高级算法 (Fall 2016)/Nonconstructive Proof of Lovász Local Lemma

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A commemoration plaque for Max Planck on his discovery of Planck's constant, in front of Humboldt University, Berlin. English translation: "Max Planck, discoverer of the elementary quantum of action h, taught in this building from 1889 to 1928."

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Max Planck, after whom the Planck constant is named

The Planck constant (Planck's constant) links the amount of energy a photon carries with the frequency of its electromagnetic wave. It is named after the physicist Max Planck. It is an important quantity in quantum physics.

The Planck constant has dimensions of physical action: energy multiplied by time, or momentum multiplied by distance. In SI units, the Planck constant is expressed in joule seconds (J⋅s) or (N⋅m⋅s) or (kg⋅m²⋅s⁻¹). The symbols are defined here.

In SI Units the Planck constant is about Template:Val J·s. Scientists have used this quantity to calculate measurements like the Planck length and the Planck time.

Background

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Symbols used in this article.

Symbol	Meaning
E	Energy
h	Planck constant
k	Boltzmann constant
c	speed of light
λ	radiation wavelength
ν	radiation frequency
T	absolute temperature

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Illustration taken from Newton's original letter to the Royal Society (1 January 1671 [Julian calendar]). S represents sunlight. The light between the planes BC and DE are in colour. These colours are recombined to form sunlight on the plane GH

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Young's double slit experiment

Between 1670 and 1900 scientists discussed the nature of light. Some scientists believed that light consisted of many millions of tiny particles. Other scientists believed that light was a wave.

Light: waves or particles?

In 1678 Christiaan Huygens wrote the book Traité de la lumiere ("Treatise on light"). He believed that light was made up of waves. He said that light could not be made up of particles because light from two beams do not bounce off each other. In 1672 Isaac Newton wrote the book Opticks. He believed that light was made up of red, yellow and blue particles which he called corpusles. Newton explained this by his "two prism experiment". The first prism broke light up into different colours. The second prism merged these colours back into white light.

During the 18th century Newton's theory was given the most attention. In 1803 Thomas Young described the "double-slit experiment". In this experiment light from two narrow slits interfere with each other. This caused a pattern which showed that light was made up of waves. For the rest of the nineteenth century wave theory of light was given the most attention. In the 1860s James Clerk Maxwell developed equations that described electromagnetic radiation as waves.

The theory of electromagnetic radiation treats light, radio waves, microwaves and many other types of wave as the same thing except that they have different wavelengths. The wavelength of light is between 400 and 600 nm, the wavelength of radio waves varies from 10 m to 1500 m and the wavelength of microwaves is about 2 cm. In a vacuum, all electromagnetic waves travel at the speed of light. The frequency of the electromagnetic wave is given by:

[math]\displaystyle{ \nu = \frac{c}{\lambda} }[/math].

The symbols are defined here.

Black body radiators

All hot bodies give off radiant heat. Radiant heat is electromagnetic radiation. Normally this radiation is in the infra-red range, but if the body is very hot (1000 °C or more), it is in the visible range. In the late 1800's many scientists studied wavelength of electromagnetic radiation from black-body radiators at different temperatures.

Rayleigh-Jeans Law

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Rayleigh-Jeans curve and Planck's curve plotted against photon wavelength.

Lord Rayleigh first published the basics of the Rayleigh-Jeans law in 1900. The theory was based on the Kinetic theory of gasses. Sir James Jeans published a more complete theory in 1905. The law relates the quantity and wavelength of electromagnetic energy given off by a black body radiator at different temperatures. The equation describing this is:

[math]\displaystyle{ B_\lambda(T) = \frac{2 c k T}{\lambda^4} }[/math].

For long-wavelength radiation, the results predicted by this equation corresponded well with practical results obtained in a laboratory. However, for short wavelengths (ultraviolet light) the difference between theory and practice were so large that it earned the nickname "the ultra-violet catastrophe".

Planck's Law

in 1895 Wien published the results of his studies into the radiation from a black body. His formula was:

[math]\displaystyle{ B_\lambda(T) = \frac{2 h c^2} {\lambda^5} e^{-\frac{hc}{\lambda kT}} }[/math].

This formula worked well for short wavelength electromagnetic radiation, but did not work well with long wavelengths.

In 1900 Max Planck published the results of his studies. He tried to develop an expression for black-body radiation expressed in terms of wavelength by assuming that radiation consisted of small quanta and then to see what happened if the quanta were made infinitely small. (This is a standard mathematical approach). The expression was:

[math]\displaystyle{ B_\lambda(T) = \frac{2 h c^2}{\lambda^5}~\frac{1}{e^\frac{hc}{\lambda kT}-1} }[/math].

If the wavelength of light is allowed to become very large, then it can be shown that the Raleigh-Jeans and the Planck relationships are almost identical.

He calculated h and k and found that

h = Template:Val erg·sec.

k = Template:Val erg·deg^-1.

The values are close to the modern day accepted values of Template:Val and Template:Val respectively. The Planck law agrees well with the experimental data, but its full significance was only appreciated several years later.

