Sunday, December 12, 2010

The Compact Muon Solenoid......

Coordinates: 46°18′34″N 6°4′37″E / 46.30944°N 6.07694°E / 46.30944; 6.07694
Large Hadron Collider (LHC)
LHC.svg
LHC experiments
ATLAS A Toroidal LHC Apparatus
CMS Compact Muon Solenoid
LHCb LHC-beauty
ALICE A Large Ion Collider Experiment
TOTEM Total Cross Section, Elastic Scattering and Diffraction Dissociation
LHCf LHC-forward
MoEDAL Monopole and Exotics Detector At the LHC
LHC preaccelerators
p and Pb Linear accelerators for protons (Linac 2) and Lead (Linac 3)
(not marked) Proton Synchrotron Booster
PS Proton Synchrotron
SPS Super Proton Synchrotron

View of the CMS endcap through the barrel sections. The ladder to the lower right gives an impression of scale.
......(CMS) experiment is one of two large general-purpose particle physics detectors built on the proton-proton Large Hadron Collider (LHC) at CERN in Switzerland and France. Approximately 3,600 people from 183 scientific institutes, representing 38 countries form the CMS collaboration who built and now operate the detector.[1] It is located in an underground cavern at Cessy in France, just across the border from Geneva.

Contents

Background

Recent collider experiments such as the now-dismantled Large Electron-Positron Collider at CERN and the (as of 2010) still running Tevatron at Fermilab have provided remarkable insights into, and precision tests of the Standard Model of Particle Physics. However, a number of questions remain unanswered.

A principal concern is the lack of any direct evidence for the Higgs Boson, the particle resulting from the Higgs mechanism which provides an explanation for the masses of elementary particles. Other questions include uncertainties in the mathematical behaviour of the Standard Model at high energies, the lack of any particle physics explanation for dark matter and the reasons for the imbalance of matter and antimatter observed in the Universe.

The Large Hadron Collider and the associated experiments are designed to address a number of these questions.

Physics goals

The main goals of the experiment are:
The ATLAS experiment, at the other side of the LHC ring is designed with similar goals in mind, and the two experiments are designed to complement each other both to extend reach and to provide corroboration of findings.

Detector summary

CMS is designed as a general-purpose detector, capable of studying many aspects of proton collisions at 14 TeV, the center-of-mass energy of the LHC particle accelerator. It contains subsystems which are designed to measure the energy and momentum of photons, electrons, muons, and other products of the collisions. The innermost layer is a silicon-based tracker. Surrounding it is a scintillating crystal electromagnetic calorimeter, which is itself surrounded with a sampling calorimeter for hadrons. The tracker and the calorimetry are compact enough to fit inside the CMS solenoid which generates a powerful magnetic field of 3.8 T. Outside the magnet are the large muon detectors, which are inside the return yoke of the magnet.




The set up of the CMS. In the middle, under the so-called barrel there is a man for scale. (HCAL=hadron calorimeter, ECAL=electromagnetic calorimeter)

CMS by layers


A slice of the CMS detector.
For full technical details about the CMS detector, please see the Technical Design Report.

The interaction point

This is the point in the centre of the detector at which proton-proton collisions occur between the two counter-rotating beams of the LHC. At each end of the detector magnets focus the beams into the interaction point. At collision each beam has a radius of 17 μm and the crossing angle between the beams is 285 μrad.
At full design luminosity each of the two LHC beams will contain 2,808 bunches of 1.15×1011 protons. The interval between crossings is 25 ns, although the number of collisions per second is only 31.6 million due to gaps in the beam as injector magnets are activated and deactivated.

At full luminosity each collision will produce an average of 20 proton-proton interactions. The collisions occur at a centre of mass energy of 14 TeV. It is worth noting that the actual interactions occur between quarks rather than protons, and so the actual energy involved in each collision will be lower, as determined by the parton distribution functions.

The first which ran in September 2008 was expected to operate at a lower collision energy of 10 TeV but this was prevented by the 19 September 2008 shutdown. When at this target level, the LHC will have a significantly reduced luminosity, due to both fewer proton bunches in each beam and fewer protons per bunch. The reduced bunch frequency does allow the crossing angle to be reduced to zero however, as bunches are far enough spaced to prevent secondary collisions in the experimental beampipe.

Layer 1 – The tracker


The silicon strip tracker of CMS.
Immediately around the interaction point the inner tracker serves to identify the tracks of individual particles and match them to the vertices from which they originated. The curvature of charged particle tracks in the magnetic field allows their charge and momentum to be measured.

The CMS silicon tracker consists of 13 layers in the central region and 14 layers in the endcaps. The innermost three layers (up to 11 cm radius) consist of 100×150 μm pixels, 66 million in total.
The next four layers (up to 55 cm radius) consist of 10 cm × 180 μm silicon strips, followed by the remaining six layers of 25 cm × 180 μm strips, out to a radius of 1.1 m. There are 9.6 million strip channels in total.
During full luminosity collisions the occupancy of the pixel layers per event is expected to be 0.1%, and 1–2% in the strip layers. The expected SLHC upgrade will increase the number of interactions to the point where over-occupancy may significantly reduce trackfinding effectiveness.

This part of the detector is the world's largest silicon detector. It has 205 m2 of silicon sensors (approximately the area of a tennis court) comprising 76 million channels.[2]

Layer 2 – The Electromagnetic Calorimeter

The Electromagnetic Calorimeter (ECAL) is designed to measure with high accuracy the energies of electrons and photons.

