# Heisenberg's uncertainty principle

Werner Heisenberg and the equation of the uncertainty relation on a German postage stamp

The Heisenberg uncertainty principle or uncertainty relation (rarely also uncertainty principle ) is the statement of quantum physics that two complementary properties of a particle are not the same any estimated. The best-known example of a pair of such properties are location and momentum .

The uncertainty relation is not the result of technically correctable inadequacies of a corresponding measuring instrument, but of a principle nature. It was formulated in 1927 by Werner Heisenberg in the context of quantum mechanics . Heisenberg's uncertainty relation can be seen as an expression of the wave character of matter . It is considered the basis of the Copenhagen interpretation of quantum mechanics.

## Quantum mechanics and classical physics

Quantum mechanics is one of the fundamental theories for describing our physical world. The conceptual structure of this theory is fundamentally different from that of classical physics .

The statements of quantum mechanics about our world are statements about the outcomes of measurements . In contrast to classical physics, only probability statements can be made in each case , so you can only predict the value distribution when measuring on an ensemble of similar systems. Heisenberg's uncertainty principle results from the fact that a physical system in quantum mechanics is described with the help of a wave function . While in classical mechanics position or momentum are simple quantities that can in principle be exactly measured, their distributions in quantum mechanics result from the square of the magnitude of the wave function or its Fourier transform , i.e. That is, they cannot be determined independently of one another. Since the distributions of position and momentum both depend on the wave function of the system, the standard deviations of the measurements are also dependent on one another. The more precisely you want to determine the location of a particle in the usual quantum mechanical description, the greater the imprecision of the momentum - and vice versa.

The following analogy illustrates the indeterminacy: Let us assume that we have a time-varying signal, e.g. B. a sound wave , and we want to measure the exact frequency of this signal at a certain point in time. This is impossible, because in order to determine the frequency more or less exactly, we have to observe the signal over a sufficiently long period of time (see Küpfmüller's uncertainty relation), and as a result we lose time precision. In other words, a tone cannot be there within an arbitrarily short period of time, such as a short scream, and at the same time have an exact frequency, such as that of an uninterrupted pure tone. The duration and the frequency of the wave are to be considered analogous to the position and momentum of a particle.

## Original wording

The first formulation of an uncertainty relation in quantum mechanics concerned the simultaneous knowledge of the position and momentum of a particle. In 1927 Heisenberg published his work on the descriptive content of quantum theoretical kinematics and mechanics and argued that the microscopic determination of the location of  a particle in general must lead to an influencing (disturbance) of the particle's momentum  . If the location of an electron is to be determined by optical observation (in the simplest case: seeing ), the particle can be illuminated so that at least one of the incident light quanta is scattered into the measuring instrument (eye, microscope). ${\ displaystyle x}$${\ displaystyle p}$

On the one hand, the inaccuracy of  the location depends on the wavelength of the light used. On the other hand, the deflection of the light quantum acts like an impact on the particle, whereby the momentum of the body experiences an indeterminacy of  ( Compton scattering ). With the help of the De Broglie relationship , Heisenberg estimated that the fundamental lower limit for these uncertainties was that the product of  and  cannot be less than the natural constant characteristic of quantum physics, Planck's constant of action . Heisenberg formulated this fundamental limit of measurability in the (symbolic) statement ${\ displaystyle \ Delta x}$${\ displaystyle \ Delta p}$${\ displaystyle \ Delta x}$${\ displaystyle \ Delta p}$ ${\ displaystyle h}$

${\ displaystyle \ Delta x \ cdot \ Delta p \ sim h}$

The initially qualitative character of this estimate stems from the fact that the statement has not been (strictly) proven and the notation used for the uncertainties is not precisely defined. With a suitable interpretation of the notation in the context of modern quantum mechanics, however, it turns out that the formula comes very close to reality.

## Uncertainty relation and everyday experience

Why these characteristic indeterminacies were not noticed earlier, neither in everyday life nor in research, can be understood if one considers the smallness of Planck's quantum of action compared to the typically achievable measurement accuracies for position and momentum. The following examples include:

The location of the vehicle is in the radar control up to be precisely determined, d. H. . The uncertainty of the velocity is assumed with and the mass with . This results in a momentum uncertainty of . Thus resulting for the product: . The restriction due to the uncertainty relation would therefore only become noticeable if the accuracy was increased by 18 decimal places each for location and speed. It is obvious that the radar signal has practically no effect on the vehicle during the measurement.${\ displaystyle \ pm 1 \, \ mathrm {m}}$${\ displaystyle \ Delta x = 2 \, \ mathrm {m}}$${\ displaystyle \ Delta v = 1 \, \ mathrm {{\ tfrac {km} {h}} = {\ tfrac {1000 \, m} {3600 \, s}} \ approx 0 {,} 3 \, { \ tfrac {m} {s}}}}$${\ displaystyle m = 1 \, \ mathrm {t} = 1000 \, \ mathrm {kg}}$${\ displaystyle \ Delta p = m \ cdot \ Delta v = 0 {,} 3 \, \ mathrm {\ tfrac {kg \ cdot km} {s}}}$${\ displaystyle \ Delta x \ cdot \ Delta p = 9 \ cdot 10 ^ {35} \, h}$
Speck of dust
In an extremely accurate microscopied speck of a mass and less blur both geographic location, as well as the speed , the result for the product: . The restriction due to the uncertainty relation would become noticeable here if the accuracy was increased by four decimal places for location and speed.${\ displaystyle m = 10 ^ {- 15} \, \ mathrm {kg}}$${\ displaystyle \ Delta x = 0 {,} 01 \, \ mathrm {\ mu m}}$${\ displaystyle \ Delta v = 1 \, \ mathrm {\ tfrac {mm} {s}}}$${\ displaystyle \ Delta x \ cdot \ Delta p = 1 {,} 5 \ cdot 10 ^ {7} \, h}$
Electron in atom
An atom is about one angstrom in diameter . With a kinetic energy of an electron bound in it of about , the result for the electron is a momentum uncertainty of about . A location determination with an inaccuracy of about 10 atomic diameters,, results for the product , which is still in the range of what is in principle possible. For location accuracy on the order of the atomic diameter with , however, the following applies: . However, this is in contradiction to the uncertainty principle, so such an accuracy of the description is in principle impossible.${\ displaystyle E _ {\ mathrm {kin}} = 10 \, \ mathrm {eV}}$${\ displaystyle \ Delta p = 1 {,} 7 \ ​​cdot 10 ^ {- 24} \, \ mathrm {kg {\ tfrac {m} {s}}}}$${\ displaystyle \ Delta x = 10 \, \ mathrm {\ AA}}$${\ displaystyle \ Delta x \ cdot \ Delta p = 2 {,} 5 \, h}$${\ displaystyle \ Delta x = 1 \, \ mathrm {\ AA}}$${\ displaystyle \ Delta x \ cdot \ Delta p = 0 {,} 25 \, h}$

## statement

The following statements are summarized under the concept of the principle of uncertainty or indefiniteness, which are related to one another but have different physical meanings. They are noted here as an example for the pair of place and momentum.

1. It is not possible to prepare a quantum mechanical state in which the location and the momentum are arbitrarily precisely defined.
2. In principle, it is impossible to measure the position and momentum of a particle at the same time with any precision.
3. The measurement of the momentum of a particle is inevitably connected with a disturbance of its position, and vice versa.

Each of these three statements can be formulated quantitatively in the form of so-called uncertainty relations, which indicate a lower limit for the achievable minimum uncertainty of the preparation or measurement.

Uncertainty relations can also apply between other pairs of physical quantities . The prerequisite for this is that the commutator of the two quantum mechanical operators assigned to the quantities is not zero. For example, Franke-Arnold et al. M. demonstrated experimentally that a corresponding relation between angular position and angular momentum applies.

## Inequalities

When formulating uncertainty relations in the context of quantum mechanics, there are different approaches that each refer to different types of measurement processes. Depending on the respective underlying measurement process, corresponding mathematical statements are then made.

### Scatter relations

In the most popular variant of uncertainty relations, the uncertainty of the location  x and the momentum  p is defined by their statistical scatter σ x and σ p . The uncertainty relation says in this case

${\ displaystyle \ sigma _ {x} \ cdot \ sigma _ {p} \ geq {\ frac {\ hbar} {2}} \ ,, \ qquad \ qquad (1)}$

where and is the circle number . ${\ displaystyle \ hbar = {\ frac {h} {2 \ pi}}}$${\ displaystyle \ pi}$

Within the framework of the formalism of quantum mechanics, the probability distributions for position and momentum measurements and thus the standard deviations result from the associated wave functions  ψ (x) and φ (p). The scattering inequality then follows from the fact that these wave functions are linked with one another with regard to position and momentum via a Fourier transformation . The Fourier transform of a spatially limited wave packet is again a wave packet, the product of the packet widths obeying a relationship which corresponds to the above inequality.

Such wave functions ψ (x) and φ (p) are called states of minimal uncertainty , for which the equality sign of the inequality results. Heisenberg and Kennard have shown that this property is achieved for Gaussian wave functions. It should be noted that the standard deviation of a Gaussian probability density is not immediately suitable as an idea for its overall width, since e.g. B. the range of values ​​in which the location or impulse are with a probability of 95% is about four times as large.

