Tuesday 3 May 2011

Zebra Puzzle

The zebra puzzle is a well-known logic puzzle. It is often called Einstein's Puzzle or Einstein's Riddle because it is said to have been invented by Albert Einstein as a boy. [1] The puzzle is also sometimes attributed to Lewis Carroll.[2] However, there is no known evidence for Einstein's or Carroll's authorship; and the original puzzle cited below mentions brands of cigarette, such as Kools, that did not exist during Carroll's lifetime or Einstein's boyhood.
There are several versions of this puzzle. The version below is quoted from the first known publication in Life International magazine on December 17, 1962. The March 25, 1963 issue contained the solution given below and the names of several hundred solvers from around the world.

Contents

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[edit] Text of the original puzzle

  1. There are five houses.
  2. The Englishman lives in the red house.
  3. The Spaniard owns the dog.
  4. Coffee is drunk in the green house.
  5. The Ukrainian drinks tea.
  6. The green house is immediately to the right of the ivory house.
  7. The Old Gold smoker owns snails.
  8. Kools are smoked in the yellow house.
  9. Milk is drunk in the middle house.
  10. The Norwegian lives in the first house.
  11. The man who smokes Chesterfields lives in the house next to the man with the fox.
  12. Kools are smoked in the house next to the house where the horse is kept. (should be "... a house ...", see Discussion section)
  13. The Lucky Strike smoker drinks orange juice.
  14. The Japanese smokes Parliaments.
  15. The Norwegian lives next to the blue house.
Now, who drinks water? Who owns the zebra? In the interest of clarity, it must be added that each of the five houses is painted a different color, and their inhabitants are of different national extractions, own different pets, drink different beverages and smoke different brands of American cigarets [sic]. One other thing: in statement 6, right means your right.
– Life International, December 17, 1962
The premises leave out some details, notably that the houses are in a row.
Since neither water nor a zebra is mentioned in the clues, there exists a reductive solution to the puzzle, namely that no one owns a zebra or drinks water. If, however, the questions are read as "Given that one resident drinks water, which is it?" and "Given that one resident owns a zebra, which is it?" then the puzzle becomes a non-trivial challenge to inferential logic. (A frequent variant of the puzzle asks "Who owns the fish?")
It is possible not only to deduce the answers to the two questions but to figure out who lives where, in what color house, keeping what pet, drinking what drink, and smoking what brand of cigarettes.
Rule 12 leads to a contradiction. It should have read "Kools are smoked in a house next to the house where the horse is kept", as opposed the the house, since the implies that there is only one house next to the house with the horse, which implies that the house with the horse is either the leftmost or the rightmost house. The text above has been kept as it is, as it is meant to be a presentation of the text of the puzzle as originally published.

[edit] Solution

House 1 2 3 4 5
Color Yellow Blue Red Ivory Green
Nationality Norwegian Ukrainian Englishman Spaniard Japanese
Drink Water Tea Milk Orange juice Coffee
Smoke Kools Chesterfield Old Gold Lucky Strike Parliament
Pet Fox Horse Snails Dog Zebra
Here are some deductive steps that can be followed to derive the solution. A useful method is to try to fit known relationships into a table and eliminate possibilities. Key deductions are in italics.

[edit] Step 1

We are told the Norwegian lives in the 1st house (10). It does not matter whether this is counted from the left or from the right. We just need to know the order, not the direction.
From (10) and (15), the 2nd house is blue. What color is the 1st house? Not green or ivory, because they have to be next to each other (6 and the 2nd house is blue). Not red, because the Englishman lives there (2). Therefore the 1st house is yellow.
It follows that Kools are smoked in the 1st house (8) and the Horse is kept in the 2nd house (12).
So what is drunk by the Norwegian in the 1st, yellow, Kools-filled house? Not tea since the Ukrainian drinks that (5). Not coffee since that is drunk in the green house (4). Not milk since that is drunk in the 3rd house (9). Not orange juice since the drinker of orange juice smokes Lucky Strikes (13). Therefore it is water (the missing beverage) that is drunk by the Norwegian.

[edit] Step 2

So what is smoked in the 2nd, blue house where we know a Horse is also kept?
Not Kools which are smoked in the 1st house (8). Not Old Gold since that house must have snails (7).
Let's suppose Lucky Strikes are smoked here, which means orange juice is drunk here (13). Then consider: Who lives here? Not the Norwegian since he lives in the 1st House (10). Not the Englishman since he lives in a red house (2). Not the Spaniard since he owns a dog (3). Not the Ukrainian since he drinks tea (4). Not the Japanese who smokes Parliaments (14). Since this is an impossible situation, Lucky Strikes are not smoked in the 2nd house.
Let's suppose Parliaments are smoked here, which means the Japanese man lives here (14). Then consider: What is drunk here? Not tea since the Ukrainian drinks that (5). Not coffee since that is drunk in the green house (4). Not milk since that is drunk in the 3rd house (9). Not orange juice since the drinker of that smokes Lucky Strike (13). Again, since this is an impossible situation, Parliaments are not smoked in the 2nd house.
Therefore, Chesterfields are smoked in the 2nd house.
So who smokes Chesterfields and keeps a Horse in the 2nd, blue house? Not the Norwegian who lives in the 1st House (10). Not the Englishman who lives in a red house (2). Not the Spaniard who owns a dog (3). Not the Japanese who smokes Parliaments (14). Therefore, the Ukrainian lives in the 2nd House, where he drinks tea (5)!

[edit] Step 3

Since Chesterfields are smoked in the 2nd house, we know from (11) that the fox is kept in either the 1st house or the 3rd house.
Let us first assume that the fox is kept in the 3rd house. Then consider: what is drunk by the man who smokes Old Golds and keeps snails (7)? We have already ruled out water and tea from the above steps. It cannot be orange juice since the drinker of that smokes Lucky Strikes (13). It cannot be milk because that is drunk in the 3rd house (9), where we have assumed a fox is kept. This leaves coffee, which we know is drunk in the green house (4).
So if the fox is kept in the 3rd house, then someone smokes Old Golds, keeps snails and drinks coffee in a green house. Who can this person be? Not the Norwegian who lives in the 1st house (10). Not the Ukrainian who drinks tea (5). Not the Englishman who lives in a red house (2). Not the Japanese who smokes Parliaments (14). Not the Spaniard who owns a dog (3).
This is impossible. So it follows that the fox is not kept in the 3rd house, but in the 1st house.