Quantum theory of light

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Solway Conference 1911. Planck, Einstein and Jeans are standing. Planck is second from the left. Einstein is second from the right. Jeans is fifth from the right. Wien is seated, third from the right

It turns out that electrons are dislodged by the photoelectric effect if light reaches a threshold frequency. Below this no electrons can be emitted from the metal. In 1905 Albert Einstein published a paper explaining the effect. Einstein proposed that a beam of light is not a wave propagating through space, but rather a collection of discrete wave packets (photons), each with energy. Einstein said that the effect was due to a photon striking an electron. This demonstrated the particle nature of light.

Einstein also found that electromagnetic radiation with a long wavelength had no effect. Einstein said that this was because the "particles" did not have enough energy to disturb the electrons.

Einstein realized that the energy of each photon was related to the photon frequency by the Planck constant. This could be written mathematically as:

[math]\displaystyle{ E = h\nu = \frac {hc} {\lambda} }[/math].

In 1921 Einstein received the Nobel Prize for linking the Planck constant to the photoelectric effect.

Application

The Planck constant is of importance in many applications. A few are listed below.

Bohr model of the atom

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Bohr's model of the atom. An electron falling from the n=3 shell to the n=2 shell loses energy. This energy is carried away as a single photon.

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Visible spectrum of Neon. Each line represents a different pair of energy levels.

In 1913 Niels Bohr published the Bohr model of the structure of an atom. Bohr said that the angular momentum of the electrons going around the nucleus can only have certain values. These values are given by the equation

[math]\displaystyle{ L=n\frac{h}{2\pi} }[/math]

where

L = angular momentum associated with a level.

n = positive integer.

h = Planck constant.

The Bohr model of the atom can be used to calculate the energy of electrons at each level. Electrons will normally fill up the lowest numbered states of an atom. If the atom receives energy from, for example, an electric current, electrons will be excited into a higher state. The electron will then drop back to a lower state and will loose their extra energy by giving off a photon. Because the energy levels have specific values, the photons will have specific energy levels. Light emitted in this way can be split into different colours using a prism. Each element has its own pattern. The pattern for neon is shown alongside.

Heisenberg's uncertainty principle

In 1927 Werner Heisenberg published the uncertainty principle. The principle states that it is not possible to make a measurement without disturbing the thing being measured. It also puts a limit on the minimum disturbance caused by making a measurement.

In the macroscopic world these disturbances make very little difference. For example, if the temperature of a flask of liquid is measured, the thermometer will absorb a small amount of energy as it heats up. This will cause a small error in the final reading, but this error is small and not important.

In quantum mechanics things are different. Some measurements are made by looking at the pattern of scattered photons. One such example is Compton scattering. If both the position and momentum of a particle is being measured, the uncertainty principle states that there is a trade-off between the accuracy with which the momentum is measured and the accuracy with which the position is measured. The equation that describes this trade-off is:

[math]\displaystyle{ \Delta x \, \Delta p\gtrsim h\qquad\qquad\qquad }[/math]

where

Δp = uncertainty in momentum.

Δx = uncertainty in position.

h = Planck constant.

Colour of light emitting diodes

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Simple LED circuit that illustrates use of the Planck constant. The colour of the light emitted depends on the voltage drop across the diode. The wavelength of the light can be calculated using the Planck constant.

In the electric circuit shown on the right, the voltage drop across the light emitting diode (LED) depends on the material of the LED. For silicon diodes the drop is 0.6 V. However for LEDs it is between 1.8 V and 2.7 V. This information enables a user to calculate the Planck constant.

The energy needed for one electron to jump the potential barrier in the LED material is given by

[math]\displaystyle{ E = Q_eV_L \, }[/math]

where

Q_e is the charge on one electron.

V_L is the voltage drop across the LED.

When the electron decays back again, it emits one photon of light. The energy of the photon is given by the same equation used in the photoelectric effect. If these equations are combined, the wavelength of light and the voltage are related by

[math]\displaystyle{ \lambda = \frac {hc} {V_L Q_e} \, }[/math]

The table below can be calculated from this relationship.

Colour	Wavelength (nm)	Voltage
red light	650	1.89
green light	550	2.25
blue light	470	2.62

Value of the Planck constant and the kilogram redefinition

Since its discovery, measurements of h have become much better. Planck first quoted the value of h to be Template:Val erg·sec. This value is within 5% of the current value.

As of 3 March 2014, the best measurements of h in SI units is Template:Val J·s. The equivalent figure in cgs units is Template:Val erg·sec. The is relative uncertainty of h is Template:Val.

The reduced Planck constant (ħ) is a value that is sometimes used in quantum mechanics. It is defined by

[math]\displaystyle{ \hbar = \frac{h}{2 \pi} }[/math].

Planck units are sometimes used in quantum mechanics instead of SI. In this system the reduced Planck constant has a value of 1, so the value of the Planck constant is 2π.

Plancks constant can now be measured with very high precision. This has caused the BIPM to consider a new definition for the kilogram. They have proposed that the international prototype kilogram is no longer used to define the kilogram. Instead the BIPM will define the Planck constant to have an exact value. Scientists will use this value and the definitions of the metre and the second to define the kilogram.

Related pages

• Wave–particle duality