Planck constant

Planck's constant is the ratio of energy ( ) and frequency ( ) of a photon or a particle, according to the formula.

It is the foundation of quantum physics and linked since its discovery by Max Planck in 1899 and 1900 properties that have been attributed in classical physics either particles or waves only. Thus, it is the basis of the wave -particle duality of modern physics.

Planck discovered that the effect of a physical process, ie the product of unreacted energy and time, can only take on discrete values ​​, namely integer multiples of, so the quantum of action.

The quantum of action is in addition to the gravitational constant and the speed of light, one of the three fundamental constants of physics and the basis of primordial ( primordial = " first-order " ) natural system of units, the Planck units.

Definition

Planck's constant is for each physical system that can oscillate harmonically, the constant ratio of the smallest possible energy expenditure to the oscillation frequency. In quantum mechanics, this applies to any physical system in terms of its total energy content and the frequency of its quantum phase.

Planck's constant has the dimension of energy times time, the effect is called. It obtained its universal significance through its occurrence in the basic equations of quantum physics ( Schrödinger 's equation, Heisenberg equation of motion, the Dirac equation). Some general consequences:

• Each harmonic oscillator ( with frequency, angular frequency ) can absorb or release that are integer multiples of the vibrational quanta are energy only in discrete amounts.
• Every physical system can change its angular momentum ( *) only by integer multiples of.

• Each physical system is associated with a pulse wave of a wavelength matter.
• For each physical system energy and angular frequency of its quantum mechanical phase satisfy the equation.
• The two variables of a physical system which are canonically conjugate to each other (eg, position and momentum of a particle, or generalized location and generalized impulse, such as rotation angle and angular momentum ) that satisfy an uncertainty relation, according to which they in no state of system both at the same time can have well-defined values ​​. Rather, for the scattering of the values ​​of both variables.

Values

The value of

Where the numbers in parenthesis the estimated uncertainty ( 1 standard deviation) specify for the mean and refer to the last two decimal digits specified.

Because vibrations instead are often expressed as a frequency as angular frequency, will also be held the reduced Planck constant (pronounced "h bar", formerly also by Paul Dirac as a " Dirac's constant ") is used. Since a period of oscillation corresponds to a phase angle, the angular frequency of the frequency difference due to the factor. The reduced quantum of action is therefore equal to the quantum of action shared by:

The energy of a photon is ω the angular frequency by

In quantum mechanics, the quantum of angular momentum.

Often the product of the speed of light c is needed, the energy times length expresses a universal relationship between energy and length scale because of its dimension. In the units commonly used in the nuclear physics applies:

In Unicode, the symbols for the Planck constant and the reduced Planck's constant lying position U 210 E ( ℎ ) or U 210 F ( ℏ ).

Historical Perspective on the discovery and reception

Thermal radiation I ( Planck 1899)

Max Planck in 1899 found a new constant of nature, as he developed a thermodynamic description of the thermal radiation of black bodies, also called black body radiation. As an outstanding unsolved problem of theoretical physics at that time was the shape of the emitted spectrum and its change with increasing temperature, as seen in the transition from red heat to incandescence. According to Kirchhoff's law, the spectrum and its temperature dependence for all black body should be completely independent of its other attributes exactly the same.

In the short-wave range the measurements showed a characteristic decrease towards higher frequencies, which are well reproduced in the Wien's radiation law discovered by Wilhelm Wien by an exponential factor (frequency, temperature, a fixed parameter ). Planck was able to justify this formula theoretically. , He measured the thermal equilibrium between the electromagnetic waves inside a cavity and the assumed model as emitting and absorbing oscillators in its walls. It was based on the concept of entropy and chose for the entropy of an oscillator a suitable approach with two parameters and which, because of this general derivation is now played an universal meaning. proved to be the parameter in the Wien 's radiation law, as the product of the Boltzmann constant. Planck gave his value with only 4% above the current market value for. Planck also realized that these new constants, together with the gravitational constant and the speed of light form a system of universal physical constants, from which can be arranged universal units for length, mass, time and temperature, the Planck units.

Thermal radiation II ( Planck 1900)

New measurements in the longwave ( infrared ) part of the heat radiation disagreed Wien's radiation law, so that its found by Planck interpretation. They showed an increasing intensity at higher frequencies that could be good play by the Rayleigh-Jeans law, which also had the advantage that it could be derived without further assumptions from classical electrodynamics and the equipartition theorem of statistical mechanics. However, this law predicted an unlimited increase in the intensity with increasing frequency, which contradicts the reality and is called the ultraviolet catastrophe. Planck was (literally ) " a happy guessed interpolating formula ", which now measurements corresponded well with any (including only newly hired ). He could theoretically derive this referred to as Planck's radiation law result only by tentatively interpreted the exponential factor of the Wien 's law as known from the kinetic theory of gases, Boltzmann factor and the fact ansetzte for different depending on the frequency of discrete energy levels. The letter he took from auxiliary variable. The comparison with the Wien's formula showed that it is just about the products mentioned: is.

