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:

$\Delta X\Delta P\geq {\hbar \over 2}$

The mathematical statement implies the physical statement. Once an observer measures the position of a particle with accuracy $\Delta X$, the state of the particle immediately after the measurement has $\scriptstyle \Delta P\geq \hbar /(2\Delta X)$.

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. A precise position measurement must alter the momentum by a large indeterminate amount, and vice-versa.

While this is true in all interpretations, in many modern interpretations of quantum mechanics (many-worlds and variants), the quantum state itself is the fundamental physical quantity, not the position 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

Main article: Introduction to quantum mechanics

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:

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

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, between X and P. 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.

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

Heisenberg's gamma-ray microscope for locating an electron (shown in blue). The incoming gamma ray (shown in green) is scattered by the electron up into the microscope's aperture angle θ. The scattered gamma-ray is shown in red. Classical optics shows that the electron position can be resolved only up to an uncertainty Δx that depends on θ and the wavelength λ of the incoming light.

Main article: Heisenberg's microscope

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

Modern understanding

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

Suppose that a physicist has a way to prepare an electron in a particular quantum state. The physicist repeats this procedure 200 times, and for 100 times measures the position, and 100 times measures the momentum. The answers will be different in each of the first 100 and second 100 experiments, and they will cluster around some mean with some spread, measured by the standard deviation.

The standard deviation of the position times the standard deviation of the momentum is never less than $\scriptstyle \hbar /2$.

Richard Feynman gives a simple argument for understanding the uncertainty in momentum and position. He starts with a particle which passes through a small hole, and then encounters a N detectors, covering each possible exit direction arranged in concentric hemispheres. He notes that when the innermost hemisphere of those counters records a particle, then the momentum of the particle and its past trajectory is known, in particular, it may be deduced that the particle left the small hole with the momentum that will be inferred later.

This is actually not in contradition with the uncertainty relation. Rather, once the detector detects the particle, the next detection involves a different counter further out, and to determine which one is impossible. This is the Heisenberg uncertainty relation, and the uncertainty translates to an uncertainty of prediction.

Derivations

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

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

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

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

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

Rearranging this formula leads to:

${\begin{aligned}\|Ax\|^{2}\|Bx\|^{2}\geq \left|\langle Bx|Ax\rangle \right|^{2}&\geq \left|\mathrm {Im} \{\langle Bx|Ax\rangle \}\right|^{2}\\&={\frac {1}{4}}\left|2\,\mathrm {Im} \{\langle Bx|Ax\rangle \}\right|^{2}\\&={\frac {1}{4}}\left|\langle Bx|Ax\rangle -\langle Bx|Ax\rangle ^{*}\right|^{2}\\&={\frac {1}{4}}\left|\langle Bx|Ax\rangle -\langle Ax|Bx\rangle \right|^{2}\\&={\frac {1}{4}}\left|\langle ABx|x\rangle -\langle BAx|x\rangle \right|^{2}\\&={\frac {1}{4}}|\langle (AB-BA)x|x\rangle |^{2}\end{aligned}}$

This gives one form of the Robertson-Schrödinger relation, a general form of the Uncertainty Principle:

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

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

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

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

where

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

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 $A-\langle A\rangle _{\psi }$ for A and $B-\langle B\rangle _{\psi }$ for B, and using the fact that

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

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:

$|(\langle \psi |{\hat {X}})({\hat {P}}|\psi \rangle )|^{2}\leq \Delta X^{2}\Delta P^{2}$

so, rigorously, for any state:

$\Delta X\Delta P\geq \langle \psi |{\hat {X}}{\hat {P}}|\psi \rangle$

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:

$\Delta X\Delta P\geq |\langle \psi |{\hat {X}}{\hat {P}}|\psi \rangle |\geq |\langle \psi |\mathrm {Im} ({\hat {X}}{\hat {P}})|\psi \rangle |={\hbar \over 2}.$

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.

