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Encyclopedia > Uncertainty principle
Quantum mechanics $Delta x , Delta p ge frac{hbar}{2}$
Uncertainty principle
Introduction to...

Mathematical formulation of... For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ... This box:      Werner Heisenberg and Erwin SchrÃ¶dinger, founders of Quantum Mechanics. ... The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. ...

Fundamental concepts
Quantum state · Wave function
Superposition · Entanglement

Measurement · Uncertainty
Exclusion · Duality
Decoherence · Ehrenfest theorem · Tunneling Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... The Bohr model of the atom The Bohr Model is a physical model that depicts the atom as a small positively charged nucleus with electrons in orbit at different levels, similar in structure to the solar system. ... For other uses, see Interference (disambiguation). ... Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ... The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ... Probability densities for the electron at different quantum numbers (l) In quantum mechanics, the quantum state of a system is a set of numbers that fully describe a quantum system. ... A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. ... Quantum superposition is the application of the superposition principle to quantum mechanics. ... It has been suggested that Quantum coherence be merged into this article or section. ... The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications. ... The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. ... This box:      In physics and chemistry, waveâ€“particle duality is the concept that all matter exhibits both wave-like and particle-like properties. ... In quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior - a feature of classical physics - and give the appearance of wavefunction collapse. ... The Ehrenfest theorem, named after Paul Ehrenfest, relates the time derivative of the expectation value for a quantum mechanical operator to the commutator of that operator with the Hamiltonian of the system. ... In quantum mechanics, quantum tunnelling is a micro and nanoscopic phenomenon in which a particle violates principles of classical mechanics by penetrating or passing through a potential barrier or impedance higher than the kinetic energy of the particle. ...

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In quantum mechanics, the position and momentum of particles do not have precise values, but have a probability distribution. There are no states in which a particle has both a definite position and a definite momentum. The narrower the probability distribution is in position, the wider it is in momentum.

Physically, the uncertainty principle requires that when the position of an atom is measured with a photon, the reflected photon will change the momentum of the atom by an uncertain amount inversely proportional to the accuracy of the position measurement. The amount of uncertainty can never be reduced below the limit set by the principle, regardless of the experimental setup. In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ...

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): In mathematics, root mean square (abbreviated RMS or rms), also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. ... In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of statistical dispersion of its values. ... $Delta X = sqrt{langle X^2 rangle-langle X rangle ^2 } ,$

times the RMS deviation of the momentum from its mean (the standard deviation of P): $Delta P = sqrt{langle P^2 rangle-langle P rangle ^2} ,$

and can never be smaller than a small fixed multiple of Planck's constant: A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... $Delta x Delta p ge frac{h}{4pi} = {hbar over 2}$

The uncertainty principle 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 this observer effect can be. Any measurement of the position with accuracy Δx collapses the quantum state making the standard deviation of the momentum Δp larger than $scriptstyle hbar/2Delta x$. Observer Effect is the name of the 87th episode from the television series Star Trek: Enterprise. ... Early twentieth century studies of the physics of very small-scale phenomena led to the Copenhagen interpretation. ... In certain interpretations of quantum mechanics, wavefunction collapse is one of two processes by which quantum systems apparently evolve according to the laws of quantum mechanics. ...

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 or momentum. Taking this perspective, while the momentum and position are still uncertain, the uncertainty is an effect caused not just by observation, but by any entanglement with the environment. The many-worlds interpretation or MWI (also known as relative state formulation, theory of the universal wavefunction, many-universes interpretation, Oxford interpretation or many worlds), is an interpretation of quantum mechanics that claims to resolve all the paradoxes of quantum theory by allowing every possible outcome to every event to... Entanglement can refer to the process which results in felt from fibers and dust bunnies from hairs etc. ...

Werner Heisenberg formulated the uncertainty principle in Niels Bohr's institute at Copenhagen, while working on the mathematical foundations of quantum mechanics. This box:      Werner Heisenberg and Erwin SchrÃ¶dinger, founders of Quantum Mechanics. ... Werner Karl Heisenberg (December 5, 1901 â€“ February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics and acknowledged to be one of the most important physicists of the twentieth century. ... Niels Henrik David Bohr (October 7, 1885 â€“ November 18, 1962) was a Danish physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. ...

