Uncertainty principle

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Quantum mechanics
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In quantum physics, the Heisenberg uncertainty principle is the statement that any experiment which locates a particle in a small region makes the momentum of that particle uncertain, and conversely, any measurement of the momentum of a particle makes the position uncertain.

It is a consequence of the fact that in quantum mechanics, the position and momentum are not determined, but have a probability distribution. There do not exist states in quantum mechanics in which a particle has both a definite position and momentum. The narrower the probability distribution is in position, the wider it is in momentum.

The mathematical statement 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):

\Delta X ^2 = \langle (X-\langle X \rangle)^2 \rangle \,

times the RMS deviation of the momentum from its mean (the standard deviation of P):

\Delta P ^2 = \langle (P-\langle P \rangle)^2 \rangle \,

can never be smaller than a small fixed multiple of Planck's constant:

\Delta X \Delta P \ge {\hbar \over 2}

The mathematical statement implies the physical statement, since once an observer measures the position with accuracy ΔX, the state of the particle immediately after the measurement has \scriptstyle \Delta P \ge \hbar / 2\Delta X .

The uncertainty principle was an important step in extracting the physical meaning of quantum mechanics. Quantum mechanics was already largely developed when the uncertainty principle was discovered by Werner Heisenberg in 1927, but the original mathematical form of quantum mechanics, matrix mechanics, was difficult to understand because the physical quantities were represented by infinite matrices. The central result of matrix mechanics was the canonical commutation relation:

[X,P] = X P - P X = i \hbar

While this result does not have an obvious physical meaning, Heisenberg showed that it implies an uncertainty (or in Bohr's language a complementarity) between X and P. The physical content of the commutation relation is the uncertainty relation--- any two variables which do not commute have an uncertainty relation of some kind.

The uncertainty princple is related to the observer effect, with which it is often conflated. In the copenhagen interpretation of quantum mechanics, the uncertainty principle is a theoretical limitation of how small the observer effect can be. In the Copenhagen interpretation, a precise position measurement must alter the momentum by a large indeterminate amount and vice-versa.

While this is true in all interpretations, in many modern interpretations of quantum mechanics (many-worlds and variants), the quantum state itself is the fundamental physical quantity, not the position and momentum. Taking this perspective, while the momentum and position are still uncertain in any quantum state, the uncertainty in the quantities produced by measurment is not just an effect caused by observervation, but an effect caused by any entanglement with the enviroment.

Contents

Main article: Introduction to quantum mechanics

The quantum state of a system (for example, a particle) is a full, quantitative, description of that quantum system. One typically imagines some experimental apparatus and procedure which "prepares" (sets up) a certain quantum state. Once the quantum state has been prepared, some aspect of it can be measured (for example, its position, energy, or momentum). If the experiment is repeated, so as to prepare and measure the same aspect of the same quantum state, the result of the measurement will often be different.

In fact, the expected result of the measurement is in general described not by a single number, but by a probability distribution that specifies the likelihoods that the various possible results will be obtained. The measurement process is often said to be random and indeterministic. (However, there is considerable dispute over this issue; in some interpretations of quantum mechanics, the result merely appears random and indeterministic.)

Every quantum state thus has a certain probability distribution associated with measuring its position, another associated with measuring its energy, yet another associated with measuring its momentum, and so forth. Each of these distributions specifies a range of likely values for the result of that measurement, and the wider the range is, the more "uncertain" we say that aspect of the quantum state is. It is convenient to use the standard deviation of the distribution as a way to quantify this uncertainty; the higher the standard deviation is, the wider the range of possible values, and the more uncertain that property is for the quantum state.

The Heisenberg uncertainty principle gives a quantitative relation between a quantum state's uncertainty in position and its uncertainty in momentum: Their product is greater than or equal to a certain universal constant (the reduced Planck constant divided by two). In other words, there are some quantum states for which measuring position gives a very predictable result, but making a momentum measurement on any of those will give a wide range of possible results; and there are other quantum states for which measuring momentum gives a very predictable result, but making a position measurement on any of those will give a wide range of possible results. There is no quantum state which has both a very specific position and a very specific momentum.

The more fundamental fact that measurement results in quantum mechanics are unpredictable, which is built into the foundations of modern-day quantum mechanics, is not the same as the Uncertainty Principle (although of course, they are related, especially historically). For more information on unpredictibility in quantum-mechanical measurements, see the article Measurement in quantum mechanics.