[edit] Step 4

From what we have found so far, we know that coffee and orange juice are drunk in the 4th and 5th houses. It doesn't matter which is drunk in which; we will just call them the coffee house and the orange juice house.
So where does the man who smokes Old Gold and keeps snails live? Not the orange juice house since Lucky Strike is smoked there (13).
Suppose this man lives in the coffee house. Then we have someone who smokes Old Gold, keeps snails and drinks coffee in a green (4) house. Again, by the same reasoning in step 3, this is impossible.
Therefore, the Old Gold-smoking, Snail-keeping man lives in the 3rd house.
It follows that Parliaments are smoked in the green, coffee-drinking house, by the Japanese man (14). This means the Spaniard must be the one who drinks orange juice, smokes Lucky Strikes and keeps a dog. By extension, the Englishman must live in the 3rd house, which is red. By process of elimination, the Spaniard's house is the ivory one.
By now we have filled in every variable except one, and it is clear that the Japanese is the one who keeps the zebra.

[edit] Right-to-left solution

The above solution assumed that the first house is the leftmost house. If we assume that the first house is the rightmost house, we find the following solution. Again the Japanese keeps the zebra, and the Norwegian drinks water.
House 5 4 3 2 1
Color Ivory Green Red Blue Yellow
Nationality Spaniard Japanese Englishman Ukrainian Norwegian
Drink Orange juice Coffee Milk Tea Water
Smoke Lucky Strike Parliament Old Gold Chesterfield Kools
Pet Dog Zebra Snails Horse Fox

[edit] Other versions

Other versions of the puzzle have one or more of the following differences to the original puzzle:
  1. Some colors, nationalities, cigarette brands and pets are substituted for other ones and the clues are given in different order. These do not change the logic of the puzzle.
  2. One rule says that the green house is on the left of the white house, instead of on the right of it. This change has the same effect as numbering the houses from right to left instead of left to right (see section right-to-left solution above). It results only in swapping of the two corresponding houses with all their properties, but makes the puzzle a bit easier. It is also important to note that the omission of the word immediately, as in immediately to the left/right of the white house, leads to multiple solutions to the puzzle.
  3. The clue "The man who smokes Blend has a neighbor who drinks water" is redundant and therefore makes the puzzle even easier. Originally, this clue was only in form of the question "Who drinks water?"
When given to children, the cigarette brands are often replaced by snacks eaten in each house.
  1. The British person lives in the red house.
  2. The Swede keeps dogs as pets.
  3. The Dane drinks tea.
  4. The green house is on the left of the white house.
  5. The green homeowner drinks coffee.
  6. The man who smokes Pall Mall keeps birds.
  7. The owner of the yellow house smokes Dunhill.
  8. The man living in the center house drinks milk.
  9. The Norwegian lives in the first house.
  10. The man who smokes Blend lives next to the one who keeps cats.
  11. The man who keeps the horse lives next to the man who smokes Dunhill.
  12. The man who smokes Bluemaster drinks beer.
  13. The German smokes Prince.
  14. The Norwegian lives next to the blue house.
  15. The man who smokes Blend has a neighbor who drinks water.
Question: Who owns the fish?
  1. The Englishman lives in the red house.
  2. The Spaniard owns the dog.
  3. Coffee is drunk in the green house.
  4. The Ukrainian drinks tea.
  5. The green house is immediately to the right of the ivory house.
  6. The Ford driver owns the snail.
  7. A Toyota driver lives in the yellow house.
  8. Milk is drunk in the middle house.
  9. The Norwegian lives in the first house to the left.
  10. The man who drives the Chevy lives in the house next to the man with the fox.
  11. A Toyota is parked next to the house where the horse is kept.
  12. The Dodge owner drinks orange juice.
  13. The Japanese owns a Porsche.
  14. The Norwegian lives next to the blue house.

[edit] References

  1. ^ Stangroom, Jeremy (2009). Einstein's Riddle: Riddles, Paradoxes, and Conundrums to Stretch Your Mind. Bloomsbury USA. pp. 10–11. ISBN 978-1-59691-665-4. 
  2. ^ James Little, Cormac Gebruers, Derek Bridge, & Eugene Freuder. "Capturing Constraint Programming Experience: A Case-Based Approach" (PDF). Cork Constraint Computation Centre, University College, Cork, Ireland. http://www.cs.ucc.ie/~dgb/papers/Little-Et-Al-2002.pdf. Retrieved 2009-09-05. 

[edit] External links

Retrieved from "http://en.wikipedia.org/wiki/Zebra_Puzzle"
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Uncertainty principle

For other uses, see Uncertainty principle (disambiguation).
Quantum mechanics
\Delta x\, \Delta p \ge \frac{\hbar}{2}
Uncertainty principle
Introduction
Mathematical formulations
v · d · e
In quantum mechanics, the Heisenberg uncertainty principle states precise inequalities that constrain certain pairs of physical properties, such as measuring the present position while determining future momentum; both cannot be simultaneously done to arbitrarily high precision. That is, the more precisely one property is measured, the less precisely the other can be controlled or determined. On the other hand, it is possible to imagine a hypothetical apparatus that measures a history of a particular particle's successive positions and momentums while also measuring times and energies to arbitrary accuracies.
The Uncertainty Principle is often misstated so as to imply that simultaneous measurements of both the position and momentum cannot be made. There is a simple Gedanken experiment that illustrates what physics does allow. Imagine a hollow evacuated sphere where the internal surface is covered by microscopic detectors that measure the position and time of contact of a He atom. Inside the sphere is one single He atom that bounces randomly from one point to another. Each time it contacts the wall, its position is measured to arbitrary accuracy, therefore its future momentum is uncertain. The time of the contact can be measured with arbitrary accuracy, therefore the future energy is uncertain. However, at the next contact with the inner surface of the sphere another accurate measurement of position and time can be made. Knowledge of those accurate times and positions allows us to compute a history of arbitrarily accurate simultaneous positions and momentums along with times and energies.
Published by Werner Heisenberg in 1927, the principle correctly implies that it is impossible to simultaneously both measure the present position while "determining" the future momentum of an electron or any other particle with an arbitrary degree of accuracy and certainty. This is not a statement about researchers' ability to measure one quantity while determining the other quantity. Rather, it is a statement about the laws of physics. That is, a system cannot be defined to simultaneously measure one value while determining the future value of these pairs of quantities. The principle states that a minimum exists for the product of the uncertainties in these properties that is equal to or greater than one half of ħ the reduced Planck constant (ħ = h/2π).

