Uncertainty principle

In quantum physics, the Heisenberg uncertainty principle is a mathematical limit on the accuracy with which it is possible to measure everything there is to know about a physical system. In its simplest form, it applies to the position and momentum of a single particle, and implies that if we continue increasing the accuracy with which one of these is measured, there will come a point at which the other must be measured with less accuracy. Mathematically, if Δx and Δp are the uncertainties in the measurements of the position and momentum, then the product ΔxΔp is at least on the order of Planck's constant. Stated with more mathematical rigor, the uncertainty principle states that when measuring conjugate quantities, the product of their standard deviations must be at least $\hbar /2$.

In classical physics, it was believed that if one knew the initial state of a system with infinite precision, one could predict the behavior of the system infinitely far into the future. According to quantum mechanics, however, there is a fundamental limit on the ability to make such predictions, because of the inability to collect the initial data with unlimited precision. Fundamentally, the uncertainty principle is a result of wave-particle duality. If a particle is to have a position that is determined to within Δx, then its wavelength must be less than about Δx. But this short wavelength, according to the de Broglie relation, requires a high momentum, which may vary from about -h/Δx to h/Δx.

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.

Mathematics provides a positive lower bound for the product of the uncertainties of measurements of the conjugate quantities. The uncertainty principle is one of the cornerstones of quantum mechanics and was discovered by Werner Heisenberg in 1927. The uncertainty principle follows from the mathematical definition of operators in quantum mechanics; it is represented by a set of theorems of functional analysis. It is often confused with the observer effect.

Until the discovery of quantum physics, it was thought that the only source of uncertainty in a measurement was the limited precision of the measuring tool. It is now understood that no treatment of any scientific subject, experiment, or measurement is accurate until the probability distribution for the measurement is specified. Uncertainty is the characterization of the relative narrowness or broadness of the distribution function of a particular measurement and is sometimes referred to as the error in the measurement.

Illustrative of this is an experiment in which a particle is prepared in a definite state and two successive measurements are performed on the particle. The first one measures the particle's position and the second immediately after measures its momentum. Each time the experiment is performed, some value x is obtained for position and some value p is obtained for momentum. Depending upon the precision of the instrument taking the measurements, each successive measurement of the positions and momenta respectively should be nearly identical, but in practice they will exhibit some deviation owing to constraints of measurement using a real world instrument that is not infinitely precise. However, Heisenberg showed that, even in theory with a hypothetical infinitely precise instrument, no measurement could be made to arbitrary accuracy of both the position and the momentum of a physical object.

The Heisenberg uncertainty principle (developed in an essay published in 1927) provides a quantitative relationship between the uncertainties of the hypothetical infinitely precise measurements of p and x as measured by the spreads of their distributions in the following way: if the particle's state is such that the first measurement yields a dispersion of values Δx, then the second measurement will have a distribution of values whose dispersion Δp is at least inversely proportional to Δx. For the limiting case, the constant of proportionality is derivable using commutator arithmetic. It is equal to Planck's constant

This stipulates that the product of the uncertainties in position and momentum is equal to or greater than about 10⁻³⁵ joule-seconds. Therefore, the product of the uncertainties only becomes significant for regimes where the uncertainty in position or momentum measurements is small. Thus, the uncertainty principle governs the observable nature of atoms and subatomic particles while its effect on measurements in the macroscopic world is negligible and can be usually ignored.

The Heisenberg uncertainty relations are a theoretical bound over all measurements. They hold for so-called ideal measurements, sometimes called von Neumann measurements. They hold even more so for non-ideal or Landau measurements.