Main article: Wave–particle duality

A fundamental postulate of quantum mechanics, which manifests itself in the Heisenberg Uncertainty Principle, is that no physical phenomenon can be (to arbitrary accuracy) described as a "classic point particle" or as a wave but rather the microphysical situation is best described in terms of wave-particle duality.

The Heisenberg uncertainty principle is a consequence of this picture. The amplitude of the wave associated with a particle corresponds to its position, and the wavelength (more precisely, its Fourier transform) is inversely proportional to momentum. In order to localize the wave so as to have a sharp peak (i.e., a small position uncertainty), it is necessary to incorporate waves with very short wavelengths, corresponding to high momenta in all directions, and therefore a large momentum uncertainty. Indeed, the Heisenberg Uncertainty Principle is equivalent to a theorem in functional analysis that the standard deviation of the squared absolute value of a function, times the standard deviation of the squared absolute value of its Fourier transform, is at least 1/(16π²) (Folland and Sitaram, Theorem 1.1).

A helpful analogy can be drawn between the wave associated with a quantum-mechanical particle and a more familiar wave, the time-varying signal associated with, say, a sound wave. It is meaningless to ask about the frequency spectrum at a single moment in time, because the measure of frequency is the measure of a repetition recurring over a period of time. Indeed, in order for a signal to have a relatively well-defined frequency, it must persist for a long period of time, and conversely, a signal that occurs at a relatively well-defined moment in time (i.e., of short duration) will necessarily encompass a broad frequency band. This is, indeed, a close mathematical analogue of the Heisenberg uncertainty principle. See also Complementarity (physics).

The uncertainty principle is often explained as the statement that the measurement of position necessarily disturbs a particle's momentum, and vice versa—i.e., that the uncertainty principle is a manifestation of the observer effect.

This statement is misleading in a modern context, because it makes it seem that the disturbances are somehow conceptually avoidable, that there are states of the particle with definite positions and momentum, but the experimental devices we have today are just not good enough to produce those states. In fact, there do not exist quantum mechanical states of definite position and momentum, so it is not the properties of the measurement equipment that is responsible for uncertainty, it is the mathematical structure of the quantum state space.

But Heisenberg did not focus on this mathematical result, he also went to great lengths to argue that the uncertainty is actually a property of the world--- that it in fact actually is physically impossible to measure the position and momentum of a particle to better precision. To do this, he used physical arguments based on the existence of quanta, but not the full quantum mechanical formalism.

The reason is that this is a surprising prediction of the quantum formalism, which was not yet universally accepted. Many people would have considered it a flaw that there are no states of definite position and momentum. Heisenberg was trying to show that this is not a flaw, but a deep and surprising aspect of the universe. In order to do this, he could not use the mathematical formalism, because it was the mathematical formalism itself that he was trying to justify.

One way in which Heisenberg originally argued for the uncertainty principle is by using an imaginary microscope as a measuring device.[1]

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.
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

In Heisenberg's initial paper,[2] he offered an example to demonstrate his understanding of his uncertainty principle. The example is now called Heisenberg's microscope (or the gamma-ray microscope), and is as follows. An experimenter tries to measure the position and momentum of an electron by shooting a photon at it. If the photon has a small wavelength (and thus large momentum), an accurate position is measured, but the collision with the photon creates great uncertainties in the electron's momentum; if the photon has a large wavelength (and thus low momentum), the collision will not much disturb the electron's momentum, but little information will be gained about the electron's position. Likewise, if a large aperture is used, the electron's location can be well resolved (see Rayleigh criterion), but the transverse momentum of the incoming photon, and hence (by conservation of momentum) the new momentum of the electron, will be poorly resolved; if a small aperture is used, the reverse holds.

As a consequence of these tradeoffs, 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 bound, which in this case is the Planck constant divided by two.[3] This is higher (by a factor of 2π) than the minimum possible bound \hbar/2 = h/(4\pi), as derived below; thus this particular experiment produces even more uncertainty than the exact bound required by quantum mechanics. In fact, Heisenberg was not aware that there was an exact bound, and thought of the uncertainty principle as a heuristic, rather than as a precise quantitative statement.[1]

The Uncertainty Principle is a property of quantum states, corresponding to the statistical properties of measurement in quantum mechanics. To clarify this point, consider again the Heisenberg microscope experiment above.

