Единици на Планк
от Уикипедия, свободната енциклопедия
Единици на Планк във физиката е система единици основана на 5 фундаментални физически константи, показани в таблицата по-долу. Планк приема, че тези константи са равни на 1 и всички останали величини мери чрез тях.
Използването на единиците на Планк елегантно опростява много от математическите формули в съвременната физика. Тези измерителни единици произлизат по естествен път от наблюденията на природните явления и за разлика от други мерни единици не са резултат от въвеждане на нарочен измерителен стандарт.
Единиците Планк не се основават на зададен еталон, обект или частица, а на свойствата на празното пространство - вакуума.
константа | символ | измерение |
---|---|---|
скорост на светлината във вакуум | ![]() |
L T-1 |
гравитационна константа | ![]() |
M-1L3T-2 |
константа на Дирак | ![]() ![]() |
ML2T-1 |
константа на Кулон | ![]() ![]() |
Q-2 M L3 T-2 |
константа на Болцман | ![]() |
ML2T-2Θ-1 |
Някои специалисти наричат полу на шега единиците на Планк „Мерните единици на Господ“. С въвеждането на тези измерителни единици се избягват множеството произволно избрани измерителни единици. Някои физици предполагат, че това са мерните единици, които евентуално биха използвали в една напреднала цивилизация на извънземни, ако такива съществуват.
[редактиране] Основни единици на Планк
След като в системата единици на Планк горните константи са са равни на 1, то единиците за дължина, маса, време, електрически заряд и температура са производни. Удобни са кохерентните измерителни системи, като SI например, при които производните единици се получават от основните без някакви мащабиращи коефициенти. По този начин можем да изведем едно множество от производни единици на Планк:
Име | количество | изразяване | приближение в SI единици | други означения |
---|---|---|---|---|
дължина на Планк | дължина (L) | ![]() |
1,61624 × 10-35 m | |
маса на планк | маса (M) | ![]() |
2,17645 × 10-8 kg | 1,311 × 1019 u |
време на Планк | време (T) | ![]() |
5,39121 × 10-44 s | |
заряд на Планк | електрически заряд (Q) | ![]() |
1,8755459 × 10-18 C | 11,70624 e |
температура на Планк | температура (Θ) | ![]() |
1,41679 × 1032 K |
[редактиране] Производни единици на Планк
Във всички измерителни системи следните величини са производни на основните физически величини:
име | количество | изразяване | приближение в SI единици |
---|---|---|---|
импулс на Планк | импулс (MLT-1) | ![]() |
6,52485 kg m/s |
енергия на Планк | енергия (ML2T-2) | ![]() |
1,9561 × 109 J |
сила на Планк | сила (MLT-2) | ![]() |
1,21027 × 1044 N |
мощност на Планк | мощност (ML2T-3) | ![]() |
3,62831 × 1052 W |
плътност на Планк | плътност (ML-3) | ![]() |
5,15500 × 1096 kg/m3 |
ъглова честота на Планк | честота (T-1) | ![]() |
1,85487 × 1043 s-1 |
налягане на Планк | налягане (ML-1T-2) | ![]() |
4,63309 × 10113 Pa |
ток на Планк | електрически ток (QT-1) | ![]() |
3,4789 × 1025 A |
Напрежение на Планк | напрежение (ML2T-2Q-1) | ![]() |
1,04295 × 1027 V |
импеданс на Планк | съпротивление (ML2T-1Q-2) | ![]() |
29,9792458 Ω |
[редактиране] Фундаментални уравнения на физиката, добиващи безразмерен (недименсионен) вид при изразяване в мерни единици на Планк
При преминаване в измерителни единици на Планк много от известните ни уравнения във физиката се опростяват както ще видим в примерите по долу:
Стандартна дименсионна формула | Бездименсионна формула в Планк мерни единици | |
---|---|---|
Закон на Нютон за всеобщата гравитация | ![]() |
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Уравнение на Шрьодингер | ![]() |
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Енергия на частица с вълнова функция в радиани![]() |
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Уравнение на Айнщайн за маса /енергия в Специална теория на относителността | ![]() |
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Уравнения на Айнщайн за ОТО | ![]() |
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Топлинна енергия на частица | ![]() |
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Закон на Кулон | ![]() |
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Уравнения на Максуел | ![]()
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[редактиране] Дискусии
With the exception of the Planck momentum and Planck impedance and the possible exception of the Planck mass, base and derived Planck units are impractical for empirical science, engineering, and everyday use, unless rescaled by many orders of magnitude. In fact, 1 Planck unit often represents the largest or smallest value that makes sense given the current understanding of physical theory. For instance, a Planck velocity of 1 equals the speed of light in a vacuum. At lengths and times of less than approximately one Planck unit, quantum theory as presently understood no longer applies. At a Planck temperature of 1, the four fundamental forces unify and all symmetries broken since the start of the Big Bang are restored. In fact, we have no understanding of the Big Bang until the age and size of the universe exceeds one Planck unit, and its temperature falls below one Planck unit. Physical theory applicable on the scale of approximately one Planck unit of distance, time, density, or temperature, requires taking account of both quantum effects and general relativity. Doing so would require a theory of quantum gravity which does not yet exist.