The ECAL is constructed from crystals of lead tungstate, PbWO4. This is an extremely dense but optically clear material, ideal for stopping high energy particles. It has a radiation length of χ0 = 0.89 cm, and has a rapid light yield, with 80% of light yield within one crossing time (25 ns). This is balanced however by a relatively low light yield of 30 photons per MeV of incident energy.

The crystals used have a front size of 22 mm × 22 mm and a depth of 230 mm. They are set in a matrix of carbon fibre to keep them optically isolated, and backed by silicon avalanche photodiodes for readout. The barrel region consists of 61,200 crystals, with a further 7,324 in each of the endcaps.

At the endcaps the ECAL inner surface is covered by the preshower subdetector, consisting of two layers of lead interleaved with two layers of silicon strip detectors. Its purpose is to aid in pion-photon discrimination.
Preparing lead tungstate crystals for the ECAL

Layer 3 – The Hadronic Calorimeter


Half of the Hadron Calorimeter
The purpose of the Hadronic Calorimeter (HCAL) is both to measure the energy of individual hadrons produced in each event, and to be as near to hermetic around the interaction region as possible to allow events with missing energy to be identified.

The HCAL consists of layers of dense material (brass or steel) interleaved with tiles of plastic scintillators, read out via wavelength-shifting fibres by hybrid photodiodes. This combination was determined to allow the maximum amount of absorbing material inside of the magnet coil.

The high pseudorapidity region (3.0 < | η | < 5.0) is instrumented by the Hadronic Forward detector. Located 11 m either side of the interaction point, this uses a slightly different technology of steel absorbers and quartz fibres for readout, designed to allow better separation of particles in the congested forward region.
The brass used in the endcaps of the HCAL used to be Russian artillery shells.[3]

Layer 4 – The magnet

Like most particle physics detectors, CMS has a large solenoid magnet. This allows the charge/mass ratio of particles to be determined from the curved track that they follow in the magnetic field. It is 13 m long and 6 m in diameter, and its refrigerated superconducting niobium-titanium coils were originally intended to produce a 4 T magnetic field. It was recently announced that the magnet will run at 3.8 T instead of the full design strength in order to maximize longevity.[4]

The inductance of the magnet is 14 Η and the nominal current for 4 T is 19,500 A, giving a total stored energy of 2.66 GJ, equivalent to about half-a-tonne of TNT. There are dump circuits to safely dissipate this energy should the magnet quench. The circuit resistance (essentially just the cables from the power converter to the cryostat) has a value of 0.1 mΩ which leads to a circuit time constant of nearly 39 hours. This is the longest time constant of any circuit at CERN. The operating current for 3.8 T is 18,160 A, giving a stored energy of 2.3 GJ.

Layer 5 – The muon detectors and return yoke

To identify muons and measure their momenta, CMS uses three types of detector: drift tubes (DT), cathode strip chambers (CSC) and resistive plate chambers (RPC). The DTs are used for precise trajectory measurements in the central barrel region, while the CSCs are used in the end caps. The RPCs provide a fast signal when a muon passes through the muon detector, and are installed in both the barrel and the end caps.
The Hadron Calorimeter Barrel (in the foreground, on the yellow frame) waits to be inserted into the superconducting magnet (the silver cylinder in the centre of the red magnet yoke).
A part of the Magnet Yoke, with drift tubes and resistive-plate chambers in the barrel region.

Collecting and collating the data

Pattern recognition


Testing the data read-out electronics for the tracker.
New particles discovered in CMS will be typically unstable and rapidly transform into a cascade of lighter, more stable and better understood particles. Particles travelling through CMS leave behind characteristic patterns, or ‘signatures’, in the different layers, allowing them to be identified. The presence (or not) of any new particles can then be inferred.

Trigger system

To have a good chance of producing a rare particle, such as a Higgs boson, a very large number of collisions are required. Most collision events in the detector are "soft" and do not produce interesting effects. The amount of raw data from each crossing is approximately 1 MB, which at the 40 MHz crossing rate would result in 40 TB of data a second, an amount that the experiment cannot hope to store or even process properly. The trigger system reduces the rate of interesting events down to a manageable 100 per second.
To accomplish this, a series of "trigger" stages are employed. All the data from each crossing is held in buffers within the detector while a small amount of key information is used to perform a fast, approximate calculation to identify features of interest such as high energy jets, muons or missing energy. This "Level 1" calculation is completed in around 1 µs, and event rate is reduced by a factor of about thousand down to 50 kHz. All these calculations are done on fast, custom hardware using reprogrammable FPGAs.

If an event is passed by the Level 1 trigger all the data still buffered in the detector is sent over fibre-optic links to the "High Level" trigger, which is software (mainly written in C++) running on ordinary computer servers. The lower event rate in the High Level trigger allows time for much more detailed analysis of the event to be done than in the Level 1 trigger. The High Level trigger reduces the event rate by a further factor of about a thousand down to around 100 events per second. These are then stored on tape for future analysis.

Data analysis

Data that has passed the triggering stages and been stored on tape is duplicated using the Grid to additional sites around the world for easier access and redundancy. Physicists are then able to use the Grid to access and run their analyses on the data.
Some possible analyses might be:
  • Looking at events with large amounts of apparently missing energy, which implies the presence of particles that have passed through the detector without leaving a signature, such as neutrinos.
  • Looking at the kinematics of pairs of particles produced by the decay of a parent, such as the Z boson decaying to a pair of electrons or the Higgs boson decaying to a pair of tau leptons or photons, to determine the properties and mass of the parent.
  • Looking at jets of particles to study the way the quarks in the collided protons have interacted.