Contents

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[edit] Historical introduction

Main article: Introduction to quantum mechanics
Werner Heisenberg formulated the uncertainty principle at Niels Bohr's institute in Copenhagen, while working on the mathematical foundations of quantum mechanics.
In 1925, following pioneering work with Hendrik Kramers, Heisenberg developed matrix mechanics, which replaced the ad-hoc old quantum theory with modern quantum mechanics. The central assumption was that the classical concept of motion does not fit at the quantum level, and that electrons in an atom did not travel on sharply defined orbits. Rather, the motion was smeared out in a strange way: the Fourier transform of time only involving those frequencies that could be seen in quantum jumps.
Heisenberg's paper did not admit any unobservable quantities like the exact position of the electron in an orbit at any time; he only allowed the theorist to talk about the Fourier components of the motion. Since the Fourier components were not defined at the classical frequencies, they could not be used to construct an exact trajectory, so that the formalism could not answer certain overly precise questions about where the electron was or how fast it was going.
The most striking property of Heisenberg's infinite matrices for the position and momentum is that they do not commute. Heisenberg's canonical commutation relation indicates by how much:
[X,P] = X P - P X = i \hbar (see derivations below)
and this result did not have a clear physical interpretation in the beginning.
In March 1926, working in Bohr's institute, Heisenberg realized that the non-commutativity implies the uncertainty principle. This implication provided a clear physical interpretation for the non-commutativity, and it laid the foundation for what became known as the Copenhagen interpretation of quantum mechanics. Heisenberg showed that the commutation relation implies an uncertainty, or in Bohr's language a complementarity.[1] Any two variables that do not commute cannot be measured simultaneously — the more precisely one is known, the less precisely the other can be known.
One way to understand the complementarity between position and momentum is by wave-particle duality. If a particle described by a plane wave passes through a narrow slit in a wall like a water-wave passing through a narrow channel, the particle diffracts and its wave comes out in a range of angles. The narrower the slit, the wider the diffracted wave and the greater the uncertainty in momentum afterward. The laws of diffraction require that the spread in angle Δθ is about λ / d, where d is the slit width and λ is the wavelength. Reasoning based on the de Broglie relation, shows that the size of the slit and the range in momentum of the diffracted wave are related by Heisenberg's rule:
\Delta x \, \Delta p \approx h. \,
In his celebrated 1927 paper, "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik" ("On the Perceptual Content of Quantum Theoretical Kinematics and Mechanics"), Heisenberg established this expression as the minimum amount of unavoidable momentum disturbance caused by any position measurement,[2] but he did not give a precise definition for the uncertainties Δx and Δp. Instead, he gave some plausible estimates in each case separately. In his Chicago lecture[3] he refined his principle:
\Delta x \, \Delta p\gtrsim h\qquad\qquad\qquad (1)
But it was Kennard[4] in 1927 who first proved the modern inequality:
\sigma_x\sigma_p\ge\frac{\hbar}{2}\quad\qquad\qquad\qquad (2)
where ħ = h/2π, and σx, σp are the standard deviations of position and momentum. Heisenberg himself only proved relation (2) for the special case of Gaussian states.[3] However, it should be noted that σx and Δx are not the same quantities. σx and σp as defined in Kennard are obtained by making repeated measurements of position on an ensemble of systems and by making repeated measurements of momentum on an ensemble of systems and calculating the standard deviation of those measurements. The Kennard expression, therefore, says nothing about the simultaneous measurement of position and momentum.
A rigorous proof of a new inequality for simultaneous measurements in the spirit of Heisenberg and Bohr has been given recently. The measurement process is as follows: Whenever a particle is localized in a finite interval Δx > 0, then the standard deviation of its momentum satisfies
\sigma_p\,\Delta x \, \ge\,\pi\hbar\qquad\qquad\qquad (3)
while the equal sign is given for Cosine states.[5]

[edit] Terminology

Throughout the main body of his original 1927 paper, written in German, Heisenberg used the word "Unbestimmtheit" ("indeterminacy") to describe the basic theoretical principle. Only in the endnote did he switch to the word "Unsicherheit" ("uncertainty"). However, when the English-language version of Heisenberg's textbook, The Physical Principles of the Quantum Theory, was published in 1930, the translation "uncertainty" was used, and it became the more commonly used term in the English language thereafter.[6]

[edit] Heisenberg's microscope

Heisenberg's gamma-ray microscope for locating an electron (shown in blue). The incoming gamma ray (shown in green) is scattered by the electron up into the microscope's aperture angle θ. The scattered gamma-ray is shown in red. Classical optics shows that the electron position can be resolved only up to an uncertainty Δx that depends on θ and the wavelength λ of the incoming light.
Main article: Heisenberg's microscope
One way in which Heisenberg originally argued for the uncertainty principle is by using an imaginary microscope as a measuring device.[3] He imagines an experimenter trying to measure the position and momentum of an electron by shooting a photon at it.
Problem 1 - If the photon has a short wavelength, and therefore a large momentum, the position can be measured accurately. But the photon scatters in a random direction, transferring a large and uncertain amount of momentum to the electron. If the photon has a long wavelength and low momentum, the collision does not disturb the electron's momentum very much, but the scattering will reveal its position only vaguely.
Problem 2 - If a large aperture is used for the microscope, the electron's location can be well resolved (see Rayleigh criterion); but by the principle of conservation of momentum, the transverse momentum of the incoming photon and hence the new momentum of the electron resolves poorly. If a small aperture is used, the accuracy of both resolutions is the other way around.
The combination of these trade-offs imply that no matter what photon wavelength and aperture size are used, the product of the uncertainty in measured position and measured momentum is greater than or equal to a lower limit, which is (up to a small numerical factor) equal to Planck's constant.[7] Heisenberg did not care to formulate the uncertainty principle as an exact limit, and preferred to use it as a heuristic quantitative statement, correct up to small numerical factors.

