Uncertainty_principle

In quantum physics, the Heisenberg uncertainty principle states that the values of certain pairs of conjugate variables (position and momentum, for instance) cannot both be known with arbitrary precision. That is, the more precisely one variable is known, the less precisely the other is known. This is not a statement about the limitations of a researcher's ability to measure particular quantities of a system, but rather about the nature of the system itself.

In quantum mechanics, the particle is described by a wave. The position is where the wave is concentrated and the momentum, a measure of the velocity, is the wavelength. The position is uncertain to the degree that the wave is spread out, and the momentum is uncertain to the degree that the wavelength is ill-defined.

The only kind of wave with a definite position is concentrated at one point, and such a wave has an indefinite wavelength. Conversely, the only kind of wave with a definite wavelength is an infinite regular periodic oscillation over all space, which has no definite position. So in quantum mechanics, there are no states which describe a particle with both a definite position and a definite momentum. The narrower the probability distribution is for the position, the wider it is in momentum.

The uncertainty principle requires that when the position of an atom is measured, the measurement process will leave the momentum of the atom changed by an uncertain amount inversely proportional to the accuracy of the measurement. The amount of uncertainty can never be reduced below the limit, no matter what the measurement process.

This means that 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 the theoretical lower limit of how small the observer effect can be.

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):

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

Any measurement of the position with accuracy $\scriptstyle \Delta X$ collapses the quantum state making the standard deviation of the momentum $\scriptstyle \Delta P$ larger than $\scriptstyle \hbar/2\Delta x$ .

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:

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

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 his celebrated paper (1927), Heisenberg established this expression as the minimum amount of unavoidable momentum disturbance caused by any position measurement¹, 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 lecture² he refined his principle:

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

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, revealing the photon's position.

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.

One way in which Heisenberg originally argued for the uncertainty principle is by using an imaginary microscope as a measuring device.² 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.⁴ 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 experiments challenging the uncertainty principle went as follows:

Bohr's response was that the wall is quantum mechanical as well, and that to measure the recoil to accuracy

Δ 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 / Δ 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

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:

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.

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.⁵

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, and 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 mechanics⁶.

Popper's criticism

Karl Popper 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 identically prepared particles, called 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 presumption 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.

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 principle⁷. In the inequality of standard deviations, some states, like the wavefunction

have a large standard deviation of position, but are actually a superposition 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.

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

I x + log 2 (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

Δ x

the information content is

log 2 Δ 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.

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

When linear operators A and B act on a function

ψ(x)

, they don't always commute. A clear example is when operator B multiplies by x, while operator A takes the derivative with respect to x. Then

This example is important, because it is very close to the canonical commutation relation of quantum mechanics. There, the position operator multiplies the value of the wavefunction by x, while the corresponding momentum operator differentiates and multiplies by $\scriptstyle -i\hbar$ , so that:

which is a statement of the Cauchy-Schwarz inequality for the inner product of the two vectors $\scriptstyle A|\psi\rangle$ and $\scriptstyle B|\psi\rangle$ . The expectation value of the product AB is greater than the magnitude of its imaginary part:

and putting the two inequalities together for Hermitian operators gives a form of the Robertson-Schrödinger relation:

Physical interpretation

by substituting $A - \lang A\rang_\psi$ for A and $B - \lang B\rang_\psi$ for B in the general operator norm inequality, since the imaginary part of the product, the commutator, is unaffected by the shift:

The big side of the inequality is the product of the norms of $A-\lang A\rang$ and $B-\lang B\rang$ , which in quantum mechanics are the standard deviations of A and B. The small side is the norm of the commutator, which for the position and momentum is just $\scriptstyle \hbar$ .

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

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

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 $\scriptstyle |i\rangle$ :

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 $\scriptstyle \hbar$ .

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:

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:

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:

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

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,

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

Δ P = h Δ(1 / λ)

, and identifying

Δ X = L

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).

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

ψ(x) 2

, careful to divide by the normalization factor.

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 $\scriptstyle p=\hbar k$ :

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 through any singularities. So up to normalization, the answer is again a Gaussian.

The width of the distribution in k is found in the same way as before, and the answer just flips A to 1/A.

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

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:

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:

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 peaks¹¹. 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

Δ E

requires a time interval

Δ t > h / Δ 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

Δ 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 particle's 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.