Milestones

1998 Construction of surface buildings for CMS begins.
2000 LEP shut down, construction of cavern begins.
2004 Cavern completed.
10 September 2008 First beam in CMS.
23 November 2009 First collisions in CMS.
30 March 2010 First 7 TeV collisions in CMS.
The insertion of the vacuum tank, June 2002
Add cYE+1, a component of CMS weighing 1,270 tonnes, finishes its 100 m descent into the CMS cavern, January 2007aption
YE+2 descent into the cavern

Computer-generated event display of protons hitting a tungsten block just upstream of CMS on the first beam day, September 2008

See also

Book:Large Hadron Collider
Books are collections of articles that can be downloaded or ordered in print.

References

  1. ^ [1]
  2. ^ CMS installs the world's largest silicon detector, CERN Courier, Feb 15, 2008
  3. ^ CMS HCAL history - CERN
  4. ^ http://iopscience.iop.org/1748-0221/5/03/T03021/pdf/1748-0221_5_03_T03021.pdf Precise mapping of the magnetic field in the CMS barrel yoke using cosmic rays

External links

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Antarctic Muon And Neutrino Detector Array....

Diagram of IceCube. IceCube will occupy a volume of one cubic kilometer. Here we depict one of the 80 strings of opctical modules (number and size not to scale). IceTop located at the surface, comprises an array of sensors to detect air showers. It will be used to calibrate IceCube and to conduct research on high-energy cosmic rays. Author: Steve Yunck, Credit: NSF




.....(AMANDA) is a neutrino telescope located beneath the Amundsen-Scott South Pole Station. In 2005, after nine years of operation, AMANDA officially became part of its successor project, IceCube.

AMANDA consists of optical modules, each containing one photomultiplier tube, sunk in Antarctic ice cap at a depth of about 1500 to 1900 meters. In its latest development stage, known as AMANDA-II, AMANDA is made up of an array of 677 optical modules mounted on 19 separate strings that are spread out in a rough circle with a diameter of 200 meters. Each string has several dozen modules, and was put in place by "drilling" a hole in the ice using a hot-water hose, sinking the cable with attached optical modules in, and then letting the ice freeze around it.

AMANDA detects very high energy neutrinos (50+ GeV) which pass through the Earth from the northern hemisphere and then react just as they are leaving upwards through the Antarctic ice. The neutrino collides with nuclei of oxygen or hydrogen atoms contained in the surrounding water ice, producing a muon and a hadronic shower. The optical modules detect the Cherenkov radiation from these latter particles, and by analysis of the timing of photon hits can approximately determine the direction of the original neutrino with a spatial resolution of approximately 2 degrees.

AMANDA's goal was an attempt at neutrino astronomy, identifying and characterizing extra-solar sources of neutrinos. Compared to underground detectors like Super-Kamiokande in Japan, AMANDA was capable of looking at higher energy neutrinos because it is not limited in volume to a manmade tank; however, it had much less accuracy because of the less controlled conditions and wider spacing of photomultipliers. Super-Kamiokande can look at much greater detail at neutrinos from the Sun and those generated in the Earth's atmosphere; however, at higher energies, the spectrum should include neutrinos dominated by those from sources outside the solar system. Such a new view into the cosmos could give important clues in the search for Dark Matter and other astrophysical phenomena.

After two short years of integrated operation as part of IceCube[1], the AMANDA counting house (in the Martin A. Pomerantz Observatory) was finally decommissioned in July and August of 2009.

See also

References

  1. ^ http://icecube.wisc.edu/science/publications/pdd/pdd12.php

External links


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When a neutrino collides with a water molecule deep in Antarctica’s ice, the particle it produces radiates a blue light called Cerenkov radiation, which IceCube will detect (Steve Yunck/NSF)

See:Dual Nature From Microstate Blackhole Creation?
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Muons reveal the interior of volcanoes

The location of the muon detector on the slopes of the Vesuvius volcano.

Like X-ray scans of the human body, muon radiography allows researchers to obtain an image of the internal structures of the upper levels of volcanoes. Although such an image cannot help to predict ‘when’ an eruption might occur, it can, if combined with other observations, help to foresee ‘how’ it could develop and serves as a powerful tool for the study of geological structures.

Muons come from the interaction of cosmic rays with the Earth's atmosphere. They are able to traverse layers of rock as thick as one kilometre or more. During their trip, they are partially absorbed by the material they go through, very much like X-rays are partially absorbed by bones or other internal structures in our body. At the end of the chain, instead of the classic X-ray plate, is the so-called 'muon telescope', a special detector placed on the slopes of the volcano.

See: Muons reveal the interior of volcanoes

***
MU-RAY project 
MUon RAdiographY

A. Kircher (1601-1680), “The interior of Vesuvius”  
A. Kircher (1601-1680): “The interior of Vesuvius” (1638)
Read more about Athanasius Kircher on Wikipedia.