[edit] Critical reactions

Main article: Bohr–Einstein debates
The Copenhagen interpretation of quantum mechanics and Heisenberg's Uncertainty Principle were in fact seen as twin targets by detractors who believed in an underlying determinism and realism. According to the Copenhagen interpretation of quantum mechanics, there is no fundamental reality that the quantum state describes, just a prescription for calculating experimental results. There is no way to say what the state of a system fundamentally is, only what the result of observations might be.
Albert Einstein believed that randomness is a reflection of our ignorance of some fundamental property of reality, while Niels Bohr believed that the probability distributions are fundamental and irreducible, and depend on which measurements we choose to perform. Einstein and Bohr debated the uncertainty principle for many years.

[edit] Einstein's slit

The first of Einstein's thought experiments challenging the uncertainty principle went as follows:
Consider a particle passing through a slit of width d. The slit introduces an uncertainty in momentum of approximately h/d because the particle passes through the wall. But let us determine the momentum of the particle by measuring the recoil of the wall. In doing so, we find the momentum of the particle to arbitrary accuracy by conservation of momentum.
Bohr's response was that the wall is quantum mechanical as well, and that to measure the recoil to accuracy Δp the momentum of the wall must be known to this accuracy before the particle passes through. This introduces an uncertainty in the position of the wall and therefore the position of the slit equal to h / Δp, and if the wall's momentum is known precisely enough to measure the recoil, the slit's position is uncertain enough to disallow a position measurement.
A similar analysis with particles diffracting through multiple slits is given by Richard Feynman.[8]

[edit] Einstein's box

Another of Einstein's thought experiments (Einstein's box) was designed to challenge the time/energy uncertainty principle. It is very similar to the slit experiment in space, except here the narrow window the particle passes through is in time:
Consider a box filled with light. The box has a shutter that a clock opens and quickly closes at a precise time, and some of the light escapes. We can set the clock so that the time that the energy escapes is known. To measure the amount of energy that leaves, Einstein proposed weighing the box just after the emission. The missing energy lessens the weight of the box. If the box is mounted on a scale, it is naively possible to adjust the parameters so that the uncertainty principle is violated.
Bohr spent a day considering this setup, but eventually realized that if the energy of the box is precisely known, the time the shutter opens at is uncertain. If the case, scale, and box are in a gravitational field then, in some cases, it is the uncertainty of the position of the clock in the gravitational field that alters the ticking rate. This can introduce the right amount of uncertainty. This was ironic as it was Einstein himself who first discovered gravity's effect on clocks.

[edit] EPR paradox for entangled particles

Bohr was compelled to modify his understanding of the uncertainty principle after another thought experiment by Einstein. In 1935, Einstein, Podolsky and Rosen (see EPR paradox) published an analysis of widely separated entangled particles. Measuring one particle, Einstein realized, would alter the probability distribution of the other, yet here the other particle could not possibly be disturbed. This example led Bohr to revise his understanding of the principle, concluding that the uncertainty was not caused by a direct interaction.[9]
But Einstein came to much more far-reaching conclusions from the same thought experiment. He believed the "natural basic assumption" that a complete description of reality would have to predict the results of experiments from "locally changing deterministic quantities", and therefore would have to include more information than the maximum possible allowed by the uncertainty principle.
In 1964 John Bell showed that this assumption can be falsified, since it would imply a certain inequality between the probabilities of different experiments. Experimental results confirm the predictions of quantum mechanics, ruling out Einstein's basic assumption that led him to the suggestion of his hidden variables. (Ironically this fact is one of the best pieces of evidence supporting Karl Popper's philosophy of invalidation of a theory by falsification-experiments. That is to say, here Einstein's "basic assumption" became falsified by experiments based on Bell's inequalities. For the objections of Karl Popper against the Heisenberg inequality itself, see below.)
While it is possible to assume that quantum mechanical predictions are due to nonlocal hidden variables, and in fact David Bohm invented such a formulation, this resolution is not satisfactory to the vast majority of physicists. The question of whether a random outcome is predetermined by a nonlocal theory can be philosophical, and it can be potentially intractable. If the hidden variables are not constrained, they could just be a list of random digits that are used to produce the measurement outcomes. To make it sensible, the assumption of nonlocal hidden variables is sometimes augmented by a second assumption — that the size of the observable universe puts a limit on the computations that these variables can do. A nonlocal theory of this sort predicts that a quantum computer encounters fundamental obstacles when it tries to factor numbers of approximately 10,000 digits or more; an achievable task in quantum mechanics.[10]