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.¹² 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:

where

Δ ψ E

is the standard deviation of the energy operator in the state $|\psi\rangle$ ,

Δ ψ 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:

Additional Information

This article from the stanford encyclopedia of philosophy provides quite a bit more information on this topic

is made possible by a world-wide funding initiative. Please Read How You Can Help Keep the Encyclopedia Free The Uncertainty Principle First published Mon Oct 8, 2001; substantive revision Mon Jul 3, 2006 Quantum mechanics is generally regarded as the physical theory that is our best candidate for a fundamental and universal description of the physical world. The conceptual framework employed by this theory differs drastically from that of classical physics. Indeed, the transition from classical to quantum physics marks a genuine revolution in our understanding of the physical world. One striking aspect of the difference between classical and quantum physics is that whereas classical mechanics presupposes that exact simultaneous values can be assigned to all physical quantities, quantum mechanics denies this possibility, the prime example being the position and momentum of a particle. According to quantum mechanics, the more precisely the position (momentum) of a particle is given, the less precisely can one say what its momentum (position) is. This is (a simplistic and preliminary formulation of) the quantum mechanical uncertainty principle for position and momentum. 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.

1. Introduction The uncertainty principle is certainly one of the most famous and important aspects of quantum mechanics. It has often been regarded as the most distinctive feature in which quantum mechanics differs from classical theories of the physical world. Roughly speaking, the uncertainty principle (for position and momentum) states that one cannot assign exact simultaneous values to the position and momentum of a physical system. Rather, these quantities can only be determined with some characteristic ‘uncertainties’ that cannot become arbitrarily small simultaneously. But what is the exact meaning of this principle, and indeed, is it really a principle of quantum mechanics? (In his original work, Heisenberg only speaks of uncertainty relations.) And, in particular, what does it mean to say that a quantity is determined only up to some uncertainty? These are the main questions we will explore in the following, focusssing on the views of Heisenberg and Bohr. The notion of ‘uncertainty’ occurs in several different meanings in the physical literature. It may refer to a lack of knowledge of a quantity by an observer, or to the experimental inaccuracy with which a quantity is measured, or to some ambiguity in the definition of a quantity, or to a statistical spread in an ensemble of similary prepared systems. Also, several different names are used for such uncertainties: inaccuracy, spread, imprecision, indefiniteness, indeterminateness, indeterminacy, latitude, etc. As we shall see, even Heisenberg and Bohr did not decide on a single terminology for quantum mechanical uncertainties. Forestalling a discussion about which name is the most appropriate one in quantum mechanics, we use the name ‘uncertainty principle’ imply because it is the most common one in the literature. 2. Heisenberg 2.1 Heisenberg's road to the uncertainty relations Heisenberg introduced his now famous relations in an article of 1927, entitled "Ueber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik". A (partial) translation of this title is: "On the anschaulich content of quantum theoretical kinematics and mechanics". Here, the term anschaulich is particularly notable. Apparently, it is one of those German words that defy an unambiguous translation into other languages. Heisenberg's title is translated as "On the physical content …" by Wheeler and Zurek (1983). His collected works (Heisenberg, 1984) translate it as "On the perceptible content …", while Cassidy's biography of Heisenberg (Cassidy, 1992), refers to the paper as "On the perceptual content …". Literally, the closest translation of the term anschaulich is ‘visualizable’. But, as in most languages, words that make reference to vision are not always intended literally. Seeing is widely used as a metaphor for understanding, especially for immediate understanding. Hence, anschaulich also means ‘intelligible’ or ‘intuitive’.[1] Why was this issue of the Anschaulichkeit of quantum mechanics such a prominent concern to Heisenberg? This question has already been considered by a number of commentators (Jammer, 1977; Miller 1982; de Regt, 1997; Beller, 1999). For the answer, it turns out, we must go back a little in time. In 1925 Heisenberg had developed the first coherent mathematical formalism for quantum theory (Heisenberg, 1925). His leading idea was that only those quantities that are in principle observable should play a role in the theory, and that all attempts to form a picture of what goes on inside the atom should be avoided. In atomic physics the observational data were obtained from spectroscopy and associated with atomic transitions. Thus, Heisenberg was led to consider the ‘transition quantities’ as the basic ingredients of the theory. Max Born, later that year, realized that the transition quantities obeyed the rules of matrix calculus, a branch of mathematics that was not so well-known then as it is now. In a famous series of papers Heisenberg, Born and Jordan developed this idea into the matrix mechanics version of quantum theory. Formally, matrix mechanics remains close to classical mechanics. The central idea is that all physical quantities must be represented by infinite self-adjoint matrices (later identified with operators on a Hilbert space). It is postulated that the matrices q and p representing the canonical position and momentum variables of a particle satisfy the so-called canonical commutation rule