### Simultaneous measurement

Schematic representation of the diffraction at the slit. The accuracy Δx of the site preparation corresponds exactly to the width of the gap.

In the variant of the uncertainty relation originally published by Heisenberg, the concept of the uncertainty of position and momentum is not always represented by the statistical spread. An example of this is the often discussed thought experiment in which the position and momentum of particles are to be determined with the help of the single slit : a broad beam of parallel-flying electrons with the same momentum hits a screen with a slit the width of the screen (see figure on the right). When passing through the gap, the position coordinate of the electrons (in the direction across the gap) is known except for the uncertainty . The masking causes a diffraction of the beam, whereby elementary waves emanate from all points of the slit according to the Huygens principle . After passing through the gap, this leads to a widening of the jet, i. H. for each individual electron to a deflection by a certain angle . ${\ displaystyle \ Delta x}$${\ displaystyle \ Delta x}$${\ displaystyle \ alpha}$

The following requirements are now met:

• The deflection angle is a random variable which can assume a different value for each particle, the frequency distribution being given by the interference pattern.${\ displaystyle \ alpha}$
• The following applies to the de Broglie wavelength of the particle:${\ displaystyle \ lambda}$
${\ displaystyle \ lambda = {\ frac {h} {p}}}$
• So that the first interference minimum can still be seen optically on the screen, the path difference must be at least as large as the De Broglie wavelength of the particle:
${\ displaystyle \ Delta x \, \ sin (\ alpha) \; \ gtrsim \; \ lambda}$
• According to Heisenberg, only the particles in the main maximum of the diffracted beam are considered. Their angles of deflection correspond to a pulse in the x-direction which lies within the given pulse interval Δ p (no random variable) of the first diffraction minimum on the pulse scale. Formally, these are exactly those that meet the following condition:${\ displaystyle \ alpha}$
${\ displaystyle p \ cdot \ sin (\ alpha) \ leq \ Delta p}$

The last two relations, together with de Broglie's formula, result in the following restriction for the scattering angles considered:

${\ displaystyle {\ frac {h} {p \ cdot \ Delta x}} \; \ lesssim \; \ sin (\ alpha) \; \ leq \; {\ frac {\ Delta p} {p}}}$

If only the outer terms are considered in this expression, then after multiplication with p · Δ x the relation of Heisenberg results  :

${\ displaystyle \ Delta x \ cdot \ Delta p \; \ gtrsim \; h \ quad \ quad \ quad (2)}$

The main difference between the two inequalities (1) and (2) lies both in the respective preparation and in the measurement processes used. With the scattering relation (1), the measurement of the scattering σ x and σ p relates to 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 relates to the preparation (projection) through a slit in the sense of (2) and still gives an estimate for the scattering σ p of the momentum can be formulated as follows: for particles (wave functions) that are in a finite Interval Δ x , the standard deviation for the momentum satisfies the inequality:

${\ displaystyle \ sigma _ {p} \ cdot \ Delta x \ geq \ pi \ cdot \ hbar}$

The minimum possible spread of the momentum distribution is therefore dependent on the specified width Δ x of the gap. In contrast, the preparation in inequality (1) relates to those particles which are known to have had a scatter σ x before the momentum measurement . Thus, the particles of the cleavage test cannot reach the lower bound of inequality (1), since Gaussian probability densities are not equal to zero on the entire real axis and not only in a finite subrange of length Δ x.

Regardless of which preparation of the wave function is carried out in spatial space , Heisenberg's diffraction experiment shows that a previous Fourier transformation is always necessary to measure the probability density of the impulse. Here, Heisenberg understands the inevitable "disturbance of the system" to be the influence of this Fourier transformation on the quantum mechanical state in spatial space. In the experiment this disturbance is caused by the temporal propagation and the dissipation of the wave function between the slit and the screen. The latter corresponds to statement 3 of the previous chapter.

### Measurement noise and interference

Another variant of inequalities, which explicitly takes into account the influence of the interaction between the measuring object and the measuring apparatus in the context of a Von Neumann measuring process , leads to the following expression ( Ozawa inequality ):

${\ displaystyle \ varepsilon _ {x} \ cdot \ eta _ {p} + \ varepsilon _ {x} \ cdot \ sigma _ {p} + \ sigma _ {x} \ cdot \ eta _ {p} \ geq { \ frac {\ hbar} {2}}}$

The new variables ε x and η p denote the influence of the measuring apparatus on the measured variables under consideration:

• ${\ displaystyle \ varepsilon _ {x}}$the mean deviation between the location before the interaction in the measuring device and the value that is subsequently displayed (measurement noise )
• ${\ displaystyle \ eta _ {p}}$the mean change in the momentum during the time development in the measuring apparatus.
• ${\ displaystyle \ sigma _ {x}}$the pure quantum fluctuation of the place
• ${\ displaystyle \ sigma _ {p}}$ the pure quantum fluctuation of momentum

The two measures for uncertainty differ conceptually from one another, since in the second case the measured value of the impulse that would be displayed at the end is not taken into account.