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 motion was not precise at the quantum level, and electrons in an atom did not travel on sharply defined orbits. Rather, the motion was smeared out in a strange way: the time Fourier transform only involving those frequencies which could be seen in quantum jumps. Hans Kramers (center) with George Uhlenbeck and Samuel Goudsmit, circa 1928. ... Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. ... The Bohr model of the atom The Bohr Model is a physical model that depicts the atom as a small positively charged nucleus with electrons in orbit at different levels, similar in structure to the solar system. ... Properties The electron (also called negatron, commonly represented as e−) is a subatomic particle. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...

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. His central result was the canonical commutation relation: In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant . ... $[X,P] = X P - P X = i hbar ,$.

and this result does not have a clear physical interpretation.

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. Heisenberg showed that the commutation relations implies an uncertainty, or in Bohr's language a complementarity. Any two variables which do not commute cannot be measured simultaneously — the more precisely one is known, the less precisely the other can be known. Early twentieth century studies of the physics of very small-scale phenomena led to the Copenhagen interpretation. ... // In physics, complementarity is a basic principle of quantum theory closely identified with the Copenhagen interpretation, and refers to effects such as the wave-particle duality, in which different measurements made on a system reveal it to have either particle-like or wave-like properties. ...

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 will diffract, and its wave will come out in a range of angles. The narrower the slit, the wider the diffracted wave and the greater the uncertainty in momentum afterwards. 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. From de Broglie's relation, the size of the slit and the range in momentum of the diffracted wave are related by Heisenberg's rule: In physics, the de Broglie hypothesis is the statement that all matter (any object) has a wave-like nature (wave-particle duality). ... $Delta x Delta p approx h ,$

In his celebrated paper (1927), Heisenberg established this expression as the minimum amount of unavoidable momentum disturbance caused by any position measurement, 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 he refined his principle: $Delta xDelta pgtrsim h$  (1)

But it was Kennard in 1927 who first proved the modern inequality $sigma_xsigma_pgefrac{hbar}{2},$    (2)

with $scriptstyle hbar=h/2pi$, and σx, σp are the standard deviations of position and momentum. Heisenberg himself only proved relation (2) for the special case of Gaussian states..

## Uncertainty principle and observer effect

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. Observer Effect is the name of the 87th episode from the television series Star Trek: Enterprise. ...

This explanation is sometimes 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 position and momentum, but the experimental devices we have today are just not good enough to produce those states. In fact, states with both definite position and momentum just do not exist in quantum mechanics, so it is not the measurement equipment that is at fault.

It is also misleading in another way, because sometimes it is a failure to measure the particle that produces the disturbance. For example, if a perfect photographic film contains a small hole, and an incident photon is not observed, then its momentum becomes uncertain by a large amount. By not observing the photon, we discover that it went through the hole. In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ...

It is misleading in yet another way, because sometimes the measurement can be performed far away. If two photons are emitted in opposite directions from the decay of positronium, the momentum of the two photons is opposite. By measuring the momentum of one particle, the momentum of the other is determined. This case is subtler, because it is impossible to introduce more uncertainties by measuring a distant particle, but it is possible to restrict the uncertainties in different ways, with different statistical properties, depending on what property of the distant particle you choose to measure. By restricting the uncertainty in p to be very small by a distant measurement, the remaining uncertainty in x stays large. Positronium (Ps) is a system consisting of an electron and its anti-particle, a positron, bound together into an exotic atom. The orbit of the two particles and the set of energy levels is similar to that of the hydrogen atom (electron and proton). ...

But Heisenberg did not focus on the mathematics of quantum mechanics, he was primarily concerned with establishing that the uncertainty is actually a property of the world--- that it is in fact physically impossible to measure the position and momentum of a particle to a precision better than that allowed by quantum mechanics. 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 was a surprising prediction of quantum mechanics, which was not yet 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 was not a bug, but a feature--- a deep, surprising aspect of the universe. In order to do this, he could not just use the mathematical formalism, because it was the mathematical formalism itself that he was trying to justify.