where = h/2π, h denotes Planck's constant, and boldface type is used to represent matrices. The new theory scored spectacular empirical success by encompassing nearly all spectroscopic data known at the time, especially after the concept of the electron spin was included in the theoretical framework. It came as a big surprise, therefore, when one year later, Erwin Schrödinger presented an alternative theory, that became known as wave mechanics. Schrödinger assumed that an electron in an atom could be represented as an oscillating charge cloud, evolving continuously in space and time according to a wave equation. The discrete frequencies in the atomic spectra were not due to discontinuous transitions (quantum jumps) as in matrix mechanics, but to a resonance phenomenon. Schrödinger also showed that the two theories were equivalent.[2] Even so, the two approaches differed greatly in interpretation and spirit. Whereas Heisenberg eschewed the use of visualizable pictures, and accepted discontinuous transitions as a primitive notion, Schrödinger claimed as an advantage of his theory that it was anschaulich. In Schrödinger's vocabulary, this meant that the theory represented the observational data by means of continuously evolving causal processes in space and time. He considered this condition of Anschaulichkeit to be an essential requirement on any acceptable physical theory. Schrödinger was not alone in appreciating this aspect of his theory. Many other leading physicists were attracted to wave mechanics for the same reason. For a while, in 1926, before it emerged that wave mechanics had serious problems of its own, Schrödinger's approach seemed to gather more support in the physics community than matrix mechanics. Understandably, Heisenberg was unhappy about this development. In a letter of 8 June 1926 to Pauli he confessed that "The more I think about the physical part of Schrödinger's theory, the more disgusting I find it", and: "What Schrödinger writes about the Anschaulichkeit of his theory, … I consider Mist (Pauli, 1979, p. 328)". Again, this last German term is translated differently by various commentators: as "junk" (Miller, 1982) "rubbish" (Beller 1999) "crap" (Cassidy, 1992), and perhaps more literally, as "bullshit" (de Regt, 1997). Nevertheless, in published writings, Heisenberg voiced a more balanced opinion. In a paper in Die Naturwissenschaften (1926) he summarized the peculiar situation that the simultaneous development of two competing theories had brought about. Although he argued that Schrödinger's interpretation was untenable, he admitted that matrix mechanics did not provide the Anschaulichkeit which made wave mechanics so attractive. He concluded: "to obtain a contradiction-free anschaulich interpretation, we still lack some essential feature in our image of the structure of matter". The purpose of his 1927 paper was to provide exactly this lacking feature. 2.2 Heisenberg's argument Let us now look at the argument that led Heisenberg to his uncertainty relations. He started by redefining the notion of Anschaulichkeit. Whereas Schrödinger associated this term with the provision of a causal space-time picture of the phenomena, Heisenberg, by contrast, declared:

His goal was, of course, to show that, in this new sense of the word, matrix mechanics could lay the same claim to Anschaulichkeit as wave mechanics. To do this, he adopted an operational assumption: terms like ‘the position of a particle’ have meaning only if one specifies a suitable experiment by which ‘the position of a particle’ can be measured. We will call this assumption the ‘measurement=meaning principle’. In general, there is no lack of such experiments, even in the domain of atomic physics. However, experiments are never completely accurate. We should be prepared to accept, therefore, that in general the meaning of these quantities is also determined only up to some characteristic inaccuracy. As an example, he considered the measurement of the position of an electron by a microscope. The accuracy of such a measurement is limited by the wave length of the light illuminating the electron. Thus, it is possible, in principle, to make such a position measurement as accurate as one wishes, by using light of a very short wave length, e.g., γ-rays. But for γ-rays, the Compton effect cannot be ignored: the interaction of the electron and the illuminating light should then be considered as a collision of at least one photon with the electron. In such a collision, the electron suffers a recoil which disturbs its momentum. Moreover, the shorter the wave length, the larger is this change in momentum. Thus, at the moment when the position of the particle is accurately known, Heisenberg argued, its momentum cannot be accurately known:

This is the first formulation of the uncertainty principle. In its present form it is an epistemological principle, since it limits what we can know about the electron. From "elementary formulae of the Compton effect" Heisenberg estimated the ‘imprecisions’ to be of the order

He continued: “In this circumstance we see the direct anschaulich content of the relation qp − pq = i.” He went on to consider other experiments, designed to measure other physical quantities and obtained analogous relations for time and energy:

However, these generalisations are not as straightforward as Heisenberg suggested. In particular, the status of the time variable in his several illustrations of relation (3) is not at all clear (Hilgevoord 2005). See also on Section 2.5. Heisenberg summarized his findings in a general conclusion: all concepts used in classical mechanics are also well-defined in the realm of atomic processes. But, as a pure fact of experience ("rein erfahrungsgemäß"), experiments that serve to provide such a definition for one quantity are subject to particular indeterminacies, obeying relations (2)-(4) which prohibit them from providing a simultaneous definition of two canonically conjugate quantities. Note that in this formulation the emphasis has slightly shifted: he now speaks of a limit on the definition of concepts, i.e. not merely on what we can know, but what we can meaningfully say about a particle. Of course, this stronger formulation follows by application of the above measurement=meaning principle: if there are, as Heisenberg claims, no experiments that allow a simultaneous precise measurement of two conjugate quantities, then these quantities are also not simultaneously well-defined. Heisenberg's paper has an interesting "Addition in proof" mentioning critical remarks by Bohr, who saw the paper only after it had been sent to the publisher. Among other things, Bohr pointed out that in the microscope experiment it is not the change of the momentum of the electron that is important, but rather the circumstance that this change cannot be precisely determined in the same experiment. An improved version of the argument, responding to this objection, is given in Heisenberg's Chicago lectures of 1930. Here (Heisenberg, 1930, p. 16), it is assumed that the electron is illuminated by light of wavelength λ and that the scattered light enters a microscope with aperture angle ε. According to the laws of classical optics, the accuracy of the microscope depends on both the wave length and the aperture angle; Abbe's criterium for its ‘resolving power’, i.e. the size of the smallest discernable details, gives