[edit] Popper's criticism

Main article: Popper's experiment
Karl Popper approached the problem of indeterminacy as a logician and metaphysical realist.[11][12] He disagreed with the application of the uncertainty relations to individual particles rather than to ensembles of identically prepared particles, referring to them as "statistical scatter relations".[11][13] In this statistical interpretation, a particular measurement may be made to arbitrary precision without invalidating the quantum theory. This directly contrasts the Copenhagen interpretation of quantum mechanics, which is non-deterministic but lacks local hidden variables.
In 1934, Popper published Zur Kritik der Ungenauigkeitsrelationen (Critique of the Uncertainty Relations) in Naturwissenschaften,[14] and in the same year Logik der Forschung (translated and updated by the author as The Logic of Scientific Discovery in 1959), outlining his arguments for the statistical interpretation. In 1982, he further developed his theory in Quantum theory and the schism in Physics, writing:
[Heisenberg's] formulae are, beyond all doubt, derivable statistical formulae of the quantum theory. But they have been habitually misinterpreted by those quantum theorists who said that these formulae can be interpreted as determining some upper limit to the precision of our measurements.[original emphasis][15]
Popper proposed an experiment to falsify the uncertainty relations, though he later withdrew his initial version after discussions with Weizsäcker, Heisenberg, and Einstein; this experiment may have influenced the formulation of the EPR experiment.[11][16] A version of this experiment was realized in 1999.[12]

[edit] Matter wave interpretation

According to the de Broglie hypothesis, every object in our Universe is a wave, a situation which gives rise to this phenomenon. Consider the measurement of the position of a particle. The particle's wave packet has non-zero amplitude, meaning that the position is uncertain – it could be almost anywhere along the wave packet. To obtain an accurate reading of position, this wave packet must be 'compressed' as much as possible, meaning it must be made up of increasing numbers of sine waves added together. The momentum of the particle is proportional to the wavenumber of one of these waves, but it could be any of them. So a more precise position measurement – by adding together more waves – means that the momentum measurement becomes less precise (and vice versa).
The only kind of wave with a definite position is concentrated at one point, and such a wave has an indefinite wavelength (and therefore an indefinite momentum). Conversely, the only kind of wave with a definite wavelength is an infinite regular periodic oscillation over all space, which has no definite position. So in quantum mechanics, there can be no states that describe a particle with both a definite position and a definite momentum. The more precise the position, the less precise the momentum.
A mathematical statement of the principle is that every quantum state has the property that the root mean square (RMS) deviation of the position from its mean (the standard deviation of the x-distribution):
\sigma_x = \sqrt{\langle(x - \langle x\rangle)^2\rangle} \,
times the RMS deviation of the momentum from its mean (the standard deviation of p):
\sigma_p = \sqrt{\langle(p - \langle p \rangle)^2\rangle} \,
can never be smaller than a fixed fraction of Planck's constant:
\sigma_x \sigma_p \ge \hbar/2.
The uncertainty principle can be restated in terms of other measurement processes, which involves collapse of the wavefunction. When the position is initially localized by preparation, the wavefunction collapses to a narrow bump in an interval Δx > 0, and the momentum wavefunction becomes spread out. The particle's momentum is left uncertain by an amount inversely proportional to the accuracy of the position measurement:
\sigma_p \,\ge\,\pi\hbar/\Delta x.
If the initial preparation in Δx is understood as an observation or disturbance of the particles then this means that the uncertainty principle is related to the observer effect. However, this is not true in the case of the measurement process corresponding to the former inequality but only for the latter inequality.

[edit] Energy-time uncertainty principle

One well-known uncertainty relation is not an obvious consequence of the Robertson–Schrödinger relation: the energy-time uncertainty principle.
Since energy bears the same relation to time as momentum does to space in special relativity, it was clear to many early founders, Niels Bohr among them, that the following relation holds:[2][3]
\Delta E \Delta t \gtrsim h,
but it was not always obvious what Δt was, because the time at which the particle has a given state is not an operator belonging to the particle, it is a parameter describing the evolution of the system. As Lev Landau once joked "To violate the time-energy uncertainty relation all I have to do is measure the energy very precisely and then look at my watch!" [17]
Nevertheless, Einstein and Bohr understood the heuristic meaning of the principle. A state that only exists for a short time cannot have a definite energy. To have a definite energy, the frequency of the state must accurately be defined, and this requires the state to hang around for many cycles, the reciprocal of the required accuracy.
For example, in spectroscopy, excited states have a finite lifetime. By the time-energy uncertainty principle, they do not have a definite energy, and each time they decay the energy they release is slightly different. The average energy of the outgoing photon has a peak at the theoretical energy of the state, but the distribution has a finite width called the natural linewidth. Fast-decaying states have a broad linewidth, while slow decaying states have a narrow linewidth.
The broad linewidth of fast decaying states makes it difficult to accurately measure the energy of the state, and researchers have even used microwave cavities to slow down the decay-rate, to get sharper peaks.[18] The same linewidth effect also makes it difficult to measure the rest mass of fast decaying particles in particle physics. The faster the particle decays, the less certain is its mass.
One false formulation of the energy-time uncertainty principle says that measuring the energy of a quantum system to an accuracy ΔE requires a time interval Δt > h / ΔE. This formulation is similar to the one alluded to in Landau's joke, and was explicitly invalidated by Y. Aharonov and D. Bohm in 1961. The time Δt in the uncertainty relation is the time during which the system exists unperturbed, not the time during which the experimental equipment is turned on (As the position in the other version of the principle refers to where the particle has some probability to be and not where the observer might look).
Another common misconception is that the energy-time uncertainty principle says that the conservation of energy can be temporarily violated – energy can be "borrowed" from the Universe as long as it is "returned" within a short amount of time.[19] Although this agrees with the spirit of relativistic quantum mechanics, it is based on the false axiom that the energy of the Universe is an exactly known parameter at all times. More accurately, when events transpire at shorter time intervals, there is a greater uncertainty in the energy of these events. Therefore it is not that the conservation of energy is violated when quantum field theory uses temporary electron-positron pairs in its calculations, but that the energy of quantum systems is not known with enough precision to limit their behavior to a single, simple history. Thus the influence of all histories must be incorporated into quantum calculations, including those with much greater or much less energy than the mean of the measured/calculated energy distribution.
In 1936 Dirac offered a precise definition and derivation of the time-energy uncertainty relation in a relativistic quantum theory of "events".[citation needed] But a better-known, more widely used formulation of the time-energy uncertainty principle was given in 1945 by L. I. Mandelshtam and I. E. Tamm, as follows.[20] For a quantum system in a non-stationary state ψ and an observable B represented by a self-adjoint operator \hat B, the following formula holds:
\sigma_E \frac{\sigma_B}{\left | \frac{\mathrm{d}\langle \hat B \rangle}{\mathrm{d}t}\right |} \ge \frac{\hbar}{2},
where σE is the standard deviation of the energy operator in the state ψ, σB stands for the standard deviation of B. Although, the second factor in the left-hand side has dimension of time, it is different from the time parameter that enters Schrödinger equation. It is a lifetime of the state ψ with respect to the observable B. In other words, this is the time after which the expectation value \langle\hat B\rangle changes appreciably.