Cosmic ray muon radiography is a technique capable of imaging variations of density inside a hundreds of meters of rock. With resolutions up to tens of meters in optimal detection conditions, muon radiography can give us images of the top region of a volcano edifice with a resolution that is significantly better than the one typically achieved with conventional gravity methods and in this way can give us information on anomalies in the density distribution, such as expected from dense lava conduits, low density magma supply paths or the compression with depth of the overlying soil.

The MU-RAY project is aimed toward the study of the internal structure of Stromboli and Vesuvius volcanoes using this technique.

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Saturday, December 11, 2010

When Muons Collide

By Leah Hesla
photo
Illustration: Sandbox Studio

When Fermilab physicist Steve Geer agreed to perform a calculation as part of a muon collider task force 10 years ago, he imagined he would show that the collider’s technical challenges were too difficult to be solved and move on to other matters. But as he delved further into the problem, he realized that the obstacles he had envisioned could in principle be overcome.

“I started as a skeptic,” he says. “But the more I studied it, I realized it might be a solvable problem.”

A muon collider—a machine that currently exists only in computer simulation—is a relative newcomer to the world of particle accelerators. At the moment, the reception from the particle physics community to this first-of-its-kind particle smasher is “polite,” says Fermilab physicist Alan Bross.

Politeness will suffice for now: research and development on the machine are gearing up thanks to funding from the US Department of Energy. In August, a DOE review panel supported the launch of the Muon Accelerator Program, or MAP, an international initiative led by Fermilab. Scientists hope the program will receive about $15 million per year over seven years to examine the collider’s feasibility and cost effectiveness. See more on Caption of Blog Post
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shows arrival directions of cosmic rays with energies above 4 x 1019eV. Red squares and green circles represent cosmic rays with energies of > 1020eV , and (4 - 10) x 1019eV , respectively.
We observed muon components in the detected air showers and studied their characteristics. Generally speaking, more muons in a shower cascade favors heavier primary hadrons and measurement of muons is one of the methods used to infer the chemical composition of the energetic cosmic rays. Our recent measurement indicates no systematic change in the mass composition from a predominantly heavy to a light composition above 3 x 1017eV claimed by the Fly's Eye group.
***

Also see:


Akeno Giant Air Shower Array (AGASA)

Muons 
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Friday, December 10, 2010

The Penrose interpretation

 ..........is a prediction of Sir Roger Penrose about the mass scale at which standard quantum mechanics will fail. This idea is inspired by quantum gravity, because it uses both the physical constants \scriptstyle \hbar and \scriptstyle G.
Penrose's idea is a variant of objective collapse theory. In these theories the wavefunction is a physical wave, which undergoes wave function collapse as a random process, with observers playing no special role. Penrose suggests that the threshold for wave function collapse is when superpositions involve at least a Planck mass worth of matter. He then hypothesizes that some fundamental gravitational event occurs, causing the wavefunction to choose one branch of reality over another. Despite the difficulties in specifying this in a rigorous way, he mathematically described the basis states involved in the Schrödinger–Newton equations.
Accepting that wavefunctions are physically real, Penrose believes that things can exist in more than one place at one time. In his view, a macroscopic system, like a human being, cannot exist in more than one position because it has a significant gravitational field. A microscopic system, like an electron, has an insignificant gravitational field, and can exist in more than one location almost indefinitely.
In Einstein's theory, any object that has mass causes a warp in the structure of space and time around it. This warping produces the effect we experience as gravity. Penrose points out that tiny objects, such as dust specks, atoms and electrons, produce space-time warps as well. Ignoring these warps is where most physicists go awry. If a dust speck is in two locations at the same time, each one should create its own distortions in space-time, yielding two superposed gravitational fields. According to Penrose's theory, it takes energy to sustain these dual fields. The stability of a system depends on the amount of energy involved: the higher the energy required to sustain a system, the less stable it is. Over time, an unstable system tends to settle back to its simplest, lowest-energy state: in this case, one object in one location producing one gravitational field. If Penrose is right, gravity yanks objects back into a single location, without any need to invoke observers or parallel universes.[1]
Penrose speculates that the transition between macroscopic and quantum begins on the scale of dust particles (whose mass is the planck mass). Dust particles could exist in more than one location for as long as one second, and this is much longer than the time a larger object could be in a superposition. He has proposed an experiment to test this theory, called FELIX (Free-orbit Experiment with Laser Interfometry X-Rays), in which an X-ray laser in space is directed toward a tiny mirror, and fissioned by a beam splitter from thousands of miles away, with which the photons are directed toward other mirrors and reflected back. One photon will strike the tiny mirror moving en route to another mirror and move the tiny mirror back as it returns, and according to Penrose's approach, that the tiny mirror exists in two locations at one time. If gravity affects the mirror, it will be unable to exist in two locations at once because gravity holds it in place. [2]

However, because this experiment would be difficult to set up, a table-top version has been proposed instead.[3]

See also

References

  1. ^ 'Folger, Tim. "If an Electron Can Be in 2 Places at Once, Why Can't You?" Discover. Vol. 25 No. 6 (June 2005). 33.
  2. ^ Penrose, R. Road to Reality pp856-860
  3. ^ 'Folger, Tim. "If an Electron Can Be in 2 Places at Once, Why Can't You?" Discover. Vol. 25 No. 6 (June 2005). 34-35.

