# Manual Uncertainty Theory

Applications dependent on the uncertainty principle for their operation include extremely low-noise technology such as that required in gravitational wave interferometers. The uncertainty principle is not readily apparent on the macroscopic scales of everyday experience.

Two alternative frameworks for quantum physics offer different explanations for the uncertainty principle. The wave mechanics picture of the uncertainty principle is more visually intuitive, but the more abstract matrix mechanics picture formulates it in a way that generalizes more easily. Mathematically, in wave mechanics, the uncertainty relation between position and momentum arises because the expressions of the wavefunction in the two corresponding orthonormal bases in Hilbert space are Fourier transforms of one another i.

A nonzero function and its Fourier transform cannot both be sharply localized. A similar tradeoff between the variances of Fourier conjugates arises in all systems underlain by Fourier analysis, for example in sound waves: A pure tone is a sharp spike at a single frequency, while its Fourier transform gives the shape of the sound wave in the time domain, which is a completely delocalized sine wave. In matrix mechanics , the mathematical formulation of quantum mechanics , any pair of non- commuting self-adjoint operators representing observables are subject to similar uncertainty limits.

An eigenstate of an observable represents the state of the wavefunction for a certain measurement value the eigenvalue. However, the particular eigenstate of the observable A need not be an eigenstate of another observable B : If so, then it does not have a unique associated measurement for it, as the system is not in an eigenstate of that observable. Ref . According to the de Broglie hypothesis , every object in the universe is a wave , i.

The time-independent wave function of a single-moded plane wave of wavenumber k 0 or momentum p 0 is. The Born rule states that this should be interpreted as a probability density amplitude function in the sense that the probability of finding the particle between a and b is.

In other words, the particle position is extremely uncertain in the sense that it could be essentially anywhere along the wave packet. On the other hand, consider a wave function that is a sum of many waves , which we may write this as. The figures to the right show how with the addition of many plane waves, the wave packet can become more localized.

## Chad Orzel: What is the Heisenberg Uncertainty Principle? | TED Talk Subtitles and Transcript | TED

We may take this a step further to the continuum limit, where the wave function is an integral over all possible modes. Adding together all of these plane waves comes at a cost, namely the momentum has become less precise, having become a mixture of waves of many different momenta. The precision of the position is improved, i.

This is the uncertainty principle, the exact limit of which is the Kennard bound. Click the show button below to see a semi-formal derivation of the Kennard inequality using wave mechanics. Without loss of generality , we will assume that the means vanish, which just amounts to a shift of the origin of our coordinates. A more general proof that does not make this assumption is given below. This gives us the simpler form. We can define an inner product for a pair of functions u x and v x in this vector space:.

We evaluate the inverse Fourier transform through integration by parts :. Applying Parseval's theorem , we see that the variance for momentum can be written as. The Cauchy—Schwarz inequality asserts that. A similar result would hold for any pair of conjugate variables. In matrix mechanics, observables such as position and momentum are represented by self-adjoint operators. When considering pairs of observables, an important quantity is the commutator.

In the case of position and momentum, the commutator is the canonical commutation relation. The physical meaning of the non-commutativity can be understood by considering the effect of the commutator on position and momentum eigenstates. If this were true, then one could write. This implies that no quantum state can simultaneously be both a position and a momentum eigenstate. When a state is measured, it is projected onto an eigenstate in the basis of the relevant observable. For example, if a particle's position is measured, then the state amounts to a position eigenstate. This means that the state is not a momentum eigenstate, however, but rather it can be represented as a sum of multiple momentum basis eigenstates.

In other words, the momentum must be less precise.

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This precision may be quantified by the standard deviations ,. As in the wave mechanics interpretation above, one sees a tradeoff between the respective precisions of the two, quantified by the uncertainty principle. The most common general form of the uncertainty principle is the Robertson uncertainty relation. The modulus squared can also be expressed as. We now substitute the above two equations above back into Eq.