Assuming that

1. the measurement noise ε x and the disturbance η p are independent of the state ψ of the particle and
2. the scatter σ x of the local distribution of the particle is smaller than the measurement noise ε x ,

became the inequality from relation (1)

${\ displaystyle \ varepsilon _ {x} \ cdot \ eta _ {p} \ geq {\ frac {\ hbar} {2}}}$

inferred, which is interpreted by the Japanese physicist Masanao Ozawa as an expression for the measuring process of Heisenberg. However, since the present consideration is not a simultaneous measurement in the sense of Heisenberg (σ p is not taken into account), it is to be expected that the product ε x · η p can also have values ​​smaller than ħ / 2. This led some authors to say that Heisenberg was wrong.

The underlying concept, which explicitly takes into account the influence of the interaction within the measuring device on the physical observables, was verified in 2012 through experiments with neutron spins and through experiments with polarized photons .

## generalization

The inequality (1), first proven by Kennard, was formally generalized in 1929 by Howard P. Robertson . With this generalization, it is also possible to specify indistinct relationships between other physical quantities. These include, for example, inequalities with regard to different angular momentum components , between energy and momentum or energy and location.

In general, the following inequality can be formulated for two observables  A and  B in Bra-Ket notation:

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

Here are

• ${\ displaystyle {\ hat {A}}}$and the self-adjoint linear operators belonging to the observables${\ displaystyle {\ hat {B}}}$
• ${\ displaystyle [{\ hat {A}}, {\ hat {B}}] = {\ hat {A}} {\ hat {B}} - {\ hat {B}} {\ hat {A}} }$the commutator of  A and  B.

In contrast to the relation (1) for position and momentum, in Robertson's generalized relation the right-hand side of the inequality can also be explicitly dependent on the wave function. The product of the scatter of A and B can therefore even assume the value zero, not only when the observables  A and  B commute with one another, but for special even when this is not the case. ${\ displaystyle \ psi}$

For position and momentum as well as other pairs of complementary observables the commutator is proportional to the unit operator ; therefore, for complementary observables, the expectation value in Robertson's relation can never become zero. Other variables often mentioned in this context that do not interchange with one another (e.g. two different angular momentum components) are, however, not complementary to one another, because their interchange product is not a number but an operator. Such pairs of observables are called incommensurable .

In contrast, interchangeable observables are in any case, i.e. H. for all , measurable at the same time without scattering, since their commutator disappears. They are then compatible , commensurable or compatible observables. ${\ displaystyle \ psi}$

The above inequality can be proven in a few lines:

First, the variances of the operators A and B are represented with the help of two state functions f and g, i.e. i.e., it be

{\ displaystyle {\ begin {aligned} | f \ rangle &: = {\ big (} {\ hat {A}} - \ langle {\ hat {A}} \ rangle {\ big)} | \ psi \ rangle \\ | g \ rangle &: = {\ big (} {\ hat {B}} - \ langle {\ hat {B}} \ rangle {\ big)} | \ psi \ rangle. \ end {aligned}} }

This gives the following representations for the variances of the operators:

{\ displaystyle {\ begin {aligned} \ sigma _ {A} ^ {2} & = {\ Big \ langle} \ psi {\ Big |} {\ big (} {\ hat {A}} - \ langle { \ hat {A}} \ rangle {\ big)} ^ {2} {\ Big |} \ psi {\ Big \ rangle} = \ langle f | f \ rangle \\\ sigma _ {B} ^ {2} & = {\ Big \ langle} \ psi {\ Big |} {\ big (} {\ hat {B}} - \ langle {\ hat {B}} \ rangle {\ big)} ^ {2} {\ Big |} \ psi {\ Big \ rangle} = \ langle g | g \ rangle. \ End {aligned}}}

Using the Schwarz inequality we get:

${\ displaystyle \ sigma _ {A} ^ {2} \ sigma _ {B} ^ {2} = \ langle f | f \ rangle \ langle g | g \ rangle \ geq | \ langle f | g \ rangle | ^ {2}}$

In order to bring this inequality into the usual form, the right-hand side is further estimated and calculated. For this purpose, one uses that the square of the absolute value of any complex number z cannot be smaller than the square of its imaginary part, i.e. H.