External links

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Quantum superposition

Quantum mechanics
\Delta x\, \Delta p \ge \frac{\hbar}{2}
Uncertainty principle
Introduction · Mathematical formulations
[hide]Fundamental concepts
Quantum state · Wave function
Superposition · Entanglement
Complementarity · Duality · Uncertainty
Measurement · Exclusion
Decoherence · Ehrenfest theorem · Tunnelling
v • d • e
......refers to the quantum mechanical property of a particle to occupy all of its possible quantum states simultaneously. Due to this property, to completely describe a particle one must include a description of every possible state and the probability of the particle being in that state.[citation needed] Since the Schrödinger equation is linear, a solution that takes into account all possible states will be a Linear combination of the solutions for each individual state.[clarification needed] This mathematical property of linear equations is known as the superposition principle.

Contents

The Superposition principle of quantum mechanics

The principle of superposition states that if the world can be in any configuration, any possible arrangement of particles or fields, and if the world could also be in another configuration, then the world can also be in a state which is a superposition of the two, where the amount of each configuration that is in the superposition is specified by a complex number.

Examples

For an equation describing a physical phenomenon, the superposition principle states that a linear combination of solutions to an equation is also a solution. When this is true then the equation is linear and said to obey the superposition principle. Thus if functions f1, f2, and f3 solve the linear equation ψ, then ψ=c1f1+c2f2+c3f3 would also be a solution where each c is a coefficient. For example, the electrical field due to a distribution of charged particles can be described by the vector sum of the contributions of the individual particles.

Similarly, probability theory states that the probability of an event can be described by a linear combination of the probabilities of certain specific other events (see Mathematical treatment). For example, the probability of flipping two coins (coin A and coin B) and having at least one turn face up can be expressed as the sum of the probabilities for three specific events- A heads with B tails, A heads with B heads, and A tails with B heads. In this case the probability could be expressed as:
P(heads >  = 1) = P(AnotB) + P(AandB) + P(BnotA)
or even:
P(heads >  = 1) = 1 − P(notAnotB)
Probability theory, as with quantum theory, would also require that the sum of probabilities for all possible events, not just those satisfying the previous condition, be normalized to one. Thus:
P(AnotB) + P(AandB) + P(BnotA) + P(notAnotB) = 1
Probability theory also states that the probability distribution along a continuum (i.e., the chance of finding something is a function of position along a continuous set of coordinates) or among discrete events (the example above) can be described using a probability density or unit vector respectively with the probability magnitude being given by a square of the density function.

In quantum mechanics an additional layer of analysis is introduced as the probability density function is now more specifically a wave function ψ. The wave function is either a complex function of a finite set of real variables or a complex vector formed of a finite or infinite number of components. As the coefficients in the linear combination that describes our probability density are now complex, the probability must now come from the absolute value of the multiplication of the wave function by its complex conjugate \psi \psi^* = \mid \psi \mid ^2. In cases where the functions are not complex, the probability of an event occurring dependent on any member of a subset of the complete set of possible events occurring is the simple sum of the event probabilities in that subset. For example, if an observer rings a bell whenever one or more coins land hands up in the example above, then the probability of the observer ringing a bell is the same as the sum of the probabilities of each event in which at least one coin lands heads up. This is a simple sum since the square of the probability function describing this system is always positive. Using the wave equation, the multiplication of the function by its complex conjugate (square) is not always positive and may produce counterintuitive results.

For example, if a photon in a plus spin state has a 0.1 amplitude to be absorbed and take an atom to the second energy level, and if the photon in a minus spin state has a −0.1 amplitude to do the same thing, a photon which has an equal amplitude to be plus or minus would have zero amplitude to take the atom to the second excited state and the atom will not be excited. If the photon's spin is measured before it reaches the atom, whatever the answer, plus or minus, it will have a nonzero amplitude to excite the atom, plus or minus 0.1.

Assuming normalization, the probability density in quantum mechanics is equal to the square of the absolute value of the amplitude. The further the amplitude is from zero, the bigger the probability. Where probability distribution is represented as a continuous function the probability is the integral of the density function over the relevant values. Where the wave equation is represented as a complex vector, then probability will be extracted from the absolute value of an inner-product of the coefficient matrix and its complex conjugate. In the atom example above, the probability that the atom will be excited is 0. But the only time probability enters the picture is when an observer gets involved. If you look to see which way the atom is, the different amplitudes become probabilities for seeing different things. So if you check to see whether the atom is excited immediately after the photon with 0 amplitude reaches it, there is no chance of seeing the atom excited.