On the other hand, the direction of a scattered photon, when it enters the microscope, is unknown within the angle ε, rendering the momentum change of the electron uncertain by an amount

leading again to the result (2). Let us now analyse Heisenberg's argument in more detail. First note that, even in this improved version, Heisenberg's argument is incomplete. According to Heisenberg's ‘measurement=meaning principle’, one must also specify, in the given context, what the meaning is of the phrase ‘momentum of the electron’, in order to make sense of the claim that this momentum is changed by the position measurement. A solution to this problem can again be found in the Chicago lectures (Heisenberg, 1930, p. 15). Here, he assumes that initially the momentum of the electron is precisely known, e.g. it has been measured in a previous experiment with an inaccuracy δpi, which may be arbitrarily small. Then, its position is measured with inaccuracy δq, and after this, its final momentum is measured with an inaccuracy δpf. All three measurements can be performed with arbitrary precision. Thus, the three quantities δpi, δq, and δpf can be made as small as one wishes. If we assume further that the initial momentum has not changed until the position measurement, we can speak of a definite momentum until the time of the position measurement. Moreover we can give operational meaning to the idea that the momentum is changed during the position measurement: the outcome of the second momentum measurement (say pf) will generally differ from the initial value pi. In fact, one can also show that this change is discontinuous, by varying the time between the three measurements. Let us now try to see, adopting this more elaborate set-up, if we can complete Heisenberg's argument. We have now been able to give empirical meaning to the ‘change of momentum’ of the electron, pf − pi. Heisenberg's argument claims that the order of magnitude of this change is at least inversely proportional to the inaccuracy of the position measurement:

However, can we now draw the conclusion that the momentum is only imprecisely defined? Certainly not. Before the position measurement, its value was pi, after the measurement it is pf. One might, perhaps, claim that the value at the very instant of the position measurement is not yet defined, but we could simply settle this by an assignment by convention, e.g., we might assign the mean value (pi + pf)/2 to the momentum at this instant. But then, the momentum is precisely determined at all instants, and Heisenberg's formulation of the uncertainty principle no longer follows. The above attempt of completing Heisenberg's argument thus overshoots its mark. A solution to this problem can again be found in the Chicago Lectures. Heisenberg admits that position and momentum can be known exactly. He writes:

Indeed, Heisenberg says: "the uncertainty relation does not hold for the past". Apparently, when Heisenberg refers to the uncertainty or imprecision of a quantity, he means that the value of this quantity cannot be given beforehand. In the sequence of measurements we have considered above, the uncertainty in the momentum after the measurement of position has occurred, refers to the idea that the value of the momentum is not fixed just before the final momentum measurement takes place. Once this measurement is performed, and reveals a value pf, the uncertainty relation no longer holds; these values then belong to the past. Clearly, then, Heisenberg is concerned with unpredictability: the point is not that the momentum of a particle changes, due to a position measurement, but rather that it changes by an unpredictable amount. It is, however always possible to measure, and hence define, the size of this change in a subsequent measurement of the final momentum with arbitrary precision. Although Heisenberg admits that we can consistently attribute values of momentum and position to an electron in the past, he sees little merit in such talk. He points out that these values can never be used as initial conditions in a prediction about the future behavior of the electron, or subjected to experimental verification. Whether or not we grant them physical reality is, as he puts it, a matter of personal taste. Heisenberg's own taste is, of course, to deny their physical reality. For example, he writes, "I believe that one can formulate the emergence of the classical ‘path’ of a particle pregnantly as follows: the ‘path’ comes into being only because we observe it" (Heisenberg, 1927, p. 185). Apparently, in his view, a measurement does not only serve to give meaning to a quantity, it creates a particular value for this quantity. This may be called the ‘measurement=creation’ principle. It is an ontological principle, for it states what is physically real. This then leads to the following picture. First we measure the momentum of the electron very accurately. By ‘measurement= meaning’, this entails that the term "the momentum of the particle" is now well-defined. Moreover, by the ‘measurement=creation’ principle, we may say that this momentum is physically real. Next, the position is measured with inaccuracy δq. At this instant, the position of the particle becomes well-defined and, again, one can regard this as a physically real attribute of the particle. However, the momentum has now changed by an amount that is unpredictable by an order of magnitude | pf − pi | ∼ h/δq. The meaning and validity of this claim can be verified by a subsequent momentum measurement. The question is then what status we shall assign to the momentum of the electron just before its final measurement. Is it real? According to Heisenberg it is not. Before the final measurement, the best we can attribute to the electron is some unsharp, or fuzzy momentum. These terms are meant here in an ontological sense, characterizing a real attribute of the electron. 2.3 The interpretation of Heisenberg's relation The relations Heisenberg had proposed were soon considered to be a cornerstone of the Copenhagen interpretation of quantum mechanics. Just a few months later, Kennard (1927) already called them the "essential core" of the new theory. Taken together with Heisenberg's contention that they provided the intuitive content of the theory and their prominent role in later discussions on the Copenhagen interpretation, a dominant view emerged in which the uncertainty relations were regarded as a fundamental principle of the theory. The interpretation of these relations has often been debated. Do Heisenberg's relations express restrictions on the experiments we can perform on quantum systems, and, therefore, restrictions on the information we can gather about such systems; or do they express restrictions on the meaning of the concepts we use to describe quantum systems? Or else, are they restrictions of an ontological nature, i.e., do they assert that a quantum system simply does not possess a definite value for its position and momentum at the same time? The difference between these interpretations is partly reflected in the various names by which the relations are known, e.g. as ‘inaccuracy relations’, or: ‘uncertainty’, ‘indeterminacy’ or ‘unsharpness relations’. The debate between these different views has been addressed by many authors, but it has never been settled completely. Let it suffice here to make only two general observations. First, it is clear that in Heisenberg's own view all the above questions stand or fall together. Indeed, we have seen that he adopted an operational "measurement=meaning" principle according to which the meaningfulness of a physical quantity was equivalent to the existence of an experiment purporting to measure that quantity. Similarly, his "measurement=creation" principle allowed him to attribute physical reality to such quantities. Hence, Heisenberg's discussions moved rather freely and quickly from talk about experimental inaccuracies to epistemological or ontological issues and back again. However, ontological questions seemed to be of somewhat less interest to him. For example, there is a passage (Heisenberg, 1927, p. 197), where he discusses the idea that, behind our observational data, there might still exist a hidden reality in which quantum systems have definite values for position and momentum, unaffected by the uncertainty relations. He emphatically dismisses this conception as an unfruitful and meaningless speculation, because, as he says, the aim of physics