Suppose that a physicist has an experimental procedure which results in an electron being set up (perfectly) in a particular quantum state. The physicist repeats this procedure 200 times (for example). For each of the first 100 times, the physicist then measures the position of the electron, using a tiny-wavelength photon and a huge aperture. Each time, a very precise value of the position is obtained, but that value may be very different each time (see Measurement in quantum mechanics). The final 100 times, the physicist measures the momentum of the electron, using a large-wavelength photon and a tiny aperture. Each time, a very precise value of the momentum is obtained, but again, the value may be very different each time.

The position uncertainty is defined as the standard deviation of all the position measurements. This uncertainty reflects the differences from one measurement to the next, a consequence not of the measurement process (which can be made arbitrarily precise), but of the state of the electron before the measurement: an electron cloud spread out in space. Likewise, the momentum uncertainty is defined as the standard deviation of all the momentum measurements. Again, this uncertainty reflects the state of the electron before the measurement, not the measurement process itself. Finally, the product of the position uncertainty and momentum uncertainty can be computed, and the value will turn out to be at least \hbar/2.

In this experiment, the physicist never attempts to measure the position and momentum of any single electron but measures a different quantity for each different electron, each prepared in the same initial state. This means that no one measurement affects any of the other measurements, but the uncertainty principle still holds. So the uncertainty principle expresses an intrinsic statistical property of states, it is not just a limitation imposed by the measurement process.[4]


Richard Feynman, in his famous Lectures on Physics, describes the uncertainty principle as follows:

...The most interesting aspect is the idea of the uncertainty principle; making an observation affects the phenomenon.[5]

However, in context, it is clear that Feynman is using the term "uncertainty principle" not to describe that particular theorem in quantum mechanics (as the term is used in this article), but rather as a broad extension of Heisenberg's original usage, and in particular as a pedagogical tool to introduce some aspects of measurement in quantum mechanics (particularly decoherence).[6] Moreover, in other contexts, he uses the Heisenberg Uncertainty Principle to explain phenomena that are clearly unrelated to the observer effect, such as the size of atoms.[7]

Measurements of position and momentum taken in several identical copies of a system in a given state will each vary according to a probability distribution characteristic of the system’s state. This is the fundamental postulate of quantum mechanics.

If we compute the standard deviations Δx and Δp of the position and momentum measurements, then

\Delta x \Delta p \ge \frac{\hbar}{2}

where

\hbar (h-bar) is the reduced Planck's constant (Planck's constant divided by 2π).

More generally, given any Hermitian operators A and B, and a system in the state ψ, there are probability distributions associated with the measurement of each of A and B, giving rise to standard deviations ΔψA and ΔψB. Then

\Delta_\psi A \, \Delta_\psi B \geq \frac{1}{2}\left|\left\langle\left[{A},{B}\right]\right\rangle_\psi\right|

where the operator [A,B] = AB - BA denotes the commutator of A and B, and \langle X \rangle_\psi denotes expectation value. This inequality is called the Robertson-Schrödinger relation, and includes the Heisenberg Uncertainty Principle as a special case. It was first pointed out in 1930 by Howard Percy Robertson and (independently) by Erwin Schrödinger.

Due to the Robertson-Schrödinger relation above, an uncertainty relation arises between any two observable quantities that can be defined by non-commuting operators. A few of the more common examples follow:

  • There is an uncertainty relation between the position and momentum of an object:
\Delta x_i \Delta p_i \geq \frac{\hbar}{2}
  • between the energy and position of a particle in a one-dimensional potential V(x):
\Delta E \Delta x \geq \frac{\hbar}{2m} \left|\left\langle p_{x}\right\rangle\right|
  • between angular position and angular momentum of an object:
\Delta O_i \Delta J_i \geq \frac{\hbar}{2}
\Delta J_i \Delta 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.
\Delta N \Delta \phi \geq 1

Unlike the above examples, some uncertainty principles are not direct consequences of the Robertson-Schrödinger relation. The most famous of these is the energy-time uncertainty principle.