Another example: If a particle can be in position A and position B, it can also be in a state where it is an amount "3i/5" in position A and an amount "4/5" in position B. To write this, physicists usually say:
|\psi\rangle = {3\over 5} i |A\rangle + {4\over 5} |B\rangle.
In the description, only the relative size of the different components matter, and their angle to each other on the complex plane. This is usually stated by declaring that two states which are a multiple of one another are the same as far as the description of the situation is concerned.
|\psi \rangle \approx \alpha |\psi \rangle
The fundamental dynamical law of quantum mechanics is that the evolution is linear, meaning that if the state A turns into A' and B turns into B' after 10 seconds, then after 10 seconds the superposition ψ turns into a mixture of A' and B' with the same coefficients as A and B. A particle can have any position, so that there are different states which have any value of the position x. These are written:
|x\rangle
The principle of superposition guarantees that there are states which are arbitrary superpositions of all the positions with complex coefficients:
\sum_x \psi(x) |x\rangle
This sum is defined only if the index x is discrete. If the index is over \reals, then the sum is not defined and is replaced by an integral instead. The quantity ψ(x) is called the wavefunction of the particle.
If a particle can have some discrete orientations of the spin, say the spin can be aligned with the z axis |+\rangle or against it |-\rangle, then the particle can have any state of the form:
C_1 |+\rangle + C_2 |-\rangle
If the particle has both position and spin, the state is a superposition of all possibilities for both:
\sum_x \psi_+(x)|x,+\rangle + \psi_-(x)|x,-\rangle \,
The configuration space of a quantum mechanical system cannot be worked out without some physical knowledge. The input is usually the allowed different classical configurations, but without the duplication of including both position and momentum.
A pair of particles can be in any combination of pairs of positions. A state where one particle is at position x and the other is at position y is written |x,y\rangle. The most general state is a superposition of the possibilities:
\sum_{xy} A(x,y) |x,y\rangle \,
The description of the two particles is much larger than the description of one particle — it is a function in twice the number of dimensions. This is also true in probability, when the statistics of two random things are correlated. If two particles are uncorrelated, the probability distribution for their joint position P(x,y) is a product of the probability of finding one at one position and the other at the other position:
P(x,y) = P_x (x) P_y(y) \,
In quantum mechanics, two particles can be in special states where the amplitudes of their position are uncorrelated. For quantum amplitudes, the word entanglement replaces the word correlation, but the analogy is exact. A disentangled wavefunction has the form:
A(x,y) = \psi_x(x)\psi_y(y) \,
while an entangled wavefunction does not have this form. Like correlation in probability, there are many more entangled states than disentangled ones. For instance, when two particles which start out with an equal amplitude to be anywhere in a box have a strong attraction and a way to dissipate energy, they can easily come together to make a bound state. The bound state still has an equal probability to be anywhere, so that each particle is equally likely to be everywhere, but the two particles will become entangled so that wherever one particle is, the other is too.

Analogy with probability

In probability theory there is a similar principle. If a system has a probabilistic description, this description gives the probability of any configuration, and given any two different configurations, there is a state which is partly this and partly that, with positive real number coefficients, the probabilities, which say how much of each there is.

For example, if we have a probability distribution for where a particle is, it is described by the "state"
\sum_x \rho(x) |x\rangle
Where ρ is the probability density function, a positive number that measures the probability that the particle will be found at a certain location.

The evolution equation is also linear in probability, for fundamental reasons. If the particle has some probability for going from position x to y, and from z to y, the probability of going to y starting from a state which is half-x and half-z is a half-and-half mixture of the probability of going to y from each of the options. This is the principle of linear superposition in probability.

Quantum mechanics is different, because the numbers can be positive or negative. While the complex nature of the numbers is just a doubling, if you consider the real and imaginary parts separately, the sign of the coefficients is important. In probability, two different possible outcomes always add together, so that if there are more options to get to a point z, the probability always goes up. In quantum mechanics, different possibilities can cancel.

In probability theory with a finite number of states, the probabilities can always be multiplied by a positive number to make their sum equal to one. For example, if there is a three state probability system:
x |1\rangle + y |2\rangle + z |3\rangle \,
where the probabilities x,y,z are positive numbers. Rescaling x,y,z so that
x+y+z=1 \,
The geometry of the state space is a revealed to be a triangle. In general it is a simplex. There are special points in a triangle or simplex corresponding to the corners, and these points are those where one of the probabilities is equal to 1 and the others are zero. These are the unique locations where the position is known with certainty.

In a quantum mechanical system with three states, the quantum mechanical wavefunction is a superposition of states again, but this time twice as many quantities with no restriction on the sign:
A|1\rangle + B|2\rangle + C|3\rangle = (A_r + iA_i) |1\rangle + (B_r + i B_i) |2\rangle + (C_r + iC_i) |3\rangle \,
rescaling the variables so that the sum of the squares is 1, the geometry of the space is revealed to be a high dimensional sphere
A_r^2 + A_i^2 + B_r^2 + B_i^2 + C_r^2 + C_i^2 = 1 \,.
A sphere has a large amount of symmetry, it can be viewed in different coordinate systems or bases. So unlike a probability theory, a quantum theory has a large number of different bases in which it can be equally well described. The geometry of the phase space can be viewed as a hint that the quantity in quantum mechanics which corresponds to the probability is the absolute square of the coefficient of the superposition.