Wave-particle duality

Main article: Wave–particle duality

A fundamental consequence of 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 uncertainty principle, as initially considered by Heisenberg, is concerned with cases in which neither the wave nor the point particle descriptions are fully and exclusively appropriate, such as a particle in a box with a particular energy value. Such systems are characterized neither by one unique "position" (one particular value of distance from a potential wall) nor by one unique value of momentum (including its direction). Any observation that determines either a position or a momentum of such a waveparticle to arbitrary accuracy — known as wavefunction collapse — is subject to the condition that the width of the wavefunction collapse in position, multiplied by the width of the wavefunction collapse in momentum, is constrained by the principle to be greater than or equal to Planck's constant divided by 4$\pi$.

Every measured particle in quantum mechanics exhibits wavelike behavior so there is an exact, quantitative analogy between the Heisenberg uncertainty relations and properties of waves or signals. For example, in a time-varying signal such as 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. In order to determine the frequencies accurately, the signal needs to be sampled for a finite (non-zero) time. This necessarily implies that time precision is lost in favor of a more accurate measurement of the frequency spectrum of a signal. This is analogous to the relationship between momentum and position, and there is an equivalent formulation of the uncertainty principle which states that the uncertainty of energy of a wave (directly proportional to the frequency) is inversely proportional to the uncertainty in time with a constant of proportionality identical to that for position and momentum.

The uncertainty principle in quantum mechanics is sometimes erroneously explained by claiming that the measurement of position necessarily disturbs a particle's momentum. Heisenberg himself may have initially offered explanations which suggested this view. That this disturbance does not describe the essence of the uncertainty principle in current theory has been demonstrated above. The fundamentally non-classical characteristics of the uncertainty measurements in quantum mechanics were clarified by the EPR paradox which arose from Einstein attempting to show flaws in quantum measurements that used the uncertainty principle. Instead of succeeding in showing that uncertainty was flawed, Einstein guided researchers to examine more closely what uncertainty measurements meant, which led to a more refined understanding of uncertainty. Prior to the publication of the EPR paper in 1935, a measurement was often visualized as a physical disturbance inflicted directly on the measured system, being sometimes illustrated as a thought experiment called Heisenberg's microscope. For instance, when measuring the position of an electron, one imagines shining a light on it, thus disturbing the electron and producing the quantum mechanical uncertainties in its position. Such explanations, which are still encountered in popular expositions of quantum mechanics, are debunked by the EPR paradox, which shows that a "measurement" can be performed on a particle without disturbing it directly, by performing a measurement on a distant entangled particle. Heisenberg's original argument used the 'old' quantum theory (namely, the Einstein-deBroglie relations) and provided a heuristic argument that the position and momentum observables were not simultaneously observable with infinite precision. The more modern uncertainty relations deal with independent measurements being done on an ensemble of systems.

Formulation and characteristics

Measurements of position and momentum taken in several identical copies of a system in a given state will vary according to known probability distributions. This is the fundamental postulate of quantum mechanics.

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

$\Delta x\Delta p\geq {\frac {\hbar }{2}}$

where

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

Two years earlier in 1925 when Heisenberg had developed his matrix mechanics the difference in position and momentum were already showing up in the formula. In developing matrix mechanics Heisenberg was measuring amplitudes of position and momentum of particles such as the electron that have a period of 2π, like a cycle in a wave, which are called Fourier series variables. When amplitudes of position and momentum are measured and multiplied together, they give intensity. However, Heisenberg found that the order of multiplication of these two quantities affected the result he obtained (the calculated intensity of h/(2π) - position and momentum did not commute. In 1927, to develop the standard deviation for the uncertainty principle, Heisenberg took the gaussian distribution or bell curve for the imprecision in the measurement of the position q of a moving electron to the corresponding bell curve of the measured momentum p. This gave the minimum standard deviation as $\hbar /2$.

Corroborating observations

An uncertainty relation arises between any two observable quantities that can be defined by non-commuting operators. This means that the uncertainty principle arises in measuring the position and the velocity of an object, or in measuring the position and momentum of an object.

