1/2 particles, and that the latter are of spin-1.

The intrinsic symmetry and anti-symmetry present with bosons and fermions, respectively, are what allow for (somewhat) stable particles to form — protons, neutrons, etc. — and without this completely outlandish and truly bizarre phenomenon, the known universe would not exist as we know it today. Spin is but one example of how ill-equipped humans are to intuitively comprehend the fundamental processes that govern our universe, and an example of how striking it is that we remain unfazed while making deeper-penetrating progress.

Friday, July 8, 2011
The qubit, in computing, is the quantum analog of a classical bit. Whereas a classical bit takes on a binary value of 1 or 0, a qubit can take on a superposition of these values, à la quantum mechanics:
|ψ> = α|1> + β|0>Thus the qubit |ψ> can be visualized as taking on any of the continuum of values on the Bloch sphere, pictured above, with eigenstates |1> and |0>.
Even though the qubit collapses to a given eigenstate upon observation, the analog nature of the qubit allows much more powerful computation than the digital equivalent in a traditional Turing machine. Shor’s Algorithm, for example, uses a series of qubits along with a quantum Fourier transform to quickly factor large semi-primes.
Factoring large semi-primes quickly would let someone break RSA encryption, and it can be done very fast (in polynomial time) on a quantum computer; on a traditional computer, the best algorithms we have still take exponential time, meaning it would take many centuries just to crack one modern encrypted message.

The qubit, in computing, is the quantum analog of a classical bit. Whereas a classical bit takes on a binary value of 1 or 0, a qubit can take on a superposition of these values, à la quantum mechanics:

|ψ> = α|1> + β|0>

Thus the qubit |ψ> can be visualized as taking on any of the continuum of values on the Bloch sphere, pictured above, with eigenstates |1> and |0>.

Even though the qubit collapses to a given eigenstate upon observation, the analog nature of the qubit allows much more powerful computation than the digital equivalent in a traditional Turing machine. Shor’s Algorithm, for example, uses a series of qubits along with a quantum Fourier transform to quickly factor large semi-primes.

Factoring large semi-primes quickly would let someone break RSA encryption, and it can be done very fast (in polynomial time) on a quantum computer; on a traditional computer, the best algorithms we have still take exponential time, meaning it would take many centuries just to crack one modern encrypted message.