${\ displaystyle | z | ^ {2} \ geq \ operatorname {Im} (z) ^ {2} = \ left [{\ frac {zz ^ {*}} {2 \ mathrm {i}}} \ right] ^ {2},}$

where represents the imaginary part of . With the substitution , the estimate for the product of the variances results ${\ displaystyle \ operatorname {Im} (z)}$${\ displaystyle z}$${\ displaystyle z: = \ langle f | g \ rangle}$

${\ displaystyle \ sigma _ {A} ^ {2} \ sigma _ {B} ^ {2} \ geq \ left | {\ frac {1} {2 \ mathrm {i}}} {\ big (} \ langle f | g \ rangle - \ langle g | f \ rangle {\ big)} \ right | ^ {2}.}$

For the scalar products occurring therein and are obtained by further calculation ${\ displaystyle \ langle f | g \ rangle}$${\ displaystyle \ langle g | f \ rangle}$

${\ displaystyle \ langle f | g \ rangle = \ langle {\ hat {A}} {\ hat {B}} \ rangle - \ langle {\ hat {A}} \ rangle \ langle {\ hat {B}} \ rangle \ qquad {\ text {or}} \ qquad \ langle g | f \ rangle = \ langle {\ hat {B}} {\ hat {A}} \ rangle - \ langle {\ hat {A}} \ rangle \ langle {\ hat {B}} \ rangle.}$

This results in the difference in the inequality

${\ displaystyle \ langle f | g \ rangle - \ langle g | f \ rangle = \ langle {\ hat {A}} {\ hat {B}} \ rangle - \ langle {\ hat {B}} {\ hat {A}} \ rangle = {\ big \ langle} \ psi {\ big |} [{\ hat {A}}, {\ hat {B}}] {\ big |} \ psi {\ big \ rangle} ,}$

thus precisely the expected value of the commutator. This ultimately leads to the inequality

${\ displaystyle \ sigma _ {A} ^ {2} \ sigma _ {B} ^ {2} \ geq \ left | {\ frac {1} {2 \ mathrm {i}}} {\ big \ langle} \ psi {\ big |} [{\ hat {A}}, {\ hat {B}}] {\ big |} \ psi {\ big \ rangle} \ right | ^ {2}}$

and taking the root yields the inequality given above.

## Derivation of the uncertainty relation according to von Neumann

The following are assumed to be given:

• A Hilbert space , provided with the dot product and the associated standard and with the identity operator on ;${\ displaystyle {\ mathcal {H}}}$ ${\ displaystyle \ langle \ cdot, \ cdot \ rangle}$ ${\ displaystyle \ | \ cdot \ | = {\ langle \ cdot, \ cdot \ rangle} ^ {\ frac {1} {2}}}$${\ displaystyle \ mathbf {1} _ {\ mathcal {H}}}$${\ displaystyle {\ mathcal {H}} \;}$
• Two self-adjoint linear operators defined in and with for a certain scalar ;${\ displaystyle {\ mathcal {H}}}$ ${\ displaystyle {\ hat {A}} \ colon {\ mathcal {H}} \ supset D ({\ hat {A}}) \ to {\ mathcal {H}}}$${\ displaystyle {\ hat {B}} \ colon {\ mathcal {H}} \ supset D ({\ hat {B}}) \ to {\ mathcal {H}}}$${\ displaystyle [{\ hat {A}}, {\ hat {B}}] = a \ mathbf {1} _ {\ mathcal {H}}}$ ${\ displaystyle a \ in \ mathbb {C} \ setminus \ {0 \}}$
• One of the norm  .${\ displaystyle \ psi \ in {\; {\ bigl (} D ({\ hat {A}}) \ cap D ({\ hat {B}}) \ cap {\ hat {A}} ^ {- 1 } (D ({\ hat {B}})) \ cap {\ hat {B}} ^ {- 1} (D ({\ hat {A}})) {\ bigr)} \;} \ subseteq { \ mathcal {H}}}$${\ displaystyle \ | \ psi \ | = 1}$

On this basis, the following calculation steps can be carried out:

Step 1

It is:

${\ displaystyle \ operatorname {Im} {\ langle {\ hat {A}} \ psi, {\ hat {B}} \ psi \ rangle} = {\ frac {\ langle {\ hat {A}} \ psi, {\ hat {B}} \ psi \ rangle - \ langle {\ hat {B}} \ psi, {\ hat {A}} \ psi \ rangle} {2 \ mathrm {i}}}}$

So:

{\ displaystyle {\ begin {aligned} 2 \ cdot \ operatorname {Im} {\ langle {\ hat {A}} \ psi, {\ hat {B}} \ psi \ rangle} & = - \ mathrm {i} \ cdot (\ langle {\ hat {B}} {\ hat {A}} \ psi, \ psi \ rangle - \ langle {\ hat {A}} {\ hat {B}} \ psi, \ psi \ rangle ) = - \ mathrm {i} \ cdot \ langle {\ hat {B}} {\ hat {A}} \ psi - {\ hat {A}} {\ hat {B}} \ psi, \ psi \ rangle = \ mathrm {i} \ cdot \ langle ({\ hat {A}} {\ hat {B}} - {\ hat {B}} {\ hat {A}}) \ psi, \ psi \ rangle \\ & = \ mathrm {i} \ cdot \ langle [{\ hat {A}}, {\ hat {B}}] \ psi, \ psi \ rangle = \ mathrm {i} \ cdot \ langle a \ mathbf {1 } _ {\ mathcal {H}} \ psi, \ psi \ rangle = \ mathrm {i} \ cdot a \ cdot \ langle \ psi, \ psi \ rangle = \ mathrm {i} \ cdot a \ cdot {\ | \ psi \ |} ^ {2} \ end {aligned}}}

That means:

${\ displaystyle {\ | \ psi \ |} ^ {2} = - {\ frac {2 \ mathrm {i}} {a}} \ cdot \ operatorname {Im} {\ langle {\ hat {A}} \ psi, {\ hat {B}} \ psi \ rangle} \ leq {\ frac {2} {| a |}} \ cdot | \ langle {\ hat {A}} \ psi, {\ hat {B}} \ psi \ rangle |}$

So it follows with Cauchy-Schwarz :

${\ displaystyle {\ | \ psi \ |} ^ {2} \ leq {\ frac {2} {| a |}} \ cdot \ | {\ hat {A}} \ psi \ | \ cdot \ | {\ has {B}} \ psi \ |}$
step 2

If there are any two scalars, then the commutator equation applies in the same way to and . ${\ displaystyle r, s \ in \ mathbb {R}}$${\ displaystyle {\ hat {A}} _ {r} = {\ hat {A}} - r \ mathbf {1} _ {\ mathcal {H}}}$${\ displaystyle {\ hat {B}} _ {s} = {\ hat {B}} - s \ mathbf {1} _ {\ mathcal {H}}}$

Consequently, one always has very generally:

${\ displaystyle {\ | \ psi \ |} ^ {2} \ leq {\ frac {2} {| a |}} \ cdot \ left \ | {\ hat {A}} _ {r} \ psi \ right \ | \ cdot \ left \ | {\ hat {B}} _ {s} \ psi \ right \ | = {\ frac {2} {| a |}} \ cdot \ left \ | {\ hat {A} } \ psi -r \ psi \ right \ | \ cdot \ left \ | {\ hat {B}} \ psi -s \ psi \ right \ |}$
step 3

As a result of step 2, because of with and always receives${\ displaystyle \ | \ psi \ | = 1}$${\ displaystyle r = \ langle {\ hat {A}} \ psi, \ psi \ rangle}$${\ displaystyle s = \ langle {\ hat {B}} \ psi, \ psi \ rangle}$

${\ displaystyle \ left \ | {\ hat {A}} \ psi - \ langle {\ hat {A}} \ psi, \ psi \ rangle \ psi \ right \ | \ cdot \ left \ | {\ hat {B }} \ psi - \ langle {\ hat {B}} \ psi, \ psi \ rangle \ psi \ right \ | \ geq {\ frac {| a |} {2}} \ cdot}$
Step 4

For the quantum mechanically relevant case , one obtains the Heisenberg uncertainty principle${\ displaystyle \ textstyle a = {\ frac {\ hbar} {\ mathrm {i}}}}$

${\ displaystyle \ left \ | {\ hat {A}} \ psi - \ langle {\ hat {A}} \ psi, \ psi \ rangle \ psi \ right \ | \ cdot \ left \ | {\ hat {B }} \ psi - \ langle {\ hat {B}} \ psi, \ psi \ rangle \ psi \ right \ | \ geq {\ frac {\ hbar} {2}}}$
Remarks
1. According to Wintner-Wielandt's theorem, Wegen is necessarily infinite-dimensional . Likewise, in connection with Hellinger-Toeplitz's theorem, it cannot apply. see: Harro Heuser : functional analysis. Theory and Application (=  Mathematical Guidelines . Volume${\ displaystyle a \ neq 0}$ ${\ displaystyle {\ mathcal {H}}}$${\ displaystyle a \ neq 0}$${\ displaystyle {\ mathcal {H}} = D ({\ hat {A}}) = D ({\ hat {B}})}$
36 ). 4th revised edition. Teubner Verlag , Wiesbaden 2006, ISBN 978-3-8351-0026-8 , p. 102, 244, 564-565 .
2. The following is written briefly instead of . In addition, it should be noted that the superscript after the operator refers to the respective original .${\ displaystyle \ psi}$${\ displaystyle | \ psi \ rangle}$${\ displaystyle -1}$
3. Here and in the following, John von Neumann's presentation and the practice of analysis are followed, according to which the scalar product is linear in the first component and antilinear in the second component . The opposite practice is often found in physics. Which variant you follow has no influence on the result here in the article. In particular, it should be noted that purely imaginary is if and only if is purely imaginary.${\ displaystyle a}$ ${\ displaystyle {\ bar {a}}}$
4. As the second equation of step 1, has the scalar under the given conditions always purely imaginary, so real part free to be.${\ displaystyle a}$