It should also be noted that, at present, the numerical value of the gravitational constant G cannot be determined experimentally to better than about 1 part in 7000. The uncertainty in G is far greater than that of any of the four other fundamental constants. In contrast, the speed of light in SI units is no longer subject to measurement error, because the meter is now defined in such a way that the speed of light is an exact quantity.
Planck neither defined nor proposed the Planck charge. Rather, its definition is a natural extension of the definitions of the other Planck units [1]. Note that the elementary charge e, measured in terms of the Planck charge, is
where α is the fine-structure constant
.
The dimensionless fine-structure constant can be thought of as taking on the numerical value that it does because of the amount of charge, measured in Planck units, that nature has happened to have assigned to electrons, protons, and other charged particles. Because the electromagnetic force between two charged particles is proportional to the product of the charges of each particle (which would, in Planck units, be proportional to ), the strength of the electromagnetic force relative to other fundamental forces is proportional to α.
Planck units normalize the Coulomb force constant (4πε0)-1 to 1, as does the cgs system of units. Consequently, the Planck impedance, ZP, equals Z0/4π, where Z0 is the characteristic impedance of free space. If Planck units normalized the permittivity of free space ε0 instead, the 4π factors in Maxwell's equations would vanish and the Planck impedance, ZP, would be identical to Z0.
Planck units normalize the gravitational constant G in Newton's law of universal gravitation to 1. In general relativity and cosmology, G is nearly always preceded by 4π or an integer multiple thereof. This fact suggests other normalizations for G, such as:
- (4π)-1. This eliminates the factor of 4π from the gravitoelectromagnetic (GEM) counterparts to Maxwell's equations. (This normalization would also replace the ubiquitous 8π of general relativity with 2.) The GEM equations hold in weak gravitational fields or reasonably flat space-time. They have the same form as Maxwell's equations (and the Lorentz force equation) of electromagnetic interaction, with mass (or mass density) replacing charge (or charge density) and (4πG)-1 replacing the permittivity ε0. Normalizing G to (4π)-1 also sets to 1 the characteristic impedance of gravitational radiation in free space, Z0 = 4πG/c. Note that general relativity implies that gravitational radiation propagates at the same velocity c as does electromagnetic radiation.
- (8π)-1. This removes the ubiquitous factor 8π from the equations of general relativity and cosmology, e.g., the Einstein equation, Einstein-Hilbert action, Friedmann equations, and the Poisson equation for gravitation. Planck units modified so that 8πG is set to 1 are known as reduced Planck units because the Planck mass is divided by
.
- (16π)-1. This sets the coefficient of R in the Einstein-Hilbert action to 1.
[редактиране] Измерителна единица на Планк и инвариантност на величините е природата
Some theoreticians and experimentalists have conjectured that some physical "constants" might actually change over time, a proposition that introduces many difficult questions. A few such questions that are relevant here might be: How would such a change make a noticeable operational difference in physical measurement or, more basically, our perception of reality? If some physical constant had changed, would we even notice it? How would physical reality be different? Which changed constants would result in a meaningful and measureable difference?