### 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.

One way in which Heisenberg originally argued for the uncertainty principle is by using an imaginary microscope as a measuring device  he imagines an experimenter trying to measure the position and momentum of an electron by shooting a photon at it. Image File history File links Heisenberg_gamma_ray_microscope. ... Image File history File links Heisenberg_gamma_ray_microscope. ... For the book by Sir Isaac Newton, see Opticks. ... Heisenbergs microscope exists only as a thought experiment, one that was proposed by Werner Heisenberg, criticized by his mentor Neils Bohr, and subsequently served as the nucleus of some commonly held ideas, and misunderstandings, about Quantum Mechanics // Basic ideas behind the experiment The above two drawings show a view... For other uses, see Electron (disambiguation). ... In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ...

If the photon has a short wavelength, and therefore a large momentum, the position can be measured accurately. But the photon will be scattered 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 will not disturb the electron's momentum very much, but the scattering will reveal its position only vaguely. For other uses, see Wavelength (disambiguation). ... For other uses, see Wavelength (disambiguation). ...

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 will be poorly resolved. If a small aperture is used, the accuracy of the two resolutions is the other way around. a big (1) and a small (2) aperture For other uses, see Aperture (disambiguation). ... Resolving power is the ability of a microscope or telescope to measure the angular separation of images that are close together. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ...

The 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 bound, which is up to a small numerical factor equal to Planck's constant. Heisenberg did not care to formulate the uncertainty principle as an exact bound, and preferred to use it as a heuristic quantitative statement, correct up to small numerical factors. A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ...

## Critical reactions

The Copenhagen interpretation of quantum mechanics and Heisenberg's Uncertainty Principle were seen as twin targets by detractors who believed in an underlying determinism and realism. Within the Copenhagen interpretation of quantum mechanics, there is no fundamental reality which the quantum state is describing, 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. This article is about the general notion of determinism in philosophy. ... Look up realism, realist, realistic in Wiktionary, the free dictionary. ... Early twentieth century studies of the physics of very small-scale phenomena led to the Copenhagen interpretation. ...

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. â€œEinsteinâ€ redirects here. ... Niels Henrik David Bohr (October 7, 1885 â€“ November 18, 1962) was a Danish physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. ... Niels Bohr with Albert Einstein at Paul Ehrenfests home in Leiden (December 1925) The Bohr-Einstein debates is a popular name given to what was actually a series of epistemological challenges presented by Albert Einstein against what has come to be called the standard or Copenhagen interpretation of quantum...

### Einstein's Slit

The first of Einstein's thought experiment 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 will 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.

### Einstein's Box

Another of Einstein's thought experiments was designed to challenge the time/energy uncertainty principle. It is very similar to the slit experiment in space, except here the narrow window through which the particle passes is in time:

Consider a box filled with light. The box has a shutter, which opens and quickly closes by a clock 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 will lessen 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 at which the shutter opens is uncertain. In the case that the scale and the box are placed in a gravitational field, then in some cases it is the uncertainty of the position of the clock in the gravitational field that will alter the ticking rate, and this can introduce the right amount of uncertainty. This was ironic, because it was Einstein himself who first discovered gravity's effect on clocks. 15ft sculpture of Einsteins 1905 E = mcÂ² formula at the 2006 Walk of Ideas, Germany In physics, mass-energy equivalence is the concept that all mass has an energy equivalence, and all energy has a mass equivalence. ... Graphic representing the gravitational redshift of a neutron star (not exact) In physics, light or other forms of electromagnetic radiation of a certain wavelength originating from a source placed in a region of stronger gravitational field (and which could be said to have climbed uphill out of a gravity well...

### EPR Measurements

Bohr was compelled to modify his understanding of the uncertainty principle after another thought experiment by Einstein. In 1935, Einstein, Podolski and Rosen 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.. Entanglement can refer to the process which results in felt from fibers and dust bunnies from hairs etc. ...

But Einstein came to much more far reaching conclusions from the same thought experiment. He felt 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 tested, since it implies a certain inequality between the probability of different experiments. Experimental results confirm the predictions of quantum mechanics, ruling out local hidden variables. Also Nintendo emulator: 1964 (emulator). ... This article or section is not written in the formal tone expected of an encyclopedia article. ... In quantum mechanics, the EPR paradox is a thought experiment which demonstrates that the result of a measurement performed on one part of a quantum system can have an instantaneous effect on the result of a measurement performed on another part, regardless of the distance separating the two parts. ...

