Uncertainty principle

The Heisenberg uncertainty principle or indeterminacy of quantum physics is the statement that two complementary properties of a particle are arbitrarily determined precisely the same time. The best-known example of a pair of such properties are the position and momentum. The uncertainty principle is not the consequence of technically recoverable shortcomings of an appropriate measuring instrument, but of principle. It was formulated in 1927 by Werner Heisenberg in quantum mechanics. The Heisenberg uncertainty principle can be regarded as an expression of the wave nature of matter. It is regarded as part of the Copenhagen interpretation of quantum mechanics.

Quantum mechanics and classical physics

Quantum mechanics is currently one of the fundamental theories for the description of our physical world. The conceptual structure of this theory differs profoundly from that of classical physics.

The statements of quantum mechanics about our world are statements of outputs of measurements. In contrast to classical physics only probability statements can be made in any case, so you can only predict the distribution of values ​​in the measurement on an ensemble of identical systems. The Heisenberg uncertainty principle arises from the fact that a physical system is described in quantum mechanics by a wavefunction. While in classical mechanics place or pulse are simple quantities that are in principle be measured accurately, their distributions arising in quantum mechanics as the absolute square of the wave function and its Fourier transform, that is, they are not independently securable. Since the distributions of the position and momentum both depend on the wave function of the system, and the standard deviations of the measurements are dependent on one another. The more specific you want to set the position of a particle in the usual quantum mechanical description, the greater the uncertainty of the momentum - and vice versa.

The following analogy illustrates the indeterminacy: Suppose that we have a time-varying signal, such as a sound wave, and we want to measure the exact frequency of this signal at a given time. This is impossible, because the frequency to determine with some accuracy, we must observe the signal over a sufficiently long period of time, and thus we lose time precision. That is, a sound can not be only within an arbitrarily short period of time there, such as a short cry, and at the same time have an exact frequency, as it has about a continuous pure tone. The duration and the frequency of the waveform must be considered analogous to the position and momentum of the particle.

Original formulation

The first formulation of an uncertainty principle in quantum mechanics concerned the simultaneous knowledge of the position and momentum of a particle. In 1927, Heisenberg published his work via the intuitive content of quantum theoretical kinematics and mechanics and argued that the microscopic determination of the location x of a particle in general to an influence ( disorder ) of the momentum p of the particle must lead. So if the place of an electron by optical observation (in the simplest case: seeing) is to be determined, so the particle can be illuminated to at least one of the incident photons is scattered into the measuring instrument (eye, microscope).

On the one hand, the inaccuracy of the location Ax here is dependent on the wavelength of the light used. On the other hand, affects the deflection of the photon as a shock to the particles, so that the momentum of the body undergoes an uncertainty of Ap ( Compton scattering). As a fundamental lower limit for these uncertainties Heisenberg gauged using the de Broglie relation that the product of Ax and Ap can not be smaller than the characteristic of quantum physics natural constant, Planck's constant. This fundamental limit of measurability Heisenberg formulated in the (symbolic) statement

The first qualitative nature of this assessment stems from the fact that this statement is not ( strictly ) is proved, and because the notation used therein for the uncertainties no precise and unambiguous definition has been used. With an appropriate interpretation of the notation in the context of modern quantum mechanics, however, shows that this formula of reality already comes very close.

Uncertainty principle and everyday

Why this characteristic uncertainties were not noticed earlier in the research in everyday life, can be understood if one considers the smallness of Planck's constant over the typical achievable measurement accuracy for position and momentum. To this end, the following examples:

• Radar control by road:

• Mote:

• Electron in the atom:

Testify

The term of the uncertainty principle, the following statements can be summarized that, while related, but have different physical meaning. They are listed here as an example for the pair position and momentum.

Each of these three statements can be quantitatively formulated in the form of so-called blur - relations that specify a lower limit on the minimum achievable blur of preparation and measurement.

Also between other pairs of physical quantities may apply uncertainty relations. Is a prerequisite that the commutator of the two sizes associated with quantum mechanical operators, is not zero.

Inequalities

In the formulation of uncertainty relations in quantum mechanics, there are various procedures that relate to each different types of measurement processes. Depending on the respective underlying measurement process are then given appropriate mathematical statements.

Scattering relations

The most popular variant of uncertainty relations, the blur of the location x and the momentum p is σx respectively by their statistical scattering and defined? P. The uncertainty principle states in this case

Where and are the circuit number.

Within the framework of the formalism of quantum mechanics, the probability distributions for position and momentum measurements and thus the standard deviations of the corresponding wave functions ψ yield (x) and φ (p). The scattering inequality follows from the fact that these wave functions are with respect to position and momentum via a Fourier transformation linked. The Fourier transform of a spatially limited wave packet is again a wave packet, wherein the product to the package width obeys a relation which corresponds to the above inequality.

