Uncertainty principle

In quantum physics, the Heisenberg uncertainty principle is the statement that locating a particle in a small region of space makes the momentum of the particle uncertain; and conversely, that measuring the momentum of a particle precisely makes the position uncertain.

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.

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):
 * $$\Delta X = \sqrt{\langle X^2 \rangle-\langle X \rangle ^2 } \,$$

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

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

\Delta X \Delta P \ge {\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 $$\Delta x$$ collapses the quantum state making the standard deviation of the momentum $$\Delta p$$ larger than $$\scriptstyle \hbar/2\Delta x$$.

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.

Historical Introduction
Werner Heisenberg formulated the uncertainty principle in Niels Bohr's institute at Copenhagen, while working on the mathematical foundations of quantum mechanics.

In 1925, following pioneering work with Hendrik Kramers, Heisenberg developed matrix mechanics, which replaced the ad-hoc old quantum theory with modern quantum mechanics. The central assumption was that the classical 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.

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:


 * $$ [X,P] = X P - P X = i \hbar \,$$.

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.

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 $$\Delta\theta$$ is about $$\lambda/d$$, where d is the slit width and $$\lambda$$ 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:


 * $$\Delta x \Delta p \approx h \,$$

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:


 * $$\Delta x\Delta p\gtrsim h$$ &emsp; &emsp;&#12288;&emsp;&#12288; &emsp;&#12288; (1)

But it was Kennard in 1927 who first proved the modern inequality


 * $$\sigma_x\sigma_p\ge\frac{\hbar}{2}\,$$ &emsp;&#12288;  &emsp;&#12288;&emsp;&#12288; &emsp;&#12288; (2)

with $$\scriptstyle \hbar=h/2\pi$$, 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.

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.

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.

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


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.

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.

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.

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.

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.

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.

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 $$\Delta P$$ 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 $$h/\Delta P$$, 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
Einstein's most famous thought experiment 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 introduces the right amount of uncertainty. This was ironic, because it was Einstein himself who first discovered gravity's effect on clocks.

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. .

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.

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.

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.

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.

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:

\psi(x) \propto e^{-x^2\over .0001 } + e^{- (x-100)^2\over .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 inquality 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.

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 $$\epsilon$$ is equal to $$I_x + \log_2(\epsilon)$$. 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 $$\Delta x$$ the information content is $$log_2\Delta x$$. This quantity can be negative, which means that the distribution is narrower than one unit, so that learning the first few bits past the decimal point gives no information since they are not uncertain.

Taking the logarithm of Heisenberg's formulation of uncertainty in natural units.

\log_2(\Delta x \Delta p) > 0 \,$$ but the lower bound is not precise.

Everett conjectured that for all quantum states:

I_x + I_p > - \log_2({1\over e\pi}) $$

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 $$[A,B] := AB-BA$$ denote their commutator. Then, taking the operator norm, one obtains:
 * $$\|[A,B]\| = \|AB-BA\| \leq \|AB\|+\|BA\| = \|A\|\|B\|+\|B\|\|A\|=2\|A\|\|B\|.$$

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.

Detailed 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, a general form of the Uncertainty Principle:
 * $$\frac{1}{4} |\langle [A,B]x | x \rangle|^2\leq \| A x \|^2 \| B x \|^2,$$

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

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|\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. The precise position-momentum uncertainty principle is found when A be X and B be P, so that the commutator is $$\scriptstyle i\hbar$$. And one should definitely note that the inequality is rigorous.

Matrix mechanics
In matrix mechanics, the commutator of the matrices X and P is always nonzero, it is a constant multiple $$\scriptstyle i\hbar$$ 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.