Applying the ideas of special relativity to the position-momentum uncertainty principle, many physicists, like Niels Bohr, postulated that the following relation ought to exist:

\Delta E \Delta t \ge \frac{\hbar}{2} ,

but it was not immediately obvious how Δt should be defined (since time is not treated as an operator). In 1926, Dirac offered a precise definition and derivation of this uncertainty relation, as coming from a relativistic quantum theory of "events". But the better-known, more widely-used, correct formulation was given only in 1945 by L. I. Mandelshtam and I. E. Tamm, as follows. For a quantum system in a non-stationary state |\psi\rangle and an observable B represented by a self-adjoint operator \hat B, the following formula holds:

\Delta_{\psi} E \frac{\Delta_{\psi} B}{\left | \frac{\mathrm{d}\langle \hat B \rangle}{\mathrm{d}t}\right |} \ge \frac{\hbar}{2} ,

where ΔψE is the standard deviation of the energy operator in the state |\psi\rangle , ΔψB stands for the standard deviation of the operator \hat B and \langle \hat B \rangle is the expectation value of \hat B in that state. 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 |\psi\rangle with respect to the observable B. In other words, this is the time after which the expectation value \langle\hat B\rangle changes appreciably.

The energy-time uncertainty principle has important implications for spectroscopy. Since excited states have a finite lifetime, they do not all release exactly the same energy when they decay; rather, the spectroscopic peaks are actually bumps with a finite width (called natural linewidth), with the center of the bump corresponding to the true energy of the excited state. For fast-decaying states, the linewidth makes it difficult to accurately measure this true energy, and indeed, researchers have even used microwave cavities to slow down the decay-rate, in order to get sharper peaks and more accurate energy measurements[10].

One particularly famous false formulation of the energy-time uncertainty principle says that the energy of a quantum system measured over the time interval Δt has to be inaccurate, with the inaccuracy ΔE given by the inequality \Delta E \Delta t \ge \hbar/2. This formulation was explicitly invalidated by Y. Aharonov and D. Bohm in 1961. Indeed, one can actually determine the accurate energy of a quantum system in an arbitrarily short interval of time. Moreover, as recent research indicates, for quantum systems with discrete energy spectra the product ΔEΔt is bounded from above by a statistical noise that in fact vanishes if sufficiently many identical copies of the system are used. This vanishing upper bound certainly removes the possibility of a lower bound, again disproving this false formulation of the energy-time uncertainty principle.

The uncertainty principle has a straightforward mathematical derivation. The key step is an application of the Cauchy-Schwarz inequality, one of the most useful theorems of linear algebra.

For two arbitrary Hermitian operators A: H → H and B: H → H, and any element x of H, then

\langle B A x | x \rangle = \langle A x | B x \rangle = \langle B x | A x \rangle^{*}

In an inner product space the Cauchy-Schwarz inequality holds.

\left|\langle B x | A x \rangle\right |^2 \leq \|A x \|^2 \|B x \|^2

Rearranging this formula leads to:

\begin{align} \|A x \|^2 \|B x \|^2 \geq \left|\langle B x | A x \rangle\right |^2 &\geq \left|\mathrm{Im}\{\langle B x | A x \rangle\}\right |^2 \\ &= \frac{1}{4} \left|2 \, \mathrm{Im}\{\langle B x | A x \rangle\}\right |^2 \\ &= \frac{1}{4} \left| \langle B x | A x \rangle - \langle B x | A x \rangle^{*} \right |^2 \\ &= \frac{1}{4} \left| \langle B x | A x \rangle - \langle A x | B x \rangle \right |^2 \\ &= \frac{1}{4} \left| \langle A B x | x \rangle - \langle B A x | x \rangle \right |^2 \\ &= \frac{1}{4} |\langle (AB - BA)x | x \rangle|^2 \end{align}

This gives one form of the Robertson-Schrödinger relation:

\frac{1}{4} |\langle [A,B]x | x \rangle|^2\leq \| A x \|^2 \| B x \|^2,

where the operator [A,B] = AB - BA denotes the commutator of A and B.

To make the physical meaning of this inequality more directly apparent, it is often written in the equivalent form:

\Delta_{\psi} A \, \Delta_{\psi} B \ge \frac{1}{2} \left|\left\langle\left[{A},{B}\right]\right\rangle_\psi\right|

where

\left\langle X \right\rangle_\psi = \left\langle \psi | X \psi \right\rangle

is the operator mean of observable X in the system state ψ and

\Delta_{\psi} X = \sqrt{\langle {X}^2\rangle_\psi - \langle {X}\rangle_\psi ^2}

is the operator standard deviation of observable X in the system state ψ. This formulation can be derived from the above formulation by plugging in A - \lang A\rang_\psi for A and B - \lang B\rang_\psi for B, and using the fact that

[A,B]=[A - \lang A\rang, B - \lang B\rang].