## Examples

1. If in the previous chapter one chooses for the operators such as and and uses that the position and momentum apply to the commutator , then Robertson's inequality results in Kennard's relation. The right side of the relation is independent of the wave function of the particle, since the commutator is a constant in this case. ${\ displaystyle A = x}$${\ displaystyle B = p}$${\ displaystyle [x, p] = i \ hbar}$

2. An uncertainty relation for the measurement of kinetic energy and location results from the commutator : ${\ displaystyle \ textstyle T = {\ frac {1} {2}} \ cdot {\ frac {p ^ {2}} {m}}}$${\ displaystyle x}$${\ displaystyle [T, x] = - \ mathrm {i} \ hbar \ cdot p / m}$

${\ displaystyle \ sigma _ {T} \ cdot \ sigma _ {x} \ geq {\ frac {\ hbar} {2}} \ cdot {\ frac {| \ langle {\ hat {p}} \ rangle |} {m}}}$

In this case the lower bound is not constant, but depends on the mean value of the momentum and thus on the wave function of the particle.

3. When measuring the energy and momentum of a particle in a potential dependent on the location , the commutator of the total energy and momentum depends on the derivative of the potential (force): The corresponding uncertainty relation for energy and momentum is thus ${\ displaystyle V (x)}$${\ displaystyle H = T + V}$${\ displaystyle p}$${\ displaystyle [H, p] = - \ mathrm {i} \ hbar \ cdot V '.}$

${\ displaystyle \ sigma _ {H} \ cdot \ sigma _ {p} \ geq {\ frac {\ hbar} {2}} \, \ left | {\ big \ langle} V '({\ hat {x} }) {\ big \ rangle} \ right |}$

In this example, too, the right-hand side of the inequality is generally not a constant.

4. In the case of the measurement of energy and time, Robertson's generalization cannot be applied directly, since time is not defined as an operator in standard quantum theory. With the help of Ehrenfest's theorem and an alternative definition of time uncertainty, however, an analog inequality can be proven, see energy-time uncertainty relation .

5. The representation applies to the time dependence of the position operator of a free particle in the Heisenberg picture

${\ displaystyle {\ hat {x}} (t) = {\ hat {x}} (0) + {\ frac {\ hat {p}} {m}} \, t}$

Due to the momentum dependency in this representation, the commutator of two position operators at the different times 0 and does not vanish: This results in the uncertainty relation for the product of the scatter of the two position measurements in the time interval${\ displaystyle t}$${\ displaystyle [x (0), x (t)] = \ mathrm {i} \ hbar \ cdot t / m}$${\ displaystyle t}$

${\ displaystyle \ sigma _ {x} (0) \ cdot \ sigma _ {x} (t) \ geq {\ frac {\ hbar} {2}} \ cdot {\ frac {t} {m}}}$

The more time elapses between the two scatter measurements, the greater the minimally achievable blurring. For two instantaneously, i.e. H. Simultaneously performed measurements of the location, however (t = 0), the commutator disappears and the lower bound of the inequality becomes 0.

6. The minimum width of a tunnel barrier can be estimated using the uncertainty relation. If one looks at an electron with its mass and the electrical charge that tunnels through a potential difference , the spatial uncertainty and thus the minimum width of the tunnel barrier result ${\ displaystyle m_ {e}}$ ${\ displaystyle e,}$ ${\ displaystyle U}$

${\ displaystyle \ sigma _ {x} \ geq {\ frac {\ hbar} {\ sqrt {8 \ cdot m_ {e} \ cdot e \ cdot U}}}}$

With a potential difference of 100 mV, such as occurs in scanning tunneling microscopy , this relationship results in a smallest tunnel barrier of about 0.3 nm, which agrees well with experimental observations.