• The most common one is the uncertainty relation between position and momentum of a particle in space:

$\Delta x_{i}\Delta p_{i}\geq {\frac {\hbar }{2}}$

• The uncertainty relation between two orthogonal components of the total angular momentum operator of a particle is as follows:

$\Delta J_{i}\Delta J_{j}\geq {\frac {\hbar }{2}}\left|\left\langle J_{k}\right\rangle \right|$

where i, j, k are distinct and J_i denotes angular momentum along the x_i axis.

Deriving

Surprisingly the uncertainty principle is an application of one of the most important theorems of linear algebra, the Cauchy-Schwarz inequality.

For two arbitrary hermitian operators A: H → H and B: H → H, and any element x of H such that A B x and B A x are both defined (so that in particular, A x and B x are also defined), then

$\langle BAx|x\rangle =\langle Ax|Bx\rangle =\langle Bx|Ax\rangle ^{*}$

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

$\left|\langle Bx|Ax\rangle \right|^{2}\leq \|Ax\|^{2}\|Bx\|^{2}$

Rearranging this formula leads to:

$\|Ax\|^{2}\|Bx\|^{2}\geq \left|\langle Bx|Ax\rangle \right|^{2}\geq \left|im(\langle Bx|Ax\rangle )\right|^{2}={\frac {1}{4}}\left|2im(\langle Bx|Ax\rangle )\right|^{2}={\frac {1}{4}}\left|\langle Bx|Ax\rangle -\langle Bx|Ax\rangle ^{*}\right|^{2}=$

$={\frac {1}{4}}\left|\langle Bx|Ax\rangle -\langle Ax|Bx\rangle \right|^{2}={\frac {1}{4}}\left|\langle ABx|x\rangle -\langle BAx|x\rangle \right|^{2}={\frac {1}{4}}|\langle (AB-BA)x|x\rangle |^{2}$

Consequently, the following general form of the uncertainty principle, first pointed out in 1930 by Howard Percy Robertson and (independently) by Erwin Schrödinger, holds:

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

This inequality is called the Robertson-Schrödinger relation.

The operator AB - BA is called the commutator of A, B and is denoted [A, B]. It is defined on those x for which ABx and BAx are both defined.

However, the Robertson-Schrödinger uncertainty relation is not the uncertainty as stated in the Heisenberg uncertainty principle. That is, this derivation is inherently only applicable to measurements on a statistical ensemble of systems, and says nothing, contrary to most popular statements, about the simultaneous measurements of individual systems.

Considering that

$[A,B]=[A-\langle A\rangle ,B-\langle B\rangle ]$

the following formulation of the uncertainty principle is an immediate conclusion of the Robertson-Schrödinger relation (insert $A-\langle A\rangle$ for A and $B-\langle B\rangle$ for B).

Suppose A and B are two observables which are identified to self-adjoint (and in particular symmetric) operators. If B A ψ and A B ψ are defined then

$\Delta _{\psi }A\,\Delta _{\psi }B\geq {\frac {1}{2}}\left|\left\langle \left[{A},{B}\right]\right\rangle _{\psi }\right|$

where

$\left\langle X\right\rangle _{\psi }=\left\langle \psi |X\psi \right\rangle =\|X\psi \|^{2}$

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

The above definitions of mean and standard deviation are defined formally in purely operator-theoretic terms. The statement becomes more meaningful however, once we note that these actually are the mean and standard deviation for the measured distribution of values. See quantum statistical mechanics.

It may be evaluated not only for pairs of conjugate operators (for example, those defining measurements of distance and of momentum, or of duration and of energy) but generally for any pair of Hermitian operators. There is also an uncertainty relation between the field strength and the number of particles which is responsible for the phenomenon of virtual particles.

Note that it is possible to have two non-commuting self-adjoint operators A and B which share an eigenvector ψ, in this case ψ represents a pure state in which it is predictable with probability one what the result of measuring A or B will be in spite of their not being simultaneously measurable.

Energy, time and further generalizations

General arguments, connected with the theory of relativity, point out that seemingly a relation like the following should exist:

$\Delta E\Delta t\geq {\frac {\hbar }{2}}$.