This formulation acquires its physical interpretation, indicated by the suggestive terminology "mean" and "standard deviation", due to the properties of measurement in quantum mechanics. Particular uncertainty relations, such as position-momentum, can usually be derived by a straightforward application of this inequality.

Main article: Introduction to quantum mechanics

The Uncertainty Principle was developed as an answer to the question: How does one measure the location of an electron around a nucleus?

In the summer of 1922 Heisenberg met Niels Bohr, the founding father of quantum mechanics, and in September 1924 Heisenberg went to Copenhagen, where Bohr had invited him as a research associate and later as his assistant. In 1925 Werner Heisenberg laid down the basic principles of a complete quantum mechanics. In his new matrix theory he replaced classical commuting variables with non-commuting ones. Heisenberg's paper marked a radical departure from previous attempts to solve atomic problems by making use of observable quantities only. He wrote in a 1925 letter, "My entire meagre efforts go toward killing off and suitably replacing the concept of the orbital paths that one cannot observe." Rather than struggle with the complexities of three-dimensional orbits, Heisenberg dealt with the mechanics of a one-dimensional vibrating system, an anharmonic oscillator. The result was formulae in which quantum numbers were related to observable radiation frequencies and intensities. In March 1926, working in Bohr's institute, Heisenberg formulated the principle of uncertainty thereby laying the foundation of what became known as the Copenhagen interpretation of quantum mechanics.

Albert Einstein was not happy with the uncertainty principle, and he challenged Niels Bohr and Werner Heisenberg with a famous thought experiment (See the Bohr-Einstein debates for more details): we fill a box with a radioactive material which randomly emits radiation. The box has a shutter, which is opened and soon thereafter shut by a clock at a precise time, thereby allowing some radiation to escape. So the time is already known with precision. We still want to measure the conjugate variable energy precisely. Einstein proposed doing this by weighing the box before and after. The equivalence between mass and energy from special relativity will allow you to determine precisely how much energy was left in the box. Bohr countered as follows: should energy leave, then the now lighter box will rise slightly on the scale. That changes the position of the clock. Thus the clock deviates from our stationary reference frame, and by general relativity, its measurement of time will be different from ours, leading to some unavoidable margin of error. In fact, a detailed analysis shows that the imprecision is correctly given by Heisenberg's relation.

The term Copenhagen interpretation of quantum mechanics was often used interchangeably with and as a synonym for Heisenberg's Uncertainty Principle by detractors who believed in fate and determinism and saw the common features of the Bohr-Heisenberg theories as a threat. Within the widely but not universally accepted Copenhagen interpretation of quantum mechanics (i.e., it was not accepted by Einstein or other physicists such as Alfred Lande), the uncertainty principle is taken to mean that on an elementary level, the physical universe does not exist in a deterministic form — but rather as a collection of probabilities, or potentials. For example, the pattern (probability distribution) produced by millions of photons passing through a diffraction slit can be calculated using quantum mechanics, but the exact path of each photon cannot be predicted by any known method. The Copenhagen interpretation holds that it cannot be predicted by any method, not even with theoretically infinitely precise measurements.

It is this interpretation that Einstein was questioning when he said "I cannot believe that God would choose to play dice with the universe." Bohr, who was one of the authors of the Copenhagen interpretation, responded, "Einstein, don't tell God what to do." Niels Bohr himself acknowledged that quantum mechanics and the uncertainty principle were counter-intuitive when he stated, "Anyone who is not shocked by quantum theory has not understood a single word."

The basic debate between Einstein and Bohr (including Heisenberg's Uncertainty Principle) was that Einstein was in essence saying: "Of course, we can know where something is; we can know the position of a moving particle if we know every possible detail, and therefore by extension, we can predict where it will go." Bohr and Heisenberg were saying: "We can only know the probable position of a moving particle, therefore by extension, we can only know its probable destination; we can never know with absolute certainty where it will go."

Einstein was convinced that this interpretation was in error. His reasoning was that all previously known probability distributions arose from deterministic events. The distribution of a flipped coin or a rolled die can be described with a probability distribution (50% heads, 50% tails), but this does not mean that their physical motions are unpredictable. Ordinary mechanics can be used to calculate exactly how each coin will land, if the forces acting on it are known. And the heads/tails distribution will still line up with the probability distribution (given random initial forces).