[edit] Entropic uncertainty principle

Main article: Hirschman uncertainty
While formulating the many-worlds interpretation of quantum mechanics in 1957, Hugh Everett III discovered a much stronger formulation of the uncertainty principle.[21] In the inequality of standard deviations, some states, like the wavefunction
\psi(x) \propto e^{-\frac{x^2}{0.0001}}+ e^{- \frac{(x-100)^2}{0.0001 }}
have a large standard deviation of position, but are actually a superposition of a small number of very narrow bumps. In this case, the momentum uncertainty is much larger than the standard deviation inequality would suggest. A better inequality uses the Shannon information content of the distribution, a measure of the number of bits learned when a random variable described by a probability distribution has a certain value.
I_x = - \int |\psi(x)|^2 \log_2 |\psi(x)|^2 \,dx
The interpretation of I is that the number of bits of information an observer acquires when the value of x is given to accuracy ε is equal to Ix + log2(ε). The second part is just the number of bits past the decimal point, the first part is a logarithmic measure of the width of the distribution. For a uniform distribution of width Δx the information content is log2Δx. This quantity can be negative, which means that the distribution is narrower than one unit, so that learning the first few bits past the decimal point gives no information since they are not uncertain.
Taking the logarithm of Heisenberg's formulation of uncertainty in natural units.
\log_2(\Delta x \Delta p) > 0 \,
but the lower bound is not precise.
Everett (and Hirschman[22]) conjectured that for all quantum states:
I_x + I_p \ge \log_2(e\pi)\,
This was proven by Beckner in 1975.[23]

[edit] Mathematical derivations

When linear operators A and B act on a function ψ, they do not always commute.
A clear example is when operator B (acting on smooth complex functions defined over the real line) multiplies by x (that is, by the canonical immersion of the real line into the complex plane ), while operator A (defined on the same domain as B) takes the derivative (with respect to x). Then, for every wave function ψ we can write
(AB - BA) \psi = \left({d\over dx}x-x{d\over dx}\right)\psi(x) = {d\over dx} ( x \psi(x)) - x {d\over dx} \psi(x)
= \psi(x) + x {d \over dx}\psi(x) - x {d \over dx}\psi(x) = \psi(x) ,
which in operator language means that
{d\over dx} x - x {d\over dx} = 1 ,
here 1 stands for the identity operator on the common domain of the operators A and B.
This example is important, because it is very close to the canonical commutation relation of quantum mechanics. There, in the position basis, the position operator multiplies the value of the wavefunction by x, while the corresponding momentum operator differentiates and multiplies by \scriptstyle -i\hbar, so that:
[p,x] = p x - x p = -i\hbar \left( {d\over dx} x - x {d\over dx} \right) = - i \hbar.
It is the nonzero commutator that implies the uncertainty.
For any two operators A and B:
\|A\psi\|^2 \|B\psi\|^2 = \langle A\psi|A\psi\rangle\langle B\psi|B\psi\rangle \ge |\langle A\psi|B\psi\rangle|^2
which is a statement of the Cauchy–Schwarz inequality for the inner product of the two vectors \scriptstyle A|\psi\rangle and \scriptstyle B|\psi\rangle. On the other hand, the expectation value of the product AB is always greater than the magnitude of its imaginary part and this last statement can be written as
|\langle\psi|AB|\psi\rangle |^2 \ge {\left\vert{1\over 2i} \langle\psi|AB - BA |\psi\rangle\right\vert}^2 ,
when the two operators are Hermitian. Putting the above two inequalities together - again when the operators are Hermitian operators - we conclude the relation:
\langle A^2 \rangle_\psi \langle B^2 \rangle_\psi \ge {1\over 4} |\langle [A,B]\rangle_\psi|^2
and the uncertainty principle is a special case.

[edit] Physical Interpretation

The inequality above acquires its dispersion interpretation:
\sigma_A\sigma_B \ge \frac{1}{2} \left|\left\langle\left[{A},{B}\right]\right\rangle\right|
where
\left\langle X \right\rangle = \left\langle \psi | X | \psi \right\rangle
is the mean of observable X in the state ψ and
\sigma_X = \sqrt{\langle {X}^2\rangle - \langle {X}\rangle^2}
is the corresponding standard deviation of observable X.
By substituting A - \lang A\rang for A and B - \lang B\rang for B in the general operator norm inequality, since the imaginary part of the product, the commutator is unaffected by the shift:
[A - \lang A\rang, B - \lang B\rang] = [ A , B ].
The big side of the inequality is the product of the norms of A-\lang A\rang and B-\lang B\rang, which in quantum mechanics are the standard deviations of A and B. The small side is the norm of the commutator, which for the position and momentum is just \scriptstyle \hbar.
A further generalization is due to Schrödinger:
Given any two Hermitian operators A and B, and a system in the state ψ, there are probability distributions for the value of a measurement of A and B, with standard deviations σA and σB. Then
\sigma_A \sigma_B \geq \sqrt{ \frac{1}{4}\left|\left\langle\left[{A},{B}\right]\right\rangle\right|^2 +{1\over 4} \left|\left\langle\left\{ A-\langle A\rangle,B-\langle B\rangle \right\} \right\rangle \right|^2}
where [A, B] = AB − BA is the commutator of A and B, {A,B}= AB+BA is the anticommutator.
This inequality is sometimes called the Robertson–Schrödinger relation, and includes the Heisenberg uncertainty principle as a special case but for a different measurement process. The inequality with the commutator term only was developed in 1930 by Howard Percy Robertson, and Erwin Schrödinger added the anticommutator term a little later.