The idea that there could be a quantization of seemingly continuous variables, the physics was when the atomic hypothesis has been attacked violently, totally foreign. But all attempts to find a theoretical derivation without the assumption of discrete energy conversion failed. Planck initially did not consider the non-continuous nature of energy exchange for a property of the supposedly well-understood light waves, but wrote it exclusively for the emission and absorption processes in the material to the cavity walls. With long delay he was awarded in 1918 for the discovery of the quantization of the Nobel Prize.

H and the light quanta

Albert Einstein, one of the few physicists who recognized the fundamental importance of Planck's work early, analyzed the 1905 is incompatible with classical physics, the photoelectric effect and could explain it by using the light -quantum hypothesis, according to which even the light has quantum properties. Accordingly, there is, in contrast to Planck's view, the electromagnetic radiation itself of particulate objects, the photons whose energy is provided, depending on the frequency of the light wave by means of the equation. Later this equation was called the Einstein equation for the light quantum. With that he led as a new problem the wave -particle duality in physics. Not least because this analysis also needed years to prevail. In 1921 she took Einstein a Nobel Prize.

H and the specific heat of solids

The quantization of the vibrational energy was also the key to the explanation of another misunderstood phenomenon, the decrease towards the specific heat of solids at low temperatures for Albert Einstein in 1907. At higher temperatures, the measured values ​​usually agreed well with the predicted by the Dulong -Petit, according to classical physics value. Einstein assumed that the heat energy in the solid body in the form of vibrations of the atoms present to its rest position, and that this purely mechanical type of oscillations can be excited only in energy levels. Since in thermal equilibrium fluctuating between individual atoms amounts of energy are of the order of, the opportunity to choose between "high" temperatures () and " deep " was to distinguish temperatures (). Then the quantization at high temperatures has no visible effect, while the absorption of heat energy hindered at low temperatures. The formula was derived from Einstein this idea out, fit ( after appropriate down ) awarded to the then measured data. Nevertheless, it was a long time further doubts that the Planck constant in the field of mechanics may be important.

H and the phase space cell

Many laws of thermodynamics, such as the specific heat of gases and solids, but also to the irreversible increase of entropy and the shape of the thus achieved equilibrium state had, by statistical mechanics (especially by Ludwig Boltzmann and Josiah Willard Gibbs ) a mechanical interpretation experience. It is based on the assumption of the disordered motion of many atoms or molecules and extremely determined using statistical methods, the most probable values ​​of macroscopically measurable quantities ( such as density, pressure, etc.) in order to characterize the equilibrium state. This requires first the total amount of all possible states of all particles are mathematically captured in a state or phase space. If we specify a particular macroscopic state, then make all particle states in which the system displays this macroscopic state in the phase space, a partial volume. From the size of each such volume is determined, will occur with the probability that the relevant macroscopic state. Mathematically, thus to form a volume integral, and you need to temporarily and as an auxiliary variable, the definition of a volume element, also called phase space cell. In the final result but the phase space cell should no longer appear. If possible, can you its size in the resulting formula to zero shrink ( such as differential sizes generally in calculus ), if not, they are seen as unwanted parameters ( eg the measures an unknown additive constant ) and tries only to consider such conclusions, which are independent of the phase space cell (eg, differences in which the constant lifting off ). Is calculated in this manner the entropy of a gas is the chemical constant constant. Otto Sackur remarked in 1913, to his surprise, that you have to give the phase space cell a certain size, so that the chemical constant coincides with the measured values ​​. The phase space cell ( per particle and per space dimension of its movement ) just need to have the size. His publication he gave the title of " The universal significance of the so-called Planck's constant ", and Max Planck called it of " fundamental importance " when the daring hypothesis would be true, that this is a result independent of the nature of the gas. This was the case.

Fundamental to this result is in particular that here begins a deeper reason for the phenomenon of quantization to show, however, was only years later fully with the quantum statistics of radiation clear. The same size of the phase space cell per state, one can deduce from the fact Einstein 's formula for the light quantum. Which is decisive for the size of the phase space cell physical quantity here is the effect in which a vibration (such as the electromagnetic wave), the effect is the product of energy and period.

H, and the size of the atoms

Classical physics has to fail in explaining the stable size of the atoms. Because if they could explain a certain size, would be a, for example, half as large atom then by the same laws just as well possible. In other words, the basic formulas of classical physics do not contain enough natural constants, that one could win a formula for a certain length of them. The quantum of action can close this gap, as already remarked Planck himself in 1899 when he introduced the Planckian units for the first time (see above). However, because the quantum of action according to the majority opinion should not be introduced into the mechanics, the first attempt to use it to explain the atomic radius until 1910 came into being and was then made ​​even partly ridiculous. Here, Arthur Erich Haas participated in an electron circles in a positive -charge field, and set the rotation frequency and the binding energy of this system into consideration. This gives a radius results in the area of ​​known from the chemistry and the kinetic theory of gases atomic radii.