While it is possible to assume that quantum mechanical predictions are due to nonlocal hidden variables, in fact David Bohm invented such a formulation, this is not a satisfactory resolution for the vast majority of physicists. The question of whether a random outcome is predetermined by a nonlocal theory can be philosophical, and 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 will encounter fundamental obstacles when it tries to factor numbers of approximately 10000 digits or more, an achievable task in quantum mechanics . David Bohm. ... The Bloch sphere is a representation of a qubit, the fundamental building block of quantum computers. ... Shors algorithm is a quantum algorithm for factoring an integer N in O((log N)3) time and O(log N) space, named after Peter Shor. ...

### Popper's Criticism

Popper has criticized Heisenberg's form of the uncertainty principle, that a measurement of position disturbs the momentum, based on the following observation: If a particle with definite momentum passes through a narrow slit, the diffracted wave has some amplitude to go in the original direction of motion. If the momentum of the particle is measured after it goes through the slit, there is always some probability, however small, that the momentum will be the same as it was before.

Popper thinks of these rare events as falsifications of the uncertainty principle in Heisenberg's original formulation. In order to preserve the principle, he concludes that Heisenberg's relation does not apply to individual particles or measurements, but only to many many identically prepared particles, to ensembles. Popper's criticism applies to nearly all probabilistic theories, since a probabilistic statement requires many measurements to either verify or falsify. Falsifiability (or refutability or testability) is the logical possibility that an assertion can be shown false by an observation or a physical experiment. ... The word ensemble can refer to a musical ensemble (This, along with ensemble cast are the most commonly used ways to describe an ensemble, though obviously not the only ways) an ensemble cast (drama) (This, along with musical ensemble are the most commonly used ways to describe an ensemble, though...

Popper's criticism does not trouble physicists. Popper's presumes is that the measurement is revealing some preexisting information about the particle, the momentum, which the particle already possesses. In the quantum mechanical description the wavefunction is not a reflection of ignorance about the values of some more fundamental quantities, it is the complete description of the state of the particle. In this philosophical view, the Copenhagen interpretation, Popper's example is not a falsification, since after the particle diffracts through the slit and before the momentum is measured, the wavefunction is changed so that the momentum is still as uncertain as the principle demands. Early twentieth century studies of the physics of very small-scale phenomena led to the Copenhagen interpretation. ...

## Refinements

### Everett's uncertainty principle

While formulating the many-worlds interpretation of quantum mechanics in 1957, Hugh Everett III discovered a much stronger formulation of the uncertainty principle . In the inequality of standard deviations, some states, like the wavefunction: The many-worlds interpretation or MWI (also known as relative state formulation, theory of the universal wavefunction, many-universes interpretation, Oxford interpretation or many worlds), is an interpretation of quantum mechanics that claims to resolve all the paradoxes of quantum theory by allowing every possible outcome to every event to... $psi(x) propto e^{-x^2over .0001 } + e^{- (x-100)^2over .0001 }$

have a large standard deviations of position, but are actually a superpositions 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 which are learned when a random variable described by a probability distribution is found to have a certain value. Not to be confused with information technology, information science, or informatics. ... $I_x = - int |psi(x)|^2 log_2 |psi(x)|^2 dx$

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. In physics, natural units are physical units of measurement defined in terms of universal physical constants in such a manner that some chosen physical constants take on the numerical value of one when expressed in terms of a particular set of natural units. ... $log_2(Delta x Delta p) > 0 ,$

but the lower bound is not precise.

Everett conjectured that for all quantum states: $I_x + I_p > - log_2({1over epi})$

He did not prove this, but he showed that Gaussian states are minima in function space for the left hand side, and that they saturate the inequality. Similar relations with less restrictive right hand sides were rigorously proven many decades later.

## Derivations

The uncertainty principle has a straightforward mathematical derivation.