is only to describe observable data. Similarly, in the Chicago Lectures (Heisenberg 1930, p. 11), he warns against the fact that the human language permits the utterance of statements which have no empirical content at all, but nevertheless produce a picture in our imagination. He notes, "One should be especially careful in using the words ‘reality’, ‘actually’, etc., since these words very often lead to statements of the type just mentioned." So, Heisenberg also endorsed an interpretation of his relations as rejecting a reality in which particles have simultaneous definite values for position and momentum. The second observation is that although for Heisenberg experimental, informational, epistemological and ontological formulations of his relations were, so to say, just different sides of the same coin, this is not so for those who do not share his operational principles or his view on the task of physics. Alternative points of view, in which e.g. the ontological reading of the uncertainty relations is denied, are therefore still viable. The statement, often found in the literature of the thirties, that Heisenberg had proved the impossibility of associating a definite position and momentum to a particle is certainly wrong. But the precise meaning one can coherently attach to Heisenberg's relations depends rather heavily on the interpretation one favors for quantum mechanics as a whole. And because no agreement has been reached on this latter issue, one cannot expect agreement on the meaning of the uncertainty relations either. 2.4 Uncertainty relations or uncertainty principle? Let us now move to another question about Heisenberg's relations: do they express a principle of quantum theory? Probably the first influential author to call these relations a ‘principle’ was Eddington, who, in his Gifford Lectures of 1928 referred to them as the ‘Principle of Indeterminacy’. In the English literature the name uncertainty principle became most common. It is used both by Condon and Robertson in 1929, and also in the English version of Heisenberg's Chicago Lectures (Heisenberg, 1930), although, remarkably, nowhere in the original German version of the same book (see also Cassidy, 1998). Indeed, Heisenberg never seems to have endorsed the name ‘principle’ for his relations. His favourite terminology was ‘inaccuracy relations’ (Ungenauigkeitsrelationen) or ‘indeterminacy relations’ (Unbestimmtheitsrelationen). We know only one passage, in Heisenberg's own Gifford lectures, delivered in 1955-56 (Heisenberg, 1958, p. 43), where he mentioned that his relations "are usually called relations of uncertainty or principle of indeterminacy". But this can well be read as his yielding to common practice rather than his own preference. But does the relation (2) qualify as a principle of quantum mechanics? Several authors, foremost Karl Popper (1967), have contested this view. 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. (The argument being that one can never derive any equation, say, the Schrödinger equation, or the commutation relation (1), from an inequality.) Popper's argument is, of course, correct but we think it misses the point. There are many statements in physical theories which are called principles even though they are in fact derivable from other statements in the theory in question. A more appropriate departing point for this issue is not the question of logical priority but rather Einstein's distinction between ‘constructive theories’ and ‘principle theories’. Einstein proposed this famous classification in (Einstein, 1919). Constructive theories are theories which postulate the existence of simple entities behind the phenomena. They endeavour to reconstruct the phenomena by framing hypotheses about these entities. Principle theories, on the other hand, start from empirical principles, i.e. general statements of empirical regularities, employing no or only a bare minimum of theoretical terms. The purpose is to build up the theory from such principles. That is, one aims to show how these empirical principles provide sufficient conditions for the introduction of further theoretical concepts and structure. The prime example of a theory of principle is thermodynamics. Here the role of the empirical principles is played by the statements of the impossibility of various kinds of perpetual motion machines. These are regarded as expressions of brute empirical fact, providing the appropriate conditions for the introduction of the concepts of energy and entropy and their properties. (There is a lot to be said about the tenability of this view, but that is not the topic of this entry.) Now obviously, once the formal thermodynamic theory is built, one can also derive the impossibility of the various kinds of perpetual motion. (They would violate the laws of energy conservation and entropy increase.) But this derivation should not misguide one into thinking that they were no principles of the theory after all. The point is just that empirical principles are statements that do not rely on the theoretical concepts (in this case entropy and energy) for their meaning. They are interpretable independently of these concepts and, further, their validity on the empirical level still provides the physical content of the theory. A similar example is provided by special relativity, another theory of principle, which Einstein deliberately designed after the ideal of thermodynamics. Here, the empirical principles are the light postulate and the relativity principle. Again, once we have built up the modern theoretical formalism of the theory (the Minkowski space-time) it