- "[An] important lesson we learn from the way that pure numbers like α define the world is what it really means for worlds to be different. The pure number we call the fine structure constant and denote by α is a combination of the electron charge, e, the speed of light, c, and Planck's constant, h. At first we might be tempted to think that a world in which the speed of light was slower would be a different world. But this would be a mistake. If c, h, and e were all changed so that the values they have in metric (or any other) units were different when we looked them up in our tables of physical constants, but the value of α remained the same, this new world would be observationally indistinguishable from our world. The only thing that counts in the definition of worlds are the values of the dimensionless constants of Nature. If all masses were doubled in value [including the Planck mass mP] you cannot tell because all the pure numbers defined by the ratios of any pair of masses are unchanged." (Barrow 2002)
Referring to Michael Duff Comment on time-variation of fundamental constants and Duff, Okun, and Veneziano Trialogue on the number of fundamental constants (The operationally indistinguishable world of Mr. Tompkins), if all physical quantities (masses and other properties of particles) were expressed in terms of Planck units, those quantities would be dimensionless numbers (mass divided by the Planck mass, length divided by the Planck length, etc.) and the only quantities that we ultimately measure in physical experiments or in our perception of reality are dimensionless numbers. When one commonly measures a length with a ruler or tape-measure, that person is actually counting tick marks on a given standard or is measuring the length relative to that given standard, which is a dimensionless value. It is no different for physical experiments, as all physical quantities are measured relative to some other like dimensioned values.
We can notice a difference if some dimensionless physical quantity such as α or the proton/electron mass ratio changes (atomic structures would change) but if all dimensionless physical quantities remained constant (this includes all possible ratios of identically dimensioned physical quantity), we could not tell if a dimensionful quantity, such as the speed of light, c, has changed. And, indeed, the Tompkins concept becomes meaningless in our existence if a dimensionful quantity such as c has changed, even drastically.
If the speed of light c, were somehow suddenly cut in half and changed to c/2, (but with all dimensionless physical quantities continuing to remain constant), then the Planck Length would increase by a factor of from the point-of-view of some unaffected "god-like" observer on the outside. But then the size of atoms (approximately the Bohr radius) are related to the Planck length by an unchanging dimensionless constant:
Then atoms would be bigger (in one dimension) by , each of us would be taller by
, and so would our meter sticks be taller (and wider and thicker) by a factor of
and we would not know the difference. Our perception of distance and lengths relative to the Planck length is, by axiom, an unchanging dimensionless constant.
Our clocks would tick slower by a factor of (from the point-of-view of this unaffected "god-like" observer) because the Planck time has increased by
but we would not know the difference (our perception of durations of time relative to the Planck time is, by axiom, an unchanging dimensionless constant). This hypothetical god-like observer on the outside might observe that light now travels at half the speed that it used to (as well as all other observed velocities) but it would still travel 299792458 of our new meters in the time elapsed by one of our new seconds. We would not notice any difference.
This in one sense contradicts George Gamow in Mr. Tompkins who suggests that if a dimensionful universal constant such as c changed, we would easily notice the difference; however, as noted, the disagreement is better thought of as the ambiguity in the phrase "changing a physical constant", when one does not specify whether one does so keeping all other dimensionless constants the same, or does so keeping all other dimensionful constants the same. The latter is a somewhat confusing possibility since most of our unit definitions are related to the outcomes of physical experiments which themselves depend on the constants, the only exception being the kilogram. Gamow does not address this subtlety; the thought experiments he conducts in his popular works assume the latter.
[редактиране] Откриване на измерителната единица Планк
Макс Планк е първият физик, който достига по експериментален път до универсалната константа. Той публикува стойността на намерената константа в списанието на Пруската Академия на Науките през май 1899 г. До този момент такава измерителна единица не съществува. Квантовата механика все още не е изобретена. За откритието на константата на Планк – h – той по-късно получава Нобелова награда.
[редактиране] Вижте също
- atomic units
- geometrized units
- dimensional analysis
- physical constants
- George Stoney
- John D. Barrow, 2002. The Constants of Nature; From Alpha to Omega - The Numbers that Encode the Deepest Secrets of the Universe. Pantheon Books. ISBN 0375422218.
- The NIST website(National Institute of Standards and Technology) is a convenient source of data on the commonly recognized constants, including Planck units.
- Planck's original paper
- A note on h and h-bar - Blaze Labs Research Argues that h-bar should not be used to derive Planck's units