Given two bounded operators A and B on H, let [A,B]: = ABBA denote their commutator. Then, taking the operator norm, one obtains: In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ... In mathematics, the operator norm is a means to measure the size of certain linear operators. ... $|[A,B]| = |AB-BA| leq |AB|-|BA| = |A||B|+|B||A|=2|A||B|.$

This gives one form of the Robertson-Schrödinger relation, a general form of the Uncertainty Principle: $|[A,B]| leq 2|A||B|.$

In the case that A and B are Hermitian operators, this admits a physical interpretation, as below. On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ...

### Physical interpretation

To make the physical meaning of this inequality more directly apparent, it is often written in the equivalent form: $Delta_{psi} A , Delta_{psi} B ge frac{1}{2} left|leftlangleleft[{A},{B}right]rightrangle_psiright|$

where $leftlangle X rightrangle_psi = leftlangle psi | X psi rightrangle$ $Delta_{psi} X = sqrt{langle {X}^2rangle_psi - langle {X}rangle_psi ^2}$

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 $A - lang Arang_psi$ for A and $B - lang Brang_psi$ for B, and using the fact that In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of statistical dispersion of its values. ... $[A,B]=[A - lang Arang, B - lang Brang].$

This formulation acquires its physical interpretation, indicated by the suggestive terminology "mean" and "standard deviation", due to the properties of measurement in quantum mechanics. The precise position-momentum uncertainty principle is found when A be X and B be P, so that the commutator is $scriptstyle ihbar$. And one should definitely note that the inequality is rigorous. The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications. ...

### Matrix mechanics

In matrix mechanics, the commutator of the matrices X and P is always nonzero, it is a constant multiple $scriptstyle ihbar$ of the identity matrix. This means that it is impossible for a state to have a definite values x for X and p for P, since then XP would be equal to the number xp and would equal PX. Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. ... In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ...

The commutator of two matrices is unchanged when they are shifted by a constant multiple of the identity--- for any two real numbers x and p $[X-x, P- p] = [X,P] = ihbar ,$

Given any quantum state ψ, define the number x $x=langle psi|X|psirangle = sum_{ij} psi^*_i X_{ij} psi_j$

to be the expected value of the position, and $p=langle psi|P|psirangle= sum_{ij} psi^*_i P_{ij} psi_j$

to be the expected value of the momentum. The quantities $scriptstyle hat X = X-x$ and $scriptstyle hat P = P-p$ are only nonzero to the extent that the position and momentum are uncertain, to the extent that the state contains some values of X and P which deviate from the mean. The expected value of the commutator $langle psi| hat X hat P - hat P hat X |psirangle = langle psi| [ hat X, hat P ] |psirangle = i hbar langle psi|psi rangle = i hbar ,$

can only be nonzero if the deviations in X in the state $scriptstyle |psirangle$ times the deviations in P are large enough.

The size of the typical matrix elements can be estimated by summing the squares over the energy states $scriptstyle |irangle$: $sum_i |langle psi| hat X |irangle |^2 = sum_i langle psi|hat X |iranglelangle i|hat X |psirangle = langle psi| hat X^2 |psirangle = Delta X^2 ,$

and this is equal to the square of the deviation, matrix elements have a size approximately given by the deviation.

So in order to produce the canonical commutation relations, the product of the deviations in any state has to be about $scriptstyle hbar$. $Delta X Delta P gtrapprox hbar$

This heuristic estimate can be made into a precise inequality using the Cauchy-Schwartz inequality, exactly as before. The inner product of the two vectors in parentheses: In mathematics, the Cauchy-Schwarz inequality, also known as the Schwarz inequality, or the Cauchy-Bunyakovski-Schwarz inequality, is a useful inequality encountered in many different settings, such as linear algebra applied to vectors, in analysis applied to infinite series and integration of products, and in probability theory, applied to... $(langle psi| hat X ) (hat P |psirangle)$

is bounded above by the product of the lengths of each vector: $|(langle psi|hat X)(hat P |psirangle)|^2 le Delta X^2 Delta P^2$

so, rigorously, for any state: $Delta X Delta P ge langle psi | hat X hat P |psi rangle$

the real part of a matrix M is $scriptstyle (M+M^dagger)/2$, so that the real part of the product of two Hermitian matrices $scriptstyle hat X hat P$ is: $mathrm{Re} (hat X hat P) = { hat X hat P + hat X hat P over 2 } = {{X,P}over 2}$

while the imaginary part is $mathrm{Im} (hat X hat P) = {hat X hat P - hat X hat P over 2i } = { [hat X,hat P] over 2i }= { hbar over 2}.$