States minimal blur are doing such wave functions ψ (x) and φ (p ), called for the results in the equal sign of the inequality. Heisenberg and Kennard have shown that this property is achieved for certain Gaussian wave functions. It should be noted here is that statistical variations in accordance with their definition, only describe the half-width of the Gaussian probability density. We instead used in each case the entire width, ie 2σx and 2σp as a definition for the blurring, so would the value 2H instead of ħ / 2 on the right side of the inequality form the lower bound.

Simultaneous measurement

In the Heisenberg originally published version of the uncertainty principle, the concept of unsharpness of position and momentum is not always represented by the statistical scatter. An example of this is the often -discussed thought experiment for the determination of the position and momentum of particles by means of the single gap: A ray of possible electron orbits is hidden by a screen with a slit of width ( see figure at right ). The particles that pass coming through the gap, in this moment by their place in the direction parallel to the screen set with the accuracy of the gap ( preparation ). The suppression of the beam is connected to a spatial deflection of the object to the (random) angle ( diffraction ), and all points of the gap are after huygens ' principle starting points for elementary waves.

Now, the following conditions are met:

• The deflection angle is a random variable that can take on a different value for each particle.
• The de Broglie wavelength of the particle is considered:

• Thus, the first interference minimum on the screen is not visually recognizable, the speed difference must be at least about as large as the de Broglie wavelength of the particle, that is to say:

• There are considered only particles whose deflection angle corresponding to a pulse ( no random variable ) of the first diffraction minimum is within the predetermined pulse interval Ap to the pulse scale. Formally, these are precisely those which satisfy the following condition:

The last two relations together give the formula of de Broglie, the following restriction for the considered according to Heisenberg scattering angle:

Will now only the outer terms considered in this expression, we obtain after multiplication by p · Ax the relation of Heisenberg:

The essential difference between the two inequalities (1 ) and (2) is located both in the respective preparation and in the underlying measuring processes. In the scattering relation ( 1) the measurement of the scattering refers σx and? P on different samples of particles, which is why one can not speak of simultaneous measurements in this case. The physical content of the Heisenberg relation ( 2 ) can therefore not be described by the Kennard relation ( 1).

A statement that refers to the preparation ( projection) through a gap in the sense of ( 2) and yet gives an estimate for the variance? P of the pulse, can be formulated as follows: For particles ( wave functions ), in a finite interval were prepared ∂ x, the standard deviation of the pulse meets the inequality

The minimum possible spread of the momentum distribution is therefore dependent on the predetermined width Ax of the gap. By contrast, the preparation in inequality (1) refers to such particles of which it is known that they have a variance σx before the measurement pulse. Thus, the lower limit of inequality (1), the particles of the cracking experiment can not reach, as Gaussian probability densities on the entire real axis is equal to zero and not only in a finite portion of the length Ax.

Irrespective of the preparation of the wave function is performed in the spatial domain, that is, the diffraction experiment Heisenberg, that for the measurement of the probability density of a previous pulse Fourier transform is always necessary. Heisenberg understood here so under the inevitable " failure of the system " the influence of this Fourier transform on the quantum mechanical state in the spatial domain. In the experiment, this error is caused by the propagation time and the wave function of the bleed between the gap and the screen. The latter just corresponds statement 3 of the previous chapter.

Measurement noise and disturbance

Another variant of inequalities that the influence of the interaction between the DUT and measurement equipment as part of a considered explicitly von Neumann measurement process, leads to the following expression:

Εx The new variables and ηp variables designate the influence of the measuring apparatus on the considered metrics. Where:

• The mean deviation between the place before the interaction in the measuring instrument and the value is subsequently displayed ( measurement noise ),
• The mean change of the pulse during the period in the development of measuring apparatus.

The two measures for uncertainty, therefore, differ conceptually from each other. In the first case the deviation between the place is considered to x before the measurement and the measured value of the place which appears at the end of the meter. In contrast, only the difference between the pulse before and after the temporal evolution of the meter is in the second case considered, but the different therefrom measured value of the pulse would appear at the end, disregarded.

Are on the assumption that first the measurement noise and the disturbance εx ηp regardless of the state of the particle and ψ 2, the dispersion σx of the spatial distribution of the particle is smaller than the measurement noise εx was from relation ( 1 ), the following inequality

Inferred, which is interpreted by the Japanese physicist Masanao Ozawa as an expression of the measurement process by Heisenberg. Since, however, when considering present here not about a simultaneous measurement in the sense of Heisenberg (? P is not considered), it is expected that the product εx · ηp values ​​smaller than ħ / 2 can assume. This has led some authors to claim that Heisenberg was wrong.