Einstein assumed that there are similar hidden variables in quantum mechanics which underlie the observed probabilities and that these variables, if known, would show that there was what Einstein termed "local realism," a description opposite to the uncertainty principle, being that all objects must already have their properties before they are observed or measured. For the greater part of the twentieth century, there were many such hidden variable theories proposed, but in 1964 John Bell theorized the Bell inequality to counter them, which postulated that although the behavior of an individual particle is random, it is also correlated with the behavior of other particles. Therefore, if the uncertainty principle is the result of some deterministic process in which a particle has local realism, it must be the case that particles at great distances instantly transmit information to each other to ensure that the correlations in behavior between particles occur. The interpretation of Bell's theorem explicitly prevents any local hidden variable theory from holding true because it shows the necessity of a system to describe correlations between objects. The implication is, if a hidden local variable is the cause of particle 1 being at a position, then a second hidden local variable would be responsible for particle 2 being in its own position — and there is no system to correlate the behavior between them. Experiments have demonstrated that there is a correlation. In the years following, Bell's theorem was tested and has held up experimentally time and time again, and these experiments are in a sense the clearest experimental confirmation of quantum mechanics. It is worth noting that Bell's theorem only applies to local hidden variable theories; non-local hidden variable theories can still exist (which some, including Bell, think is what can bridge the conceptual gap between quantum mechanics and the observable world).

Whether Einstein's view or Heisenberg's view is true or false is not a directly empirical matter. One criterion by which we may judge the success of a scientific theory is the explanatory power it gives us, and to date it seems that Heisenberg's view has been the better at explaining physical subatomic phenomena.

The uncertainty principle is stated in popular culture in many ways, for example, by some stating that it is impossible to know both where an electron is and where it is going at the same time. This is roughly correct, although it fails to mention an important part of the Heisenberg principle, which is the quantitative bounds on the uncertainties. Heisenberg stated that it is impossible to determine simultaneously and with unlimited accuracy the position and momentum of a particle, but due to Planck's Constant being so small, the Uncertainty Principle was intended to apply only to the motion of atomic particles. However, culture often misinterprets this to mean that it is impossible to make a completely accurate measurement.

  • In the Coen Brothers film The Man Who Wasn't There, lawyer Freddy Riedenschneider (Tony Shalhoub) uses the Uncertainty Principle as a defense for his client. Riedenschneider can't remember Heisenberg's name, calling him "Fritz something-or-other. Or is it. Maybe it's Werner."
  • In the The Luck of the Fryrish episode of the animated sci-fi sitcom Futurama the Professor loses at the horse track when his horse is narrowly beat out in a "quantum finish". He complains, "No fair! You changed the outcome by measuring it!" (this may refer instead to wavefunction collapse).
  • In the science fiction television series Star Trek: The Next Generation, the fictional transporters used to "beam" characters to different locations overcame the sampling limitations due to the Uncertainty Principle with the use of "Heisenberg compensators." When asked, "How do the Heisenberg compensators work?" by Time magazine on 28 November 1994, Michael Okuda, technical advisor on Star Trek, famously responded, "They work just fine, thank you."[11]
  • In the 1997 film The Lost World: Jurassic Park, chaostician Ian Malcolm claims that the effort "to observe and document, not interact" with the dinosaurs is a scientific impossibility because of "the Heisenberg Uncertainty Principle, whatever you study, you also change." This is an inaccurate confusion with the observer effect, as explained above.
  • The Michael Frayn play Copenhagen (1998) highlights some of the processes that went into the formation of the Uncertainty Principle. The play dramatizes the meetings between Werner Heisenberg and Niels Bohr. It highlights, as well, the discussion of the work that both did to help build nuclear bombs - Heisenberg for Germany and Bohr for the United States and allied forces.
  • In an episode of the television show Aqua Teen Hunger Force, Meatwad (who was temporarily made into a genius) tries to incorrectly explain Heisenberg's Uncertainty Principle to Frylock to explain his new found intelligence. "Heisenberg's Uncertainty Principle tells us that at a specific curvature of space, knowledge can be transferred into energy, or — and this is key now — matter."
  • In an episode of Stargate SG-1, Samantha Carter explains, using the Uncertainty Principle, that the future is not predetermined, that one can only calculate possibilities.
  • Horror novelist Dennis Etchison makes mention of the Heisenberg principle in a handful of his stories and books.
  • On the television show "CSI: Crime Scene Investigation" in the episode Living Doll, Gil Grissom says that he lives "by the uncertainty principle. The mere act of observing a phenomenon changes its nature." This is an inaccurate confusion with the observer effect, as explained above.
  • In Episode 16 (No Need for Hiding) of the English-dubbed version of the Japanese anime Tenchi Universe, Washu gives a rapid explanation of the Uncertainty Principle while singing karaoke.
  • In A Sound of Thunder (2005) A space travel technician says, incorrectly, that the Heisenberg Uncertainty Principle states that nothing is ever 100% certain.
  • The French electronic music group Télépopmusik recorded a song called "dp.dq>=h/4pi" for their album Genetic World (2001).