The magnitude of $scriptstyle langle psi | hat X hat P |psi rangle$ is bigger than the magnitude of its imaginary part, which is the expected value of the imaginary part of the matrix: $Delta X Delta P ge | langle psi | hat X hat P |psi rangle | ge | langle psi | mathrm{Im} (hat X hat P ) |psirangle | = {hbar over 2}.$

Note that the uncertainty product is for the same reason bounded below by the expected value of the anticommutator, which adds a term to the uncertainty relation. The extra term is not as useful for the uncertainty of position and momentum, because it has zero expected value in a gaussian wavepacket, like the ground state of a harmonic oscillator. The anticommutator term is useful for bounding the uncertainty of spin operators though.

### Wave mechanics

In Schrödinger's wave mechanics The quantum mechanical wavefunction contains information about both the position and the momentum of the particle. The position of the particle is where the wave is concentrated, while the momentum is the typical wavelength. In physics, the SchrÃ¶dinger equation, proposed by the Austrian physicist Erwin SchrÃ¶dinger in 1925, describes the time-dependence of quantum mechanical systems. ...

The wavelength of a localized wave cannot be determined very well. If the wave extends over a region of size L and the wavelength is approximately λ, the number of cycles in the region is approximately L / λ. The inverse of the wavelength can be changed by about 1 / L without changing the number of cycles in the region by a full unit, and this is approximately the uncertainty in the inverse of the wavelength, $Delta left({1over lambda}right) = {1over L}$

This is an exact counterpart to a well known result in signal processing --- the shorter a pulse in time, the less well defined the frequency. The width of a pulse in frequency space is inversely proportional to the width in time. It is a fundamental result in Fourier analysis, the narrower the peak of a function, the broader the Fourier transform. Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... Fourier analysis, named after Joseph Fouriers introduction of the Fourier series, is the decomposition of a function in terms of a sum of sinusoidal basis functions (vs. ...

Multiplying by h, and identifying ΔP = hΔ(1 / λ), and identifying ΔX = L. $Delta P Delta X gtrapprox h$

The uncertainty Principle can be seen as a theorem in Fourier analysis: 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). Fourier analysis, named after Joseph Fouriers introduction of the Fourier series, is the decomposition of a function in terms of a sum of sinusoidal basis functions (vs. ...

An instructive example is the (unnormalized) gaussian wave-function $langle x | psi rangle = psi(x) = e^{- {Ax^2 over 2}}.$

The expectation value of X is zero by symmetry, and so the variance is found by averaging X2 over all positions with the weight ψ(x)2, careful to divide by the normalization factor. $langle X^2 rangle = {int_{-infty}^infty e^ {- A x^2} x^2 dx over int_{-infty}^infty e^{- Ax^2} dx } = - {dover dA} log ( int_{-infty}^infty e^{- A x^2} dx ) = - {dover dA} log(sqrt{piover A} ) = {1 over 2A}$

The fourier transform of the gaussian is the wavefunction in k-space, where k is the wavenumber and is related to the momentum by DeBroglie's relation $scriptstyle p=hbar k$: $langle k | psi rangle = psi(k) = int_{-infty}^{infty} e^{- {Ax^2over 2} + i p x} = int_{-infty}^{infty} e^{ - {Aover 2}(x - ip/A)^2 - {p^2over 2A} } = e^{-{p^2over 2A}} int_{-infty}^{infty} e^{- {Aover 2}(x- ip/A)^2}$

The last integral does not depend on p, because there is a continuous change of variables $xrightarrow x-ip/A$ which removes the dependence, and this deformation of the integration path in the complex plane does not pass any singularities. So up to normalization, the answer is again a Gaussian. $langle k | psi rangle = e^{- p^2 over 2A} :$

The width of the distribution in k is found in the same way as before, and the answer just flips A to 1/A. $Delta k^2 = {Delta P^2 over hbar^2} = {A over 2}$

so that for this example $Delta X Delta P = sqrt{1over 2A}sqrt{hbar^2 Aover 2} = {hbar over 2}$

which shows that the uncertainty relation inequality is tight. There are wavefunctions which saturate the bound.