The underlying concept that takes into account the influence of the interaction within the meter to the physical observables explicitly, 2012 has been verified by experiments with neutron spins and by experiments with polarized photons.

Generalization

The first of Kennard proven inequality (1) in 1929 formally generalized by Robertson. This generalization can also be blur relationships between other physical quantities specify. These include inequalities with respect to different angular momentum components between energy and momentum or place and energy. In Bra- Ket notation can be used for two observables A and B generally be formulated the following inequality:

Here, A and B, the Hermitian observables associated linear operators. The expression [ A, B ] = AB -BA designates the commutator of A and B. In contrast to the existing for the position and momentum ratio (1), in the generalized relation Robertson and the right side of the inequality explicitly dependent on the wave function be. The product of the dispersions of A and B can, therefore, even assume the value zero, not only when the A and B observables commutate with each other but for special even if this is not the case. For position and momentum, and other so-called " complementary " Observablenpaare, the commutator is proportional to the unit but each operator. The expected value in the relation of Robertson can therefore be zero for complementary observables never. Other often cited in this connection variable that does not interchange with each other (eg two different angular momentum components), however, are not complementary to each other, because their Vertauschungsprodukt is not a number but an operator. Commuting observables are, however, in any case, that is for all, at the same time free from dispersion measure, since their commutator vanishes. It then is compatible or compatible observables.

The above inequality can be proved in a few lines:

First, the variance of the operators A and B by using two state functions f and g shown, i.e., whether

Thus we obtain for the variances of the operators, the representations:

Using the inequality between black, this results in:

To bring this inequality in the usual form, the right side is further estimated and calculated. Used is that the magnitude squared of any complex number z can not be smaller than the square of the imaginary part, i.e.

Where the imaginary part of represents. With the substitution, could provide the product of the variances of the estimation

For the scalar products occurring therein and is obtained by further calculate

Thus results for the difference in the inequality

So just the expected value of the commutator. This eventually leads to the inequality

And a square root provides the above mentioned inequality.

Derivation of the uncertainty relation, according to von Neumann

As given, it is assumed:

• A Hilbert space provided on the inner product and the associated norm and as the identity operator;
• Two into defined self-adjoint operator with the linear characteristic where for the scalar.

On this basis, can perform the following calculation steps:

It is:

So the following applies:

This means:

So follows by Cauchy-Schwarz:

Now, are any two scalars, the Kommutatorgleichung applies equally for and how to easily recalculate.

Consequently, it has always very generally:

As a result of step 2 is obtained for, and always

For the quantum- mechanically relevant case to win now immediately for the Heisenberg Uncertainty Principle

Examples

1 If you select in the previous chapter for the operators as well as used and that applies to the commutator of position and momentum, so the inequality of Robertson gives the relation of Kennard. The right-hand side of the relationship is independent of the wave function of the particle, because the commutator is a constant in this case.

A second uncertainty relation for measurement of kinetic energy and place arises from the commutator that is,

In this case, the lower limit is not constant, but depends on the mean value of the pulse and thus of the wavefunction of the particle.

3 In a measurement of the energy and momentum of a particle in a potential dependent on the location of the commutator of the total energy and of the derivative of the potential (force) is dependent, that is, the corresponding uncertainty relation for energy and momentum is thus

Also in this example, the right side of the inequality is generally a constant.

4 In the case of measurement of energy and time, the generalization of Robertson does not directly apply, because the time is not defined in the standard quantum theory as an operator. With the help of honorable firmly between theorem and an alternative definition of time uncertainty, however, can prove an analogous inequality, see energy-time uncertainty relation.

5 For the time dependence of the position operator in the Heisenberg picture the illustration applies

Because of dependency of the pulse in this illustration shows that the commutator of two local operators at different points in time 0 and does not disappear, i.e., a result for the product of the deviations of the position measurements in the time interval between the uncertainty

The more time that passes between the two scattering measurements, the greater the minimum achievable blur. Note that in this example of two non- simultaneous measurements carried out there is an uncertainty. For two instantaneous measurements made of the place of the commutator and the lower bound of the inequality vanishes is equal to 0

6 The minimum width of a tunnel barrier can be estimated via the uncertainty principle. If we consider an electron with the mass and the electric charge that tunnels through a potential difference is obtained for the position uncertainty and thus the minimum width of the tunnel barrier

When a potential difference of 100 mV, as found for example in scanning tunneling microscopy, is due to this relationship a smallest tunnel barrier of about 0.3 nm, which agrees well with experimental observations.