Hamiltonian evolution

The numbers that describe the amplitudes for different possibilities define the kinematics, the space of different states. The dynamics describes how these numbers change with time. For a particle that can be in any one of infinitely many discrete positions, a particle on a lattice, the superposition principle tells you how to make a state:
\sum_n \psi_n |n\rangle \,
So that the infinite list of amplitudes \scriptstyle (... \psi_{-2},\psi_{-1},\psi_0,\psi_1,\psi_2 ...) completely describes the quantum state of the particle. This list is called the state vector, and formally it is an element of a Hilbert space, an infinite dimensional complex vector space. It is usual to represent the state so that the sum of the absolute squares of the amplitudes add up to one:
\sum \psi_n^*\psi_n = 1
For a particle described by probability theory random walking on a line, the analogous thing is the list of probabilities (...P − 2,P − 1,P0,P1,P2,...), which give the probability of any position. The quantities that describe how they change in time are the transition probabilities \scriptstyle K_{x\rightarrow y}(t), which gives the probability that, starting at x, the particle ends up at y after time t. The total probability of ending up at y is given by the sum over all the possibilities
P_y(t_0+t) = \sum_x P_x(t_0) K_{x\rightarrow y}(t) \,
The condition of conservation of probability states that starting at any x, the total probability to end up somewhere must add up to 1:
\sum_y K_{x\rightarrow y} = 1 \,
So that the total probability will be preserved, K is what is called a stochastic matrix.
When no time passes, nothing changes: for zero elapsed time \scriptstyle K{x\rightarrow y}(0) = \delta_{xy} , the K matrix is zero except from a state to itself. So in the case that the time is short, it is better to talk about the rate of change of the probability instead of the absolute change in the probability.
P_y(t+dt) = P_y(t) + dt \sum_x P_x R_{x\rightarrow y} \,
where \scriptstyle R_{x\rightarrow y} is the time derivative of the K matrix:
R_{x\rightarrow y} = { K_{x\rightarrow y}(dt) - \delta_{xy} \over dt} \,.
The equation for the probabilities is a differential equation which is sometimes called the master equation:
{dP_y \over dt} = \sum_x P_x R_{x\rightarrow y} \,
The R matrix is the probability per unit time for the particle to make a transition from x to y. The condition that the K matrix elements add up to one becomes the condition that the R matrix elements add up to zero:
\sum_y R_{x\rightarrow y} = 0 \,
One simple case to study is when the R matrix has an equal probability to go one unit to the left or to the right, describing a particle which has a constant rate of random walking. In this case \scriptstyle R_{x\rightarrow y} is zero unless y is either x+1,x, or x−1, when y is x+1 or x−1, the R matrix has value c, and in order for the sum of the R matrix coefficients to equal zero, the value of R_{x\rightarrow x} must be −2c. So the probabilities obey the discretized diffusion equation:
{dP_x \over dt } = c(P_{x+1} - 2P_{x} + P_{x-1}) \,
which, when c is scaled appropriately and the P distribution is smooth enough to think of the system in a continuum limit becomes:
{\partial P(x,t) \over \partial t} = c {\partial^2 P \over \partial x^2 } \,
Which is the diffusion equation.
Quantum amplitudes give the rate at which amplitudes change in time, and they are mathematically exactly the same except that they are complex numbers. The analog of the finite time K matrix is called the U matrix:
\psi_n(t) = \sum_m U_{nm}(t) \psi_m \,
Since the sum of the absolute squares of the amplitudes must be constant, U must be unitary:
\sum_n U^*_{nm} U_{np} = \delta_{mp} \,
or, in matrix notation,
U^\dagger U = I \,
The rate of change of U is called the Hamiltonian H, up to a traditional factor of i:
H_{mn} = i{d \over dt} U_{mn}
The Hamiltonian gives the rate at which the particle has an amplitude to go from m to n. The reason it is multiplied by i is that the condition that U is unitary translates to the condition:
(I + i H^\dagger dt )(I - i H dt ) = I \,
H^\dagger - H = 0 \,
which says that H is Hermitian. The eigenvalues of the Hermitian matrix H are real quantities which have a physical interpretation as energy levels. If the factor i were absent, the H matrix would be antihermitian and would have purely imaginary eigenvalues, which is not the traditional way quantum mechanics represents observable quantities like the energy.
For a particle which has equal amplitude to move left and right, the Hermitian matrix H is zero except for nearest neighbors, where it has the value c. If the coefficient is everywhere constant, the condition that H is Hermitian demands that the amplitude to move to the left is the complex conjugate of the amplitude to move to the right. The equation of motion for ψ is the time differential equation:
i{d \psi_n \over dt} = c^* \psi_{n+1} + c \psi_{n-1}
In the case that left and right are symmetric, c is real. By redefining the phase of the wavefunction in time, \psi\rightarrow \psi e^{i2ct}, the amplitudes for being at different locations are only rescaled, so that the physical situation is unchanged. But this phase rotation introduces a linear term.
i{d \psi_n \over dt} = c \psi_{n+1} - 2c\psi_n + c\psi_{n-1}
which is the right choice of phase to take the continuum limit. When c is very large and psi is slowly varying so that the lattice can be thought of as a line, this becomes the free Schrödinger equation:
i{ \partial \psi \over \partial t } = - {\partial^2 \psi \over \partial x^2}
If there is an additional term in the H matrix which is an extra phase rotation which varies from point to point, the continuum limit is the Schrödinger equation with a potential energy:
i{ \partial \psi \over \partial t} = - {\partial^2 \psi \over \partial x^2} + V(x) \psi
These equations describe the motion of a single particle in non-relativistic quantum mechanics.

Quantum mechanics in imaginary time

The analogy between quantum mechanics and probability is very strong, so that there are many mathematical links between them. In a statistical system in discrete time, t=1,2,3, described by a transition matrix for one time step \scriptstyle K_{m\rightarrow n}, the probability to go between two points after a finite number of time steps can be represented as a sum over all paths of the probability of taking each path:
K_{x\rightarrow y}(T) = \sum_{x(t)} \prod_t K_{x(t)x(t+1)} \,
where the sum extends over all paths x(t) with the property that x(0) = 0 and x(T) = y. The analogous expression in quantum mechanics is the path integral.