Substituting the above into Eq. This proof has an issue  related to the domains of the operators involved. In this example, the commutator is a nonzero constant—just as in the Heisenberg uncertainty relation—and yet there are states where the product of the uncertainties is zero. This issue can be overcome by using a variational method for the proof. It suffices to assume that they are merely symmetric operators.

The distinction between these two notions is generally glossed over in the physics literature, where the term Hermitian is used for either or both classes of operators. See Chapter 9 of Hall's book  for a detailed discussion of this important but technical distinction. A few of the most common relations found in the literature are given below.

These states are normalizable, unlike the eigenstates of the momentum operator on the line.

These also furnish an example of operators satisfying the canonical commutation relations but not the Weyl relations. Refs  . Consider a one-dimensional quantum harmonic oscillator QHO.

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## Uncertainty Theory

It is possible to express the position and momentum operators in terms of the creation and annihilation operators :. Through integration over the propagator , we can solve for the full time-dependent solution. After many cancelations, the probability densities reduce to. Copying the variances above and applying trigonometric identities , we can write the product of the standard deviations as. A coherent state is a right eigenstate of the annihilation operator ,. In the picture where the coherent state is a massive particle in a QHO, the position and momentum operators may be expressed in terms of the annihilation operators in the same formulas above and used to calculate the variances,.

Moreover, every squeezed coherent state also saturates the Kennard bound although the individual contributions of position and momentum need not be balanced in general. The eigenfunctions in position and momentum space are. Assume a particle initially has a momentum space wave function described by a normal distribution around some constant momentum p 0 according to. If the state is allowed to evolve in free space, then the time-dependent momentum and position space wave functions are.

On the other hand, the standard deviation of the position is. The stronger uncertainty relations proved by Maccone and Pati give non-trivial bounds on the sum of the variances for two incompatible observables. Since this positivity condition is true for all a , b , and c , it follows that all the eigenvalues of the matrix are positive.

The positive eigenvalues then imply a corresponding positivity condition on the determinant :.

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Heisenberg's original version, however, was dealing with the systematic error , a disturbance of the quantum system produced by the measuring apparatus, i. Heisenberg's uncertainty principle, as originally described in the formulation, mentions only the first term of Ozawa inequality, regarding the systematic error.

The formal derivation of the Heisenberg relation is possible but far from intuitive. It was not proposed by Heisenberg, but formulated in a mathematically consistent way only in recent years. There is increasing experimental evidence     that the total quantum uncertainty cannot be described by the Heisenberg term alone, but requires the presence of all the three terms of the Ozawa inequality. Using the same formalism,  it is also possible to introduce the other kind of physical situation, often confused with the previous one, namely the case of simultaneous measurements A and B at the same time :.

The two simultaneous measurements on A and B are necessarily  unsharp or weak. It is also possible to derive an uncertainty relation that, as the Ozawa's one, combines both the statistical and systematic error components, but keeps a form very close to the Heisenberg original inequality. By adding Robertson . For many distributions, the standard deviation is not a particularly natural way of quantifying the structure.

For example, uncertainty relations in which one of the observables is an angle has little physical meaning for fluctuations larger than one period.

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A solution that overcomes these issues is an uncertainty based on entropic uncertainty instead of the product of variances. While formulating the many-worlds interpretation of quantum mechanics in , Hugh Everett III conjectured a stronger extension of the uncertainty principle based on entropic certainty. Note that the entropies will be functions of these chosen parameters. Depending on one's choice of the x 0 p 0 product, the expression may be written in many ways. If x 0 and p 0 are chosen to be unity in whatever system of units are being used, then.

Note that these inequalities can be extended to multimode quantum states, or wavefunctions in more than one spatial dimension. The quantum entropic uncertainty principle is more restrictive than the Heisenberg uncertainty principle. From the inverse logarithmic Sobolev inequalities . In other words, the Heisenberg uncertainty principle, is a consequence of the quantum entropic uncertainty principle, but not vice versa. A few remarks on these inequalities. First, the choice of base e is a matter of popular convention in physics.