## literature

• Werner Heisenberg: About the descriptive content of quantum theoretical kinematics and mechanics. Zeitschrift für Physik, Volume 43, 1927, pp. 172-198.
• Ders .: The physical principles of quantum theory. S. Hirzel 1930, 2008.
• Ders .: The part and the whole . Piper, Munich 1969.
• Ders .: quantum theory and philosophy. Reclam, Stuttgart 1979.
• Johann v. Neumann : Mathematical foundations of quantum mechanics. Unchanged reprint of the 1st edition from 1932. Chapter III "The quantum mechanical statistics." Section 4 "Indeterminacy relations" (=  The basic teachings of the mathematical sciences in individual representations . Volume 38 ). Springer-Verlag , Berlin a. a. 1968, ISBN 3-540-04133-8 . MR0223138
• Joachim Weidmann : Linear operators in Hilbert spaces. Part 1: Basics (=  mathematical guidelines ). Teubner Verlag , Stuttgart u. a. 2000, ISBN 3-519-02236-2 . MR1887367

## Individual evidence

1. W. Heisenberg: About the descriptive content of quantum theoretical kinematics and mechanics . In: Journal of Physics . tape 43 , no. 3 , 1927, pp. 172–198 , doi : 10.1007 / BF01397280 ([ Original work as HTML ( Memento from May 10, 2013 in the Internet Archive )]).
2. See Walter Greiner : Quantum Mechanics . 6. revised and exp. Edition. Publisher Harri Deutsch , Zurich u. a. 2005, ISBN 978-3-8171-1765-9 , pp. 55-56 .
• 1) p. 55 below: “The wave character of matter ... is expressed, among other things, by the fact that in the field of microphysics there is a direct connection between determination of position and momentum. This manifests itself in the fact that the position and momentum of a particle cannot be sharply defined at the same time. The degree of uncertainty is given by the Heisenberg uncertainty principle. "
• 2) P. 56 (footnote): “In search of the correct description of the atomic phenomena, Heisenberg formulated his positivistic principle in July 1925 that only“ principally observable ”quantities should be used ... Heisenberg succeeded in close cooperation with N. Bohr to show the deeper ... physical background of the new formalism. Heisenberg's uncertainty principle of 1927 became the basis of the Copenhagen interpretation of quantum theory. "
3. a b Werner Heisenberg: Physical principles of quantum theory. S. Hirzel Verlag, Leipzig 1930.
4. Paul Busch, Teiko Heinonen, Pekka Lahti: Heisenberg's uncertainty principle . In: Physics Reports . tape 452 , no. 6 , 2007, p. 155-176 , doi : 10.1016 / j.physrep.2007.05.006 , arxiv : quant-ph / 0609185v3 .
5. Sonja Franke-Arnold et al .: Uncertainty Principle for angular position and angular momentum , in: New Journal of Physics Vol. 6 (2004) p. 103, [1]
6. a b E. H. Kennard: On the quantum mechanics of simple types of movement . In: Journal of Physics . tape 44 , no. 4 , 1927, pp. 326-352 , doi : 10.1007 / BF01391200 .
7. LE Ballentine: The Statistical Interpretation of Quantum Mechanics . In: Reviews of Modern Physics . tape 42 , no. 4 , 1970, pp. 358-381 .
8. JBM Uffink, J. Hilgevoord: Uncertainty principle and uncertainty relations . In: Foundations of Physics . tape 15 , no. 9 , 1985, pp. 925-944 , doi : 10.1007 / BF00739034 .
9. T. Schürmann, I. Hoffmann: A closer look at the uncertainty relation of position and momentum . In: Foundations of Physics . tape 39 , no. 8 , 2009, p. 958-963 , doi : 10.1007 / s10701-009-9310-0 , arxiv : 0811.2582 .
10. a b Masanao Ozawa: Physical content of Heisenberg's uncertainty relation: Limitation and reformulation . In: Phys. Lett. A . tape 318 , 2003, p. 21-29 , arxiv : quant-ph / 0210044 .
11. Quantum Uncertainty: Are You Certain, Mr. Heisenberg? In: Science Daily. January 18, 2012.
12. Geoff Brumfiel: Common Interpretation of Heisenberg's Uncertainty Principle Is Proved False. Scientific American, September 11, 2012.
13. Compare also Rainer Scharf: Quantum Physics. The great Heisenberg was wrong. In: FAZ.NET of November 17, 2012.
14. ^ A b H. P. Robertson: The Uncertainty Principle . In: Physical Review . tape 34 , no. 1 , 1929, p. 163-164 , doi : 10.1103 / PhysRev.34.163 .
15. Johann v. Neumann : Mathematical foundations of quantum mechanics. Unchanged reprint of the 1st edition from 1932. Chapter III "The quantum mechanical statistics". Section 4 “Relations of indeterminacy” (=  The basic teachings of the mathematical sciences in individual representations . Volume 38 ). 1st edition. Springer-Verlag , Berlin a. a. 1968, ISBN 3-540-04133-8 , pp. 123-124 .
16. Markus Bautsch: Scanning tunnel microscopic investigations on metals atomized with argon. Chapter 2.1: Vacuum tunneling - uncertainty principle in tunneling. Page 10, Verlag Köster, Berlin (1993), ISBN 3-929937-42-5 .