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


 * $$[X-x, P- p] = [X,P] = i\hbar \,$$

Given any quantum state $$\psi$$, define the number x


 * $$x=\langle \psi|X|\psi\rangle = \sum_{ij} \psi^*_i X_{ij} \psi_j $$

to be the expected value of the position, and


 * $$p=\langle \psi|P|\psi\rangle= \sum_{ij} \psi^*_i P_{ij} \psi_j$$

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 |\psi\rangle = \langle \psi| [ \hat X, \hat P ] |\psi\rangle = i \hbar \langle \psi|\psi \rangle = i \hbar \, $$

can only be nonzero if the deviations in X in the state $$\scriptstyle |\psi\rangle$$ 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 $$\scriptstyle |i\rangle$$:

\sum_i |\langle \psi| \hat X |i\rangle |^2 = \sum_i \langle \psi|\hat X |i\rangle\langle i|\hat X |\psi\rangle = \langle \psi| \hat X^2 |\psi\rangle = \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:



(\langle \psi| \hat X ) (\hat P |\psi\rangle) $$

is bounded above by the product of the lengths of each vector:



$$
 * (\langle \psi|\hat X)(\hat P |\psi\rangle)|^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 ) |\psi\rangle | = {\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.

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 $$\lambda$$, the number of cycles in the region is approximately $$ L/\lambda $$. 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 wavelength.


 * $$ \Delta \lambda = {1\over L} $$

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.

multiplying by $$h$$, and identifying $$\Delta P = h \Delta \lambda$$, and identifying $$\Delta X = L$$.


 * $$ \Delta P \Delta X \gtrapprox h $$

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).

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 $$X^2$$ over all positions with the weight $$\psi(x)^2$$, 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 } = - {d\over dA} \log ( \int_{-\infty}^\infty e^{- A x^2} dx ) = - {d\over dA} \log(\sqrt{\pi\over 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^2\over 2} + i p x} = \int_{-\infty}^{\infty} e^{ - {A\over 2}(x - ip/A)^2 - {p^2\over 2A} } = e^{-{p^2\over 2A}} \int_{-\infty}^{\infty} e^{- {A\over 2}(x- ip/A)^2} $$

The last integral does not depend on p, because there is a continuous change of variables $$x\rightarrow x-ip/A $$ 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


 * $$ \Delta X \Delta P = \sqrt{1\over 2A}\sqrt{\hbar^2 A\over 2} = {\hbar \over 2} $$

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
 * $$\Delta_\psi A \, \Delta_\psi B \geq \sqrt{ \frac{1}{4}\left|\left\langle\left[{A},{B}\right]\right\rangle_\psi\right|^2

+{1\over 4} \left|\left\langle\left\{ A-\langle A\rangle_\psi,B-\langle B\rangle_\psi \right\} \right\rangle_\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.

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):


 * $$\Delta E \Delta x \geq {\hbar\over 2m} \left|\left\langle p_{x}\right\rangle\right| $$


 * between angular position and angular momentum of an object with small angular uncertainty:


 * $$\Delta \Theta_i \Delta J_i \gtrapprox \frac{\hbar}{2} $$


 * between two orthogonal components of the total angular momentum operator of an object:


 * $$ \Delta J_i \Delta J_j \geq \frac{\hbar}{2} \left|\left\langle J_k\right\rangle\right|$$
 * where i, j, k are distinct and Ji denotes angular momentum along the xi axis.


 * between the number of electrons in a superconductor and the phase of its Ginzburg-Landau order parameter


 * $$ \Delta N \Delta \phi \geq 1$$

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:


 * $$ \Delta E \Delta t \gtrapprox \hbar $$,

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.

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.

One false formulation of the energy-time uncertainty principle says that measuring the energy of a quantum system to an accuracy $$\Delta E$$ requires a time interval $$\Delta t > h/\Delta E$$. 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 $$\Delta t$$ in the uncertainty relation is the time during which the system exists unperturbed, not the time during which the experimental equipment is turned on.

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 formualations, 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.

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 $$|\psi\rangle$$ 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 $$\Delta_{\psi} E$$ 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.

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:


 * 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 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 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.
 * 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.