## Robertson-Schrödinger relation

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 ΔψA and ΔψB. Then On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ... The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications. ... $Delta_psi A , Delta_psi B geq sqrt{ frac{1}{4}left|leftlangleleft[{A},{B}right]rightrangle_psiright|^2 +{1over 4} left|leftlangleleft{ A-langle Arangle_psi,B-langle Brangle_psi right} rightrangle_psi right|^2}$

where [A,B] = AB - BA is the commutator of A and B, {A,B}= AB+BA is the anticommutator, and $langle X rangle_psi$ is the expectation value. This inequality is called the Robertson-Schrödinger relation, and includes the Heisenberg uncertainty principle as a special case. 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. In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. ... In quantum physics, the Heisenberg uncertainty principle states that one cannot assign with full precision values for certain pairs of observable variables, including the position and momentum, of a single particle at the same time. ... Year 1930 (MCMXXX) was a common year starting on Wednesday (link will display 1930 calendar) of the Gregorian calendar. ... Howard Percy Robertson (January 27, 1903 - August 26, 1961) was a scientist known for contributions related to cosmology and the uncertainty principle. ... SchrÃ¶dinger in 1933, when he was awarded the Nobel Prize in Physics Bust of SchrÃ¶dinger, in the courtyard arcade of the main building, University of Vienna, Austria. ...

### Other uncertainty principles

The Robertson Schrödinger relation gives the uncertainty relation for any two observables that do not commute:

• 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): $Delta E Delta x geq {hbarover 2m} left|leftlangle p_{x}rightrangleright|$
• between angular position and angular momentum of an object with small angular uncertainty: $Delta Theta_i Delta J_i gtrapprox frac{hbar}{2}$ $Delta J_i Delta J_j geq frac{hbar}{2} left|leftlangle J_krightrangleright|$
where i, j, k are distinct and Ji denotes angular momentum along the xi axis. $Delta N Delta phi geq 1$

This box:      This gyroscope remains upright while spinning due to its angular momentum. ... Superconductivity is a phenomenon occurring in certain materials at low temperatures, characterised by the complete absence of electrical resistance and the damping of the interior magnetic field (the Meissner effect. ... There are very few or no other articles that link to this one. ... In physics, Ginzburg-Landau theory is a mathematical theory used to model superconductivity. ...

## 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: For a generally accessible and less technical introduction to the topic, see Introduction to special relativity. ... Niels Henrik David Bohr (October 7, 1885 â€“ November 18, 1962) was a Danish physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. ... $Delta E Delta t gtrapprox hbar$,

but it was not obvious what Δt is, 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!"

Nevertheless, Einstein and Bohr understood the heuristic meaning of the principle. A state which only exists for a short time cannot have a definite energy. In order to have a definite energy, the frequency of the state needs to be accurately 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. Electromagnetic spectroscopy a. ... The spectral linewidth characterises the width of a peak in the spectrum of a system, e. ...

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 . 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. The term mass in special relativity is used in a couple of different ways, occasionally leading to a great deal of confusion. ... Thousands of particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ...

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. Professor Yakir Aharonov BSc PhD is a physicist specialising in Quantum Physics and holds a joint professorship at Tel Aviv University, Israel and the University of South Carolina, America. ... David Bohm. ...

In 1936, Dirac offered a precise definition and derivation of the time-energy uncertainty relation, in a relativistic quantum theory of "events". In this formulation, particles followed a trajectory in space time, and each particles trajectory was parametrized independently by a different proper time. The many-times formulation of quantum mechanics is mathematically equivalent to the standard formulations, but it was in a form more suited for relativistic generalization. It was the inspiration for Shin-Ichiro Tomonaga's to covariant perturbation theory for quantum electrodynamics. Sin-Itiro Tomonaga or Shinichirō Tomonaga (朝永 振一郎 Tomonaga Shinichirō, March 31, 1906–July 8, 1979) was a Japanese physicist, influential in the development of quantum electrodynamics, work for which he was jointly awarded the Nobel Prize in Physics in 1965 along with Richard Feynman and Julian Schwinger. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ...