  1. ^ a b Hilgevoord, Jan and Jos Uffink, "The Uncertainty Principle", The Stanford Encyclopedia of Philosophy (Fall 2006 Edition), Edward N. Zalta (ed.), URL = [1]
  2. ^ Heisenberg W. (1930) Die Physikalischen Prinzipien der Quantenmechanik (Leipzig: Hirzel). English translation The Physical Principles of Quantum Theory (Chicago: University of Chicago Press, 1930). (As cited in the Stanford Encyclopedia of Philosophy.)
  3. ^ Tipler, Paul A.; Ralph A. Llewellyn (1999). "5-5", Modern Physics, 3rd Ed., W. H. Freeman and Co.. ISBN 1-5725-9164-1. 
  4. ^ Cite error: No text given.
  5. ^ The Feynman Lectures on Physics (with Leighton and Sands), 1970 paperback three-volume set. ISBN 0-201-02115-3. Volume III, Ch. 2, page 8.
  6. ^ For example, in Vol. III, Ch. 1, page 9, he says that the "uncertainty principle" states that one cannot measure which slit, in the double-slit experiment, a photon passes through, without destroying the interference pattern. This is the Englert-Greenberger duality relation, and it is a consequence not of the "Heisenberg uncertainty principle" (the theorem relating position and momentum uncertainties, as described in this article), but of decoherence-related phenomena: See Berthold-Georg Englert, Marlan O Scully & Herbert Walther, Quantum Optical Tests of Complementarity, Nature, Vol 351, pp 111-116 (9 May 1991).
  7. ^ Volume I, Ch. 2, page 6
  8. ^ 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. 
  9. ^ Anderson, P.W. (1964), "Special Effects in Superconductivity", in Caianiello, E.R., Lectures on the Many-Body Problem, Vol. 2, New York: Academic Press
  10. ^ Gabrielse, Gerald; H. Dehmelt (1985). "Observation of Inhibited Spontaneous Emission". Physical Review Letters 55: 67-70. 
  11. ^ "Reconfigure the Modulators!", Time Magazine, November 28, 1994. 

  • W. Heisenberg, "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik", Zeitschrift für Physik, 43 1927, pp. 172-198. English translation: J. A. Wheeler and H. Zurek, Quantum Theory and Measurement Princeton Univ. Press, 1983, pp. 62-84.
  • L. I. Mandelshtam, I. E. Tamm "The uncertainty relation between energy and time in nonrelativistic quantum mechanics", Izv. Akad. Nauk SSSR (ser. fiz.) 9, 122-128 (1945). English translation: J. Phys. (USSR) 9, 249-254 (1945).
  • G. Folland, A. Sitaram, "The Uncertainty Principle: A Mathematical Survey", Journal of Fourier Analysis and Applications, 1997 pp 207-238.

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Top Matching Results may highlight information from other Search.com pages, content from the CNET Network of sites, or third party content. The listings are based purely on relevance. Search.com does not receive payment for listings in this section but our partners that provide this data may get paid for listing these products.

Sponsored Links

This section contains paid listings which have been purchased by companies that want to have their sites appear for specific search terms and related content. These listings are administered, sorted and maintained by a third party and are not endorsed by Search.com.

Search Results

Search.com sends your search query to several search engines at one time and integrates the results into one list which has been sorted by relevance using Search.com's proprietary algorithm. You can customize the list of search engines included in your metasearch from the preferences.

The search engines that are used in your metasearch may allow companies to pay to have their Web sites included within the results. To view the Paid Inclusion policy for a specific search engine, please visit their Web site. Search.com does not accept payment or share revenue with any search engine partner for listings in this section.