More success came in 1913 Niels Bohr, who went out in his atomic model of the same image, but also circular orbits of different energy and, in particular, the emission of photons in the quantum leap from one introduced to the other track. The agreement with the measured wavelengths, which he, however, received only by a hardly to be justified quantum condition ( with the new principal quantum number ), the model made ​​famous fast. The supporting role of the quantum of action at the inner structure of atoms was proved. Quantum condition is quickly detected as angular momentum quantization, because the circular track for the principal quantum number can be defined by the condition that the angular momentum of the electron has the value.

This great progress made ​​Bohr's atomic model, the relevant starting point for further developments, although similar great progress then stayed away for years. In particular the experiments have failed to understand atoms with several electrons.

H and the matter waves

The success of the Bohr model of the atom since 1913 owed ​​itself to the good part of the quantum condition which engages external hard into the mechanics by allowing the electron few of the mechanically possible orbits. Due to the ongoing difficulties with the further development of the atomic theory has been looking for ways to reshape the mechanism itself so that it takes into account the quantum condition from the outset. It should be the current quantum theory be replaced by a veritable quantum mechanics. The biggest step before the real beginning of quantum mechanics made ​​Louis de Broglie in 1924, by attributing even material particles, such as electrons, wave properties. He transferred the relation found for photons between momentum and wavelength of the imaginary of him matter-wave of the electron. He extended the wave -particle duality on particles. As an immediate success shows that the Bohr orbit to the principal quantum number n even has the scope, therefore, matter-wave of the electron can form a standing wave it. Without saying about this matter wave can be much, Erwin Schrödinger took place in early 1926 a formula for the propagation of this wave in a force field, with whom he founded the wave mechanics. For the stationary states of the hydrogen atom, he was able to calculate precisely the known results with this Schrödinger equation without additional quantum condition. In addition, known errors of Bohr's model have been fixed, such as that the atom was flat or that the angular momentum can not be. The only constant of nature enters into the Schrödinger equation on the quantum of action. The same applies to the equation that Werner Heisenberg few months earlier won from a " quantum- theoretical reinterpretation of kinematic and mechanical relations " by which he founded the matrix mechanics. Both approaches are mathematically equivalent and are considered basic equations of quantum mechanics itself. What is still left is the difficulty to get a compatible with the wave -particle duality picture of the quantum mechanical concepts and procedures.

Angular momentum

Planck's motivation for the term " quantum of action " was at first alone, the physical dimension of energy times time constant. The classical mechanical orbital angular momentum has the same dimension. In the 1913 set out by Niels Bohr atomic model of the hydrogen atom of the orbital angular momentum of the electron appeared as quantized size in appearance. He can take the amount by only integer multiples of: the angular momentum quantum number. In Bohr's model, they can be used for pathways to principal quantum number have all values.

In quantum mechanics, the operator belonging to the orbital angular momentum satisfies the three commutation relations

Also, applies ( as in classical mechanics) for the scalar product (zero operator). This results in general:

• The quantum number of the orbital angular momentum is a non-negative integer. The amount of the angular momentum is a little greater than:

• The component of angular momentum along any axis can take any integer multiple of Planck's constant, these magnetic quantum number is the amount stipulated by limited. There are therefore different values ​​, one always odd number.

According to quantum mechanical calculation can take the integer values ​​in the hydrogen atom to the principal quantum number, the orbital angular momentum quantum number, which is consistent with all observations.

Also for the spin ( intrinsic angular momentum of a particle around its own center of gravity, often referred to ) is relevant. However, he is quantized in units. The corresponding operator satisfies the same commutation relations as, but is not the zero operator. It follows that the spin quantum numbers can also be half-integer. So the values ​​are for the magnetic quantum number possible again at half-integer spin an even number. Depending on whether the spin is integer or half-integer, we distinguish the two basic types of particles of bosons and fermions in physics.

Uncertainty principle

In the Heisenberg commutation relation, the (reduced ) Planck constant occurs as the value of the commutator between the position and momentum operator:

As a result applies to the product of position and momentum blur the Heisenberg uncertainty principle

Von Klitzing constant

The Von Klitzing constant is the size that occurs when the quantum Hall effect. It has the well-known unit of electrical resistors ohms, its value is. This constant can be extremely accurately measured. You could serve analogous to the modern definition of the speed of light towards restoring the determination of Planck's constant to very accurate resistance measurements.

H and the definition of the kilogram

The Bureau International des Poids et Mesures, which for the definition of the International System of Units ( SI units ) is responsible, has in 2010 adopted a proposal for a revamped system of units, in which the definition of the kilogram is attributed to the Planck Wirkumsquantum. Since the unit of the Wirkumsquantums is thus the kg, together with the definition of m and second due to a physical constant of nature.