$\langle X^{2}\rangle ={\int _{-\infty }^{\infty }e^{-Ax^{2}}x^{2}dx \over \int _{-\infty }^{\infty }e^{-Ax^{2}}dx}=-{d \over dA}\log(\int _{-\infty }^{\infty }e^{-Ax^{2}}dx)=-{d \over dA}\log({\sqrt {\pi \over A}})={1 \over 2A}$

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 $\scriptstyle p=\hbar k$:

$\langle k|\psi \rangle =\psi (k)=\int _{-\infty }^{\infty }e^{-{Ax^{2} \over 2}+ipx}=\int _{-\infty }^{\infty }e^{-{A \over 2}(x-ip/A)^{2}-{p^{2} \over 2A}}=e^{-{p^{2} \over 2A}}\int _{-\infty }^{\infty }e^{-{A \over 2}(x-ip/A)^{2}}$

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 Δ_ψA and Δ_ψB. 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 J_i denotes angular momentum along the x_i 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:

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

where $\Delta _{\psi }E$ is the standard deviation of the energy operator in the state $|\psi \rangle$, $\Delta _{\psi }B$ stands for the standard deviation of the operator ${\hat {B}}$ and $\langle {\hat {B}}\rangle$ is the expectation value of ${\hat {B}}$ in that state. Although, the second factor in the left-hand side has dimension of time, it is different from the time parameter that enters Schrödinger equation. It is a lifetime of the state $|\psi \rangle$ with respect to the observable $B$. In other words, this is the time after which the expectation value $\langle {\hat {B}}\rangle$ changes appreciably.

Critical reactions

Albert Einstein was not happy with the uncertainty principle, and he challenged Niels Bohr and Werner Heisenberg with many thought experiments designed to test it. The most famous went as follows (see also Bohr-Einstein debates):

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 scale and the box are placed in a gravitational field, then the uncertainty of the position of the clock in the gravitational field will alter the rate, and this introduces the right amount of uncertainty.

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.

While Einstein believed that randomness is a reflection of our ignorance of some fundamental property of reality, Bohr believed that the probability distributions are fundamental and irreducible, and depend on which measurements we choose to perform. There is no objective reality underneath determining the outcome. Not only is there a veil hiding the clockwork, but the clockwork is different depending on how you lift the veil.

But 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 .

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 Coen Brothers film The Man Who Wasn't There, lawyer Freddy Riedenschneider (Tony Shalhoub) uses the Uncertainty Principle as a defense for his client. Riedenschneider can't remember Heisenberg's name, calling him "Fritz something-or-other. Or is it. Maybe it's Werner."
• In the The Luck of the Fryrish episode of the animated sci-fi sitcom Futurama the Professor loses at the horse track when his horse is narrowly beat out in a "quantum finish". He complains, "No fair! You changed the outcome by measuring it!" (this may refer instead to wavefunction collapse).
• In the science fiction television series Star Trek: The Next Generation, the fictional transporters used to "beam" characters to different locations overcame the sampling limitations due to the Uncertainty Principle with the use of "Heisenberg compensators." When asked, "How do the Heisenberg compensators work?" by Time magazine on 28 November 1994, Michael Okuda, technical advisor on Star Trek, famously responded, "They work just fine, thank you."
• 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 the genius 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.
• Horror novelist Dennis Etchison makes mention of the Heisenberg principle in a handful of his stories and books.
• On the television show "CSI: Crime Scene Investigation" in the episode Living Doll, Gil Grissom says that he lives "by the uncertainty principle. The mere act of observing a phenomenon changes its nature." 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.
• In A Sound of Thunder (2005) A space travel technician says that the Heisenberg Uncertainty Principle states that nothing is ever 100% certain.
• The French electronic music group Télépopmusik recorded a song called "dp.dq>=h/4pi" for their album Genetic World (2001).
• In the webcomic Questionable Content, Pintsize tries to explain his lateness using relativity and the Heisenberg Uncertainty Principle.
• In an episode of the West wing, season 5 episode 18 "Access" aired 3-31-2004, a documentary reporter asks the character CJ Craig if she is, "..familiar with the Heisenberg principle?"
• In the television show "Numb3rs" in the episode Uncertainty Principle, Charlie uses Uncertainy Principle to help predict a team of bank robbers actions.

See also

• Quantum indeterminacy
• Introduction to quantum mechanics
• Correspondence principle

References

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

External links

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