But its correct mathematical formulation was given only in 1945 by L. I. Mandelshtam and I. E. Tamm. This is the only known formulation of the time-energy uncertainty relation rigorously derived from the mathematical structure of quantum mechanics. For a quantum system in a non-stationary state $|\psi \rangle$ and an observable $B$ represented by a self-adjoint operator ${\hat {B}}$, the formula takes the following form:

$\Delta _{\psi }E{\frac {\Delta _{\psi }B}{\left|{\frac {{\mbox{d}}\langle {\hat {B}}\rangle }{{\mbox{d}}t}}\right|}}\geq {\frac {\hbar }{2}}$,

where $\Delta _{\psi }E$ is the standard deviation of the energy operator in the state $|\psi \rangle$, $\Delta _{\psi }B$ 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 relation has an important implication for spectroscopy. As excited states have a short lifetime their energy uncertainty is not negligible. For this reason sharp lines cannot be obtained even under ideal conditions. This relation helps also to give an idea of the "chaotic" behavior of the space-time, wherein very small time steps authorize huge energy variations.

Historically, the time-energy uncertainty relation was motivated by many physicists, like Niels Bohr, especially in the vaguer form $\Delta E\Delta t\approx h$, simply by relating E and t to x and p in the original relation.

In particular, one false formulation is still present. It says that the energy of a quantum system measured over the time interval $\Delta t$ has to be inaccurate, with the inaccuracy $\Delta E$ given by the inequality $\Delta E\Delta t\geq {\frac {\hbar }{2}}$. This formulation has been invalidated by Y. Aharonov and D. Bohm in 1961. One can actually determine 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 $\Delta E\Delta t$ is bounded from above by a statistical noise that vanishes if sufficiently many identical copies of the system are used. This vanishing upper bound removes the possibility of lower bounds including many false formulations of the time-energy uncertainty.

In 1926 Dirac offered an alternative rigourous derivation of the uncertainty relation

$\Delta E\Delta t\geq {\frac {\hbar }{2}}$

as coming from a relativistic quantum theory of "events".

History and interpretations

Main article: Interpretation of 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 immediately 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 thereby by extension, we can predict where it will go." Bohr and Heisenberg were saying the opposite: "There is no way to know where a moving particle is ever even given every possible detail, and thereby by extension, we can never predict 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 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.

Popular culture

The uncertainty principle is stated in popular culture in many ways, for example by 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.

The uncertainty principle is sometimes connected to the "observer effect," wherein the observation of an event changes the event. The observer effect is an important effect in many fields, from electronics to psychology and social science. Following the optical paradigm used by Heisenberg in the 1920s, many (but not all) commentators have interpreted Heisenberg's original research to mean that that every observation requires an energy exchange to create observed 'data'--an intervention that changes the energy (wave state)of the observed object. Although, in recent years, new technologies have made possible less invasive forms of observation, the uncertainty principle--as originally formulated and adapted--supports the conclusion that the act of measurement itself introduces an irreducible uncertainty in certain measurements.

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."

In the science fiction television series Star Trek: The Next Generation, the fictional transporters used to "beam" characters to different locations overcome the limitations of sampling the subject 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."

There is also a British experimental drone doom metal band called Uncertainty Principle.

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 converted to energy... ,or and this is key now, matter."

See also

• Quantum indeterminacy
• Introduction to quantum mechanics
• Correspondence principle

References

Journal articles

• 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.

External links

• Matter as a Wave - a chapter from an online textbook
• The Uncertainty Relations: Description, Applications on Project PHYSNET
• Quantum mechanics: Myths and facts
• Stanford Encyclopedia of Philosophy entry
• aip.org: Quantum mechanics 1925-1927 - The uncertainty principle
• Eric Weisstein's World of Physics - Uncertainty principle
• Schrödinger equation from an exact uncertainty principle
• John Baez on the time-energy uncertainty relation
• The time-energy certainty relation