[edit] Examples

The Robertson-Schrödinger relation gives the uncertainty relation for any two observables that do not commute:
  • between position and momentum by applying the commutator relation [x,p_x]=i\hbar:
\sigma_x\sigma_p \geq \frac{\hbar}{2}
  • between the kinetic energy T and position x of a particle :
\sigma_T\sigma_x \geq {\hbar\over 2m} \left|\left\langle p_x\right\rangle\right|
\sigma_{J_i} \sigma_{J_j} \geq \frac{\hbar}{2} \left|\left\langle J_k\right\rangle\right|
where i, j, k are distinct and Ji denotes angular momentum along the xi axis. This relation implies that only a single component of a system's angular momentum can be defined with arbitrary precision, normally the component parallel to an external (magnetic or electric) field.
  • between angular position and angular momentum of an object with small angular uncertainty:
\Delta \Theta_i \Delta J_i \gtrapprox \frac{\hbar}{2}
\Delta N \Delta \phi \geq 1

[edit] Uncertainty theorems in harmonic analysis

Further information: Fourier uncertainty principle
In the context of harmonic analysis, the uncertainty principle implies that one cannot at the same time localize the value of a function and its Fourier transform; to wit, the following inequality holds
\left(\int_{-\infty}^\infty x^2 |f(x)|^2\,dx\right)\left(\int_{-\infty}^\infty \xi^2 |\hat{f}(\xi)|^2\,d\xi\right)\ge \frac{\|f\|_2^4}{16\pi^2}.
Other purely mathematical formulations of uncertainty exist between a function ƒ and its Fourier transform – see Fourier uncertainty principle. A variety of such results can be found in (Havin & Jöricke 1994) or (Folland & Sitaram 1997); for a short survey, see (Sitaram 2001).

[edit] Signal processing

In the context of signal processing, particularly time–frequency analysis, uncertainty principles are referred to as the Gabor limit, after Dennis Gabor, or sometimes the Heisenberg–Gabor limit. The basic result, which follows from Benedicks's theorem, below, is that a function cannot be both time limited and band limited (a function and its Fourier transform cannot both have bounded domain) – see bandlimited versus timelimited. Stated alternatively, "one cannot simultaneously localize a signal (function) in both the time domain (ƒ) and frequency domain (Fourier transform)". When applied to filters, the result is that one cannot achieve high temporal resolution and frequency resolution at the same time; a concrete example are the resolution issues of the short-time Fourier transform – if one uses a wide window, one achieves good frequency resolution at the cost of temporal resolution, while a narrow window has the opposite trade-off.
Alternative theorems give more precise quantitative results, and in time–frequency analysis, rather than interpreting the (1-dimensional) time and frequency domains separately, one instead interprets the limit as a lower limit on the support of a function in the (2-dimensional) time–frequency plane. In practice the Gabor limit limits the simultaneous time–frequency resolution one can achieve without interference; it is possible to achieve higher resolution, but at the cost of different components of the signal interfering with each other.

[edit] Benedicks's theorem

Amrein-Berthier (Amrein & Berthier 1977) and Benedicks's theorem (Benedicks 1985) intuitively says that the set of points where ƒ is non-zero and the set of points where \hat{f} is nonzero cannot both be small. Specifically, it is impossible for a function ƒ in L2(R) and its Fourier transform to both be supported on sets of finite Lebesgue measure. A more quantitative version is due to Nazarov (Nazarov 1994) and (Jaming 2007):
\|f\|_{L^2(\mathbf{R}^d)}\leq Ce^{C|S||\Sigma|} \bigl(\|f\|_{L^2(S^c)} + \| \hat{f} \|_{L^2(\Sigma^c)} \bigr)
One expects that the factor CeC | S | | Σ | may be replaced by Ce^{C(|S||\Sigma|)^{1/d}} which is only known if either S or Σ is convex.

[edit] Hardy's uncertainty principle

The mathematician G. H. Hardy (Hardy 1933) formulated the following uncertainty principle: it is not possible for ƒ and \hat{f} to both be "very rapidly decreasing." Specifically, if ƒ is in L2(R), is such that
|f(x)|\leq C(1+|x|)^Ne^{-a\pi x^2}
and
|\hat{f}(\xi)|\leq C(1+|\xi|)^Ne^{-b\pi \xi^2} (C > 0,N an integer)
then, if ab > 1,f = 0 while if ab = 1 then there is a polynomial P of degree \leq N such that
f(x)=P(x)e^{-a\pi x^2}. \,
This was later improved as follows: if f\in L^2(\mathbf{R}^d) is such that
\int_{\mathbf{R}^d}\int_{\mathbf{R}^d}|f(x)||\hat{f}(\xi)|\frac{e^{\pi|\langle x,\xi\rangle|}}{(1+|x|+|\xi|)^N} \, dx \, d\xi < +\infty
then
f(x)=P(x)e^{-\pi\langle Ax,x\rangle}
where P is a polynomial of degree <\frac{N-d}{2} and A is a real d\times d positive definite matrix.
This result was stated in Beurling's complete works without proof and proved in Hörmander (Hörmander 1991) (the case d = 1,N = 0) and Bonami–Demange–Jaming (Bonami, Demange & Jaming 2003) for the general case. Note that Hörmander–Beurling's version implies the case ab > 1 in Hardy's Theorem while the version by Bonami–Demange–Jaming covers the full strength of Hardy's Theorem.
A full description of the case ab < 1 as well as the following extension to Schwarz class distributions appears in Demange (Demange 2010):
Theorem. If a tempered distribution f\in\mathcal{S}'(\R^d) is such that
e^{\pi|x|^2}f\in\mathcal{S}'(\R^d)
and
e^{\pi|\xi|^2}\hat f\in\mathcal{S}'(\R^d)
then
f(x)=P(x)e^{-\pi\langle Ax,x\rangle}
for some convenient polynomial P and real positive definite matrix A of type d\times d.