A generic transition matrix in probability has a stationary distribution, which is the eventual probability to be found at any point no matter what the starting point. If there is a nonzero probability for any two paths to reach the same point at the same time, this stationary distribution does not depend on the initial conditions. In probability theory, the probability m for the stochastic matrix obeys detailed balance when the stationary distribution ρn has the property:
\rho_n K_{n\rightarrow m} = \rho_m K_{m\rightarrow n} \,
Detailed balance says that the total probability of going from m to n in the stationary distribution, which is the probability of starting at m ρm times the probability of hopping from m to n, is equal to the probability of going from n to m, so that the total back-and-forth flow of probability in equilibrium is zero along any hop. The condition is automatically satisfied when n=m, so it has the same form when written as a condition for the transition-probability R matrix.
\rho_n R_{n\rightarrow m} = \rho_m R_{m\rightarrow n} \,
When the R matrix obeys detailed balance, the scale of the probabilities can be redefined using the stationary distribution so that they no longer sum to 1:
p'_n = \sqrt{\rho_n}\;p_n \,
In the new coordinates, the R matrix is rescaled as follows:
\sqrt{\rho_n} R_{n\rightarrow m} {1\over \sqrt{\rho_m}} = H_{nm} \,
and H is symmetric
H_{nm} = H_{mn} \,
This matrix H defines a quantum mechanical system:
i{d \over dt} \psi_n = \sum H_{nm} \psi_m \,
whose Hamiltonian has the same eigenvalues as those of the R matrix of the statistical system. The eigenvectors are the same too, except expressed in the rescaled basis. The stationary distribution of the statistical system is the ground state of the Hamiltonian and it has energy exactly zero, while all the other energies are positive. If H is exponentiated to find the U matrix:
U(t) = e^{-iHt} \,
and t is allowed to take on complex values, the K' matrix is found by taking time imaginary.
K'(t) = e^{-Ht} \,
For quantum systems which are invariant under time reversal the Hamiltonian can be made real and symmetric, so that the action of time-reversal on the wave-function is just complex conjugation. If such a Hamiltonian has a unique lowest energy state with a positive real wave-function, as it often does for physical reasons, it is connected to a stochastic system in imaginary time. This relationship between stochastic systems and quantum systems sheds much light on supersymmetry.

Formal interpretation

Applying the superposition principle to a quantum mechanical particle, the configurations of the particle are all positions, so the superpositions make a complex wave in space. The coefficients of the linear superposition are a wave which describes the particle as best as is possible, and whose amplitude interferes according to the Huygens principle.

For any physical quantity in quantum mechanics, there is a list of all the states where the quantity has some value. These states are necessarily perpendicular to each other using the Euclidean notion of perpendicularity which comes from sums-of-squares length, except that they also must not be i multiples of each other. This list of perpendicular states has an associated value which is the value of the physical quantity. The superposition principle guarantees that any state can be written as a combination of states of this form with complex coefficients.
Write each state with the value q of the physical quantity as a vector in some basis \psi^q_n, a list of numbers at each value of n for the vector which has value q for the physical quantity. Now form the outer product of the vectors by multiplying all the vector components and add them with coefficients to make the matrix
A_{nm} = \sum_q q \psi^{*q}_n \psi^q_m
where the sum extends over all possible values of q. This matrix is necessarily symmetric because it is formed from the orthogonal states, and has eigenvalues q. The matrix A is called the observable associated to the physical quantity. It has the property that the eigenvalues and eigenvectors determine the physical quantity and the states which have definite values for this quantity.

Every physical quantity has a Hermitian linear operator associated to it, and the states where the value of this physical quantity is definite are the eigenstates of this linear operator. The linear combination of two or more eigenstates results in quantum superposition of two or more values of the quantity. If the quantity is measured, the value of the physical quantity will be random, with a probability equal to the square of the coefficient of the superposition in the linear combination. Immediately after the measurement, the state will be given by the eigenvector corresponding to the measured eigenvalue.

It is natural to ask why "real" (macroscopic, Newtonian) objects and events do not seem to display quantum mechanical features such as superposition. In 1935, Erwin Schrödinger devised a well-known thought experiment, now known as Schrödinger's cat, which highlighted the dissonance between quantum mechanics and Newtonian physics, where only one configuration occurs, although a configuration for a particle in Newtonian physics specifies both position and momentum.
In fact, quantum superposition results in many directly observable effects, such as interference peaks from an electron wave in a double-slit experiment. The superpositions, however, persist at all scales, absent a mechanism for removing them. This mechanism can be philosophical as in the Copenhagen interpretation, or physical.

Recent research indicates that chlorophyll within plants appears to exploit the feature of quantum superposition to achieve greater efficiency in transporting energy, allowing pigment proteins to be spaced further apart than would otherwise be possible.[1][2]

If the operators corresponding to two observables do not commute, they have no simultaneous eigenstates and they obey an uncertainty principle. A state where one observable has a definite value corresponds to a superposition of many states for the other observable.

See also

References

  1. ^ Scholes, Gregory; Elisabetta Collini, Cathy Y. Wong, Krystyna E. Wilk, Paul M. G. Curmi, Paul Brumer & Gregory D. Scholes (4 February 2010). "Coherently wired light-harvesting in photosynthetic marine algae at ambient temperature". Nature 463 (463): 644–647. http://www.nature.com/nature/journal/v463/n7281/full/nature08811.html. 
  2. ^ Moyer, Michael (September 2009). "Quantum Entanglement, Photosynthesis and Better Solar Cells". Scientific American. http://www.scientificamerican.com/article.cfm?id=quantum-entanglement-and-photo. Retrieved 12 May 2010.
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