is straightforward to prove the validity of these principles. But again this does not count as an argument for claiming that they were no principles after all. So the question whether the term ‘principle’ is justified for Heisenberg's relations, should, in our view, be understood as the question whether they are conceived of as empirical principles. One can easily show that this idea was never far from Heisenberg's intentions. We have already seen that Heisenberg presented the relations as the result of a "pure fact of experience". A few months after his 1927 paper, he wrote a popular paper with the title "Ueber die Grundprincipien der Quantenmechanik" ("On the fundamental principles of quantum mechanics") where he made the point even more clearly. Here Heisenberg described his recent break-through in the interpretation of the theory as follows: "It seems to be a general law of nature that we cannot determine position and velocity simultaneously with arbitrary accuracy". Now actually, and in spite of its title, the paper does not identify or discuss any ‘fundamental principle’ of quantum mechanics. So, it must have seemed obvious to his readers that he intended to claim that the uncertainty relation was a fundamental principle, forced upon us as an empirical law of nature, rather than a result derived from the formalism of the theory. This reading of Heisenberg's intentions is corroborated by the fact that, even in his 1927 paper, applications of his relation frequently present the conclusion as a matter of principle. For example, he says "In a stationary state of an atom its phase is in principle indeterminate" (Heisenberg, 1927, p. 177, [emphasis added]). Similarly, in a paper of 1928, he described the content of his relations as: "It has turned out that it is in principle impossible to know, to measure the position and velocity of a piece of matter with arbitrary accuracy. (Heisenberg, 1984, p. 26, [emphasis added])" So, although Heisenberg did not originate the tradition of calling his relations a principle, it is not implausible to attribute the view to him that the uncertainty relations represent an empirical principle that could serve as a foundation of quantum mechanics. In fact, his 1927 paper expressed this desire explicitly: "Surely, one would like to be able to deduce the quantitative laws of quantum mechanics directly from their anschaulich foundations, that is, essentially, relation [(2)]" (ibid, p. 196). This is not to say that Heisenberg was successful in reaching this goal, or that he did not express other opinions on other occasions. Let us conclude this section with three remarks. First, if the uncertainty relation is to serve as an empirical principle, one might well ask what its direct empirical support is. In Heisenberg's analysis, no such support is mentioned. His arguments concerned thought experiments in which the validity of the theory, at least at a rudimentary level, is implicitly taken for granted. Jammer (1974, p. 82) conducted a literature search for high precision experiments that could seriously test the uncertainty relations and concluded they were still scarce in 1974. Real experimental support for the uncertainty relations in experiments in which the inaccuracies are close to the quantum limit have come about only more recently. (See Kaiser, Werner and George 1983, Uffink 1985, Nairz, Andt, and Zeilinger, 2001.) A second point is the question whether the theoretical structure or the quantitative laws of quantum theory can indeed be derived on the basis of the uncertainty principle, as Heisenberg wished. Serious attempts to build up quantum theory as a full-fledged Theory of Principle on the basis of the uncertainty principle have never been carried out. Indeed, the most Heisenberg could and did claim in this respect was that the uncertainty relations created "room" (Heisenberg 1927, p. 180) or "freedom" (Heisenberg, 1931, p. 43) for the introduction of some non-classical mode of description of experimental data, not that they uniquely lead to the formalism of quantum mechanics. A serious proposal to construe quantum mechanics as a theory of principle was provided only recently by Bub (2000). But, remarkably, this proposal does not use the uncertainty relation as one of its fundamental principles. Third, it is remarkable that in his later years Heisenberg put a somewhat different gloss on his relations. In his autobiography Der Teil und das Ganze of 1969 he described how he had found his relations inspired by a remark by Einstein that "it is the theory which decides what one can observe" -- thus giving precedence to theory above experience, rather than the other way around. Some years later he even admitted that his famous discussions of thought experiments were actually trivial since "… if the process of observation itself is subject to the laws of quantum theory, it must be possible to represent its result in the mathematical scheme of this theory" (Heisenberg, 1975, p. 6). 2.5 Mathematical elaboration When Heisenberg introduced his relation, his argument was based only on qualitative examples. He did not provide a general, exact derivation of his relations.[3] Indeed, he did not even give a definition of the uncertainties δq, etc., occurring in these relations. Of course, this was consistent with the announced goal of that paper, i.e. to provide some qualitative understanding of quantum mechanics for simple experiments. The first mathematically exact formulation of the uncertainty relations is due to Kennard. He proved in 1927 the theorem that for all normalized state vectors |ψ> the following inequality holds:

Here, Δψp and Δψq are standard deviations of position and momentum in the state vector |ψ>, i.e.,

ψ)², (Δψq)² = <q²>ψ − (<q>ψ)².(10) where <·>ψ = <ψ|·|ψ> denotes the expectation value in state |ψ>. The inequality (9) was generalized in 1929 by Robertson who proved that for all observables (self-adjoint operators) A and B ΔψA ΔψB ≥ ½|<[A,B]> ψ|(11) where [A, B] := AB − BA denotes the commutator. This relation was in turn strengthened by Schrödinger (1930), who obtained: (ΔψA)² (ΔψB)² ≥ ¼|<[A,B]> ψ|² + ¼|<{A−<A> ψ, B− ψ}>ψ|²(12) where {A, B} := (AB + BA) denotes the anti-commutator. Since the above inequalities have the virtue of being exact and general, in contrast to Heisenberg's original semi-quantitative formulation, it is tempting to regard them as the exact counterpart of Heisenberg's relations (2)-(4). Indeed, such was Heisenberg's own view. In his Chicago Lectures (Heisenberg 1930, pp. 15-19), he presented Kennard's derivation of relation (9) and claimed that "this proof does not differ at all in mathematical content" from the semi-quantitative argument he had presented earlier, the only difference being that now "the proof is carried through exactly". But it may be useful to point out that both in status and intended role there is a difference between Kennard's inequality and Heisenberg's previous formulation (2). The inequalities discussed in the present section are not statements of empirical fact, but theorems of the quantum mechanical formalism. As such, they presuppose the validity of this formalism, and in particular the commutation relation (1), rather than elucidating its intuitive content or to create ‘room’ or ‘freedom’ for the validity of this relation. At best, one should see the above inequalities as showing that the formalism is consistent with Heisenberg's empirical principle. This situation is similar to that arising in other theories of principle where, as noted in Section 2.4, one often finds that, next to an empirical principle, the formalism also provides a corresponding theorem. And similarly, this situation should not, by itself, cast doubt on the question whether Heisenberg's relation can be regarded as a principle of quantum mechanics. There is a second notable difference between (2) and (9). Heisenberg did not give a general definition for the ‘uncertainties’ δp and δq. The most definite remark he made about them was that they could be taken as "something like the mean error". In the discussions of thought experiments, he and Bohr would always quantify uncertainties on a case-to-case basis by choosing some parameters which happened to be relevant to the experiment at hand. By contrast, the inequalities (9)-(12) employ a single specific expression as a measure for ‘uncertainty’: the standard deviation. At the time, this choice was not unnatural, given that this expression is well-known and widely used in error theory and the description of statistical fluctuations. However, there was very little or no discussion of whether this choice was appropriate for a general formulation of the uncertainty relations. A standard deviation reflects the spread or expected fluctuations in a series of measurements of an observable in a given state. It is not at all easy to connect this idea with the concept of the ‘inaccuracy’ of a measurement, such as the resolving power of a microscope. In fact, even though Heisenberg had taken Kennard's inequality as the precise formulation of the uncertainty relation, he and Bohr never relied on standard deviations in their many discussions of thought experiments, and indeed, it has been shown (Uffink and Hilgevoord, 1985; Hilgevoord and Uffink, 1988) that these discussions cannot be framed in terms of standard deviation. Another problem with the above elaboration is that the ‘well-known’ relations (5) are actually false if energy E and action J are to be positive operators (Jordan 1927). In that case, self-adjoint operators t and w do not exist and inequalities analogous to (9) cannot be derived. Also, these inequalities do not hold for angle and angular momentum (Uffink 1990). These obstacles have led to a quite extensive literature on time-energy and angle-action uncertainty relations (Muga et al. 2002, Hilgevoord 2005). 3. Bohr In spite of the fact that Heisenberg's and Bohr's views on quantum mechanics are often lumped together as (part of) ‘the Copenhagen interpretation’, there is considerable difference between their views on the uncertainty relations. 3.1 From wave-particle duality to complementarity Long before the development of modern quantum mechanics, Bohr had been particularly concerned with the problem of particle-wave duality, i.e. the problem that experimental evidence on the behaviour of both light and matter seemed to demand a wave picture in some cases, and a particle picture in others. Yet these pictures are mutually exclusive. Whereas a particle is always localized, the very definition of the notions of wavelength and frequency requires an extension in space and in time. Moreover, the classical particle picture is incompatible with the characteristic phenomenon of interference. His long struggle with wave-particle duality had prepared him for a radical step when the dispute between matrix and wave mechanics broke out in 1926-27. For the main contestants, Heisenberg and Schrödinger, the issue at stake was which view could claim to provide a single coherent and universal framework for the description of the observational data. The choice was, essentially between a description in terms of continuously evolving waves, or else one of particles undergoing discontinuous quantum jumps. By contrast, Bohr insisted that elements from both views were equally valid and equally needed for an exhaustive description of the data. His way out of the contradiction was to renounce the idea that the pictures refer, in a literal one-to-one correspondence, to physical reality. Instead, the applicability of these pictures was to become dependent on the experimental context. This is the gist of the viewpoint he called ‘complementarity’. Bohr first conceived the general outline of his complementarity argument in early 1927, during a skiing holiday in Norway, at the same time when Heisenberg wrote his uncertainty paper. When he returned to Copenhagen and found Heisenberg's manuscript, they got into an intense discussion. On the one hand, Bohr was quite enthusiastic about Heisenberg's ideas which seemed to fit wonderfully with his own thinking. Indeed, in his subsequent work, Bohr always presented the uncertainty relations as the symbolic expression of his complementarity viewpoint. On the other hand, he criticized Heisenberg severely for his suggestion that these relations were due to discontinuous changes occurring during a measurement process. Rather, Bohr argued, their proper derivation should start from the indispensability of both particle and wave concepts. He pointed out that the uncertainties in the experiment did not exclusively arise from the discontinuities but also from the fact that in the experiment we need to take into account both the particle theory and the wave theory. It is not so much the unknown disturbance which renders the momentum of the electron uncertain but rather the fact that the position and the momentum of the electron cannot be simultaneously defined in this experiment. (See the "Addition in Proof" to Heisenberg's paper.) We shall not go too deeply into the matter of Bohr's interpretation of quantum mechanics since we are mostly interested in Bohr's view on the uncertainty principle. For a more detailed discussion of Bohr's philosophy of quantum physics we refer to Scheibe (1973), Folse (1985), Honner (1987) and Murdoch (1987). It may be useful, however, to sketch some of the main points. Central in Bohr's considerations is the language we use in physics. No matter how abstract and subtle the concepts of modern physics may be, they are essentially an extension of our ordinary language and a means to communicate the results of our experiments. These results, obtained under well-defined experimental circumstances, are what Bohr calls the "phenomena". A phenomenon is "the comprehension of the effects observed under given experimental conditions" (Bohr 1939, p. 24), it is the resultant of a physical object, a measuring apparatus and the interaction between them in a concrete experimental situation. The essential difference between classical and quantum physics is that in quantum physics the interaction between the object and the apparatus cannot be made arbitrarily small; the interaction must at least comprise one quantum. This is expressed by Bohr's quantum postulate: [… the] essenc

(1)	$\Delta x\Delta p\gtrsim h.$
But it was Kennard³ in 1927 who first proved the modern inequality:
(2)	$\sigma_x\sigma_p\ge\frac{\hbar}{2}\,$

Contents

Historical Introduction