But a better-known, more widely-used formulation of the time-energy uncertainty principle was given only in 1945 by L. I. Mandelshtam and I. E. Tamm, as follows. For a quantum system in a non-stationary state $|psirangle$ and an observable B represented by a self-adjoint operator $hat B$, the following formula holds: Leonid Isaakovich Mandelshtam (Леонид Исаакович Мандельштам, last name more often spelled as Mandelstam) (May 4, 1879 - November 27, 1944) was a Russian/Soviet physicist of Jewish background. ... Igor Tamm. ... $Delta_{psi} E frac{Delta_{psi} B}{left | frac{mathrm{d}langle hat B rangle}{mathrm{d}t}right |} ge frac{hbar}{2}$,

where ΔψE is the standard deviation of the energy operator in the state $|psirangle$, 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 $|psirangle$ with respect to the observable B. In other words, this is the time after which the expectation value $langlehat Brangle$ changes appreciably. This box:      For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ...

## Popular culture

The uncertainty principle appears in popular culture in many places, although it is sometimes stated imprecisely, or as a stand-in for the observer effect: Observer Effect is the name of the 87th episode from the television series Star Trek: Enterprise. ...

• 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."
• In 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!", conflating the Uncertainty principle with the observer effect.
• In the well known joke: "Heisenberg is pulled over by a policeman whilst driving down a motorway, the policeman gets out of his car, walks towards Heisenberg's window and motions with his hand for Heisenberg to wind the window down, which he does. The policeman then says ‘Do you know what speed you were driving at sir?’, to which Heisenberg responds ‘No, but I knew exactly where I was.’"
• 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 conflates the uncertainty principle with the observer effect.
• 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 on 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 explain (albeit incorrectly) 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.
• 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" again conflating it with the observer effect.
• 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.
• The French electronic music group Télépopmusik recorded a song called "dp.dq>=h/4pi" for their album Genetic World (2001).
• In the webcomic Questionable Content, Pintsize tries to explain his lateness using relativity and the Heisenberg Uncertainty Principle.

Quantum indeterminacy is the apparent necessary incompleteness in the description of a physical system, that has become one of the characteristics of the standard description of quantum physics. ... This box:      Werner Heisenberg and Erwin SchrÃ¶dinger, founders of Quantum Mechanics. ... In physics, the correspondence principle is a principle, first invoked by Niels Bohr in 1923, which states that the behavior of quantum mechanical systems reduce to classical physics in the limit of large quantum numbers. ... Results from FactBites:

 Heisenberg - Quantum Mechanics, 1925-1927: The Uncertainty Principle (966 words) Heisenberg's route to uncertainty lies in a debate that began in early 1926 between Heisenberg and his closest colleagues on the one hand, who espoused the "matrix" form of quantum mechanics, and Erwin Schrödinger and his colleagues on the other, who espoused the new "wave mechanics." His analysis showed that uncertainties, or imprecisions, always turned up if one tried to measure the position and the momentum of a particle at the same time. (Similar uncertainties occurred when measuring the energy and the time variables of the particle simultaneously.) These uncertainties or imprecisions in the measurements were not the fault of the experimenter, said Heisenberg, they were inherent in quantum mechanics.
 The Uncertainty Principle (Stanford Encyclopedia of Philosophy) (10573 words) The uncertainty principle played an important role in many discussions on the philosophical implications of quantum mechanics, in particular in discussions on the consistency of the so-called Copenhagen interpretation, the interpretation endorsed by the founding fathers Heisenberg and Bohr. This should not suggest that the uncertainty principle is the only aspect of the conceptual difference between classical and quantum physics: the implications of quantum mechanics for notions as (non)-locality, entanglement and identity play no less havoc with classical intuitions. Popper argued that the uncertainty relations cannot be granted the status of a principle on the grounds that they are derivable from the theory, whereas one cannot obtain the theory from the uncertainty relations.
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