[edit] Uncertainty principle of game theory

The uncertainty principle of game theory was formulated by Szekely and Rizzo in 2007.[26] This principle is a lower bound for the entropy of optimal strategies of players in terms of the commutator of two nonlinear operators: minimum and maximum. If the payoff matrix (aij) of an arbitrary zero-sum game is normalized (i.e. the smallest number in this matrix is 0, the biggest number is 1) and the commutator
minj maxi (aij) − maxi minj (aij) = h
then the entropy of the optimal strategy of any of the players cannot be smaller than the entropy of the two-point distribution [1/(1+h), h/(1+h)] and this is the best lower bound. (This is zero if and only if h = 0 i.e. if min and max are commutable in which case the game has pure nonrandom optimal strategies). As an application, one could optimize between these two-point strategies via considering the distribution [1/(1+h), h/(1+h)] on all pairs of pure strategies. In many practical cases we do not lose much by neglecting more complex strategies.

[edit] See also

[edit] Notes

  1. ^ Bohr, Niels (1958), Atomic Physics and Human Knowledge, New York: Wiley, p. 38 
  2. ^ a b Heisenberg, W. (1927), "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik", Zeitschrift für Physik 43 (3–4): 172–198, Bibcode 1927ZPhy...43..172H, doi:10.1007/BF01397280. 
  3. ^ a b c d Heisenberg, W. (1930), Physikalische Prinzipien der Quantentheorie, Leipzig: Hirzel  English translation The Physical Principles of Quantum Theory. Chicago: University of Chicago Press, 1930.
  4. ^ Kennard, E. H. (1927), "Zur Quantenmechanik einfacher Bewegungstypen", Zeitschrift für Physik 44 (4–5): 326, Bibcode 1927ZPhy...44..326K, doi:10.1007/BF01391200. 
  5. ^ Schürmann, T.; Hoffmann, I. (2009), "A closer look at the uncertainty relation of position and momentum", Foundations of Physics 39 (8): 958–963, arXiv:0811.2582, Bibcode 2009FoPh...39..958S, doi:10.1007/s10701-009-9310-0. 
  6. ^ Cassidy, David (2009), Beyond Uncertainty: Heisenberg, Quantum Physics, and the Bomb, New York: Bellevue Literary Press, p. 185 
  7. ^ Tipler, Paul A.; Llewellyn, Ralph A. (1999), "5-5", Modern Physics (3rd ed.), W. H. Freeman and Co., ISBN 1572591641 
  8. ^ Feynman lectures on Physics, vol 3, 2-2
  9. ^ Isaacson, Walter (2007), Einstein: His Life and Universe, New York: Simon & Schuster, p. 452, ISBN 9780743264730 
  10. ^ Gerardus 't Hooft has at times advocated this point of view.
  11. ^ a b c [[Karl Popper |Popper, Karl]] (1959). The Logic of Scientific Discovery. Hutchinson & Co.. 
  12. ^ a b Kim, Yoon-Ho; Yanhua Shih (1999). "Experimental Realization of Popper's Experiment: violation of the uncertainty principle?". Foundations of Physics 29 (12): 1849–1861. doi:10.1023/A:1018890316979. 
  13. ^ Jarvie, Ian Charles; Milford, Karl; Miller, David W (2006). Karl Popper: a centenary assessment,. 3. Ashgate Publishing. ISBN 9780754657125. 
  14. ^ Popper, Karl; Carl Friedrich von Weizsäcker (1934). "Zur Kritik der Ungenauigkeitsrelationen (Critique of the Uncertainty Relations)". Naturwissenschaften 22 (48): 807–808. Bibcode 1934NW.....22..807P. doi:10.1007/BF01496543. 
  15. ^ Popper, K. Quantum theory and the schism in Physics, Unwin Hyman Ltd, 1982, pp. 53-54.
  16. ^ Mehra, Jagdish; Rechenberg, Helmut (2001). The Historical Development of Quantum Theory. Springer. ISBN 9780387950860. 
  17. ^ The GMc-interpretation of Quantum Mechanics, by Christian Jansson, February 25, 2008
  18. ^ Gabrielse, Gerald; H. Dehmelt (1985), "Observation of Inhibited Spontaneous Emission", Physical Review Letters 55 (1): 67–70, Bibcode 1985PhRvL..55...67G, doi:10.1103/PhysRevLett.55.67, PMID 10031682 
  19. ^ Griffiths, David J. An Introduction to Quantum Mechanics Pearson / Prentice Hall (2005).
  20. ^ L. I. Mandelshtam, I. E. Tamm, The uncertainty relation between energy and time in nonrelativistic quantum mechanics, 1945
  21. ^ DeWitt, B. S.; Graham, N. (1973), The Many-Worlds Interpretation of Quantum Mechanics, Princeton: Princeton University Press, pp. 52–53, ISBN 0691081263 
  22. ^ Hirschman, I. I., Jr. (1957), "A note on entropy", American Journal of Mathematics 79 (1): 152–156, doi:10.2307/2372390, JSTOR 2372390, http://jstor.org/stable/2372390. 
  23. ^ Beckner, W. (1975), "Inequalities in Fourier analysis", Annals of Mathematics 102 (6): 159–182, doi:10.2307/1970980, JSTOR 1970980, http://jstor.org/stable/1970980. 
  24. ^ Likharev, K.K.; A.B. Zorin (1985), "Theory of Bloch-Wave Oscillations in Small Josephson Junctions", J. Low Temp. Phys. 59 (3/4): 347–382, Bibcode 1985JLTP...59..347L, doi:10.1007/BF00683782 
  25. ^ Anderson, P.W. (1964), "Special Effects in Superconductivity", in Caianiello, E.R., Lectures on the Many-Body Problem, Vol. 2, New York: Academic Press 
  26. ^ Szekely, G. J. & Rizzo, M. L. (2007), "The Uncertainty Principle of Game Theory", American Mathematical Monthly 114 (8): 688–702. 

[edit] References

[edit] External links


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