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Reword
User:Patrick0Moran/I2QM_list Related articles:

Emission spectrum

Planck's constant

Black body

Quantum indeterminacy

Rayleigh-Jeans law

Wien_approximation

Wien's displacement law

Correspondence principle

Planck was able to calculate the value of h from experimental data on black-body radiation: his result, 6.55 × 10−34 J·s, is within 1.2% of the currently accepted value.[5] He was also able to make the first determination of the Boltzmann constant kB from the same data and theory.[8]

Prior to Planck's work, it had been assumed that the energy of a body could take on any value whatsoever – that it was a continuous variable. Following pre-quantum ideas, people would have assumed that the more intense the light shone upon a photovoltaic cell, the higher both the voltage and the amperage produced by that device. People were surprised to learn that light of higher frequency produced a higher voltage across the terminals of the photovoltaic device. Classical physics could not predict the way that energy intensities were linked to frequency. Developing

Prior to Planck's work, it had been assumed that the energy of a body could take on any value whatsoever – that it was a continuous variable. Quantum mechanics was discovered in the process of trying to understand discontinuities in emission spectra that were not explained by classical physics. Heisenberg developed matrix mechanics while trying to explain the intensities of the discontinuous visual spectrum of hydrogen radiation. Earlier quantum theory could predict the locations of the isolated frequencies of the visual spectrum of hydrogen radiation, and Heisenberg succeeded in explaining their intensities. Their equations only indirectly mention the gaps between the frequencies of light present. However, classical theory expected a continuous spectrum, so in order to demonstrate how how quantum mechanics follows the Correspondence principle it is necessary to show how, for a large range of values, the gaps between frequencies produced are so small that they cannot be observed. To do so effectively requires calculation of pairs of adjacent frequencies and then calculating the difference between the members of each pair. Finding pairs for which the difference approaches zero is not possible by setting the Planck constant to zero. In fact, setting h equal to zero results in trying to divide zero by zero.

Prior to Planck's work, it had been assumed that the energy of a body could take on any value whatsoever – that it was a continuous variable. The Rayleigh-Jeans law makes close predictions for a narrow range of values at one limit of temperatures, but the results diverge more and more strongly as temperatures increase. To make Planck's law, which correctly predicts blackbody emissions, it was necessary to multiply the classical expression by a complex factor that involves h in both the numerator and the denominator. The influence of h in this complex factor would not disappear if it were set to zero or to any other value. Making an equation out of Planck's law that would reproduce the Rayleigh-Jeans law could not be done by changing the values of h, of the Boltzmann constant, or of any other constant or variable in the equation. In this case the picture given by classical physics is not duplicated by a range of results in the quantum picture.

A quantum of negative celerity
Planck's postulate:
 * $$E=nh\nu\,$$

Following his model, there were n harmonic oscillators in each black body for any particular frequency. So if the relationship between energy and frequency is:
 * $$E=h\nu\,$$

then to get the total energy of the system at this frequency you would have to multiply the "quantum" of energy for each oscillator by the number of oscillators.

If I remember correctly, even the idea of atoms was not firmly established at this time, so Planck was looking at a black box situation, knowing only what went into the box and what came out.

This postulate is formulated in such a way that we might thing that three integers were involved: (1) surely n is an integer. (It is so defined.) (2) If you ask anybody for the frequency of some wave phenomenon, they will never say something such as 33.33...3 hertz. But fractional hertz are an artifact of our arbitrary unit of time, the second. (3)Based on one true integer and one number falsely believed to be restricted to integer values of hertz, one might assume that the energy output would also be in integer units.

So what is 1/3 cycles per second? It is 1 cycle every 3 seconds. What is 4.5 cycles per second? It is 9 cycles every two seconds or 1 cycle every 2/9 second. So just make one unit of time, Uec, that is equal to 3 seconds, and the first vibration is one cycle per Uec. Make another unit of time, Wec, that is equal to 2/9 second, and the second vibration is one cycle per Wec.

equations
Planck's Law:


 * $$B_\nu(T) =\frac{ 2 h\nu^{3}}{c^2}\frac{1}{ e^{\frac{h\nu}{k_\mathrm{B}T}}-1}.$$

Alternate format for Planck's Law from Wien approximation:


 * $$I(\nu, T) = \frac{2 h \nu^3}{c^2} \frac{1}{e^{\frac{h \nu}{kT}}-1}$$ &nbsp Alternate from Rayleigh-Jeans law
 * $$B_\lambda(T) = \frac{2 c^2}{\lambda^5}~\frac{h}{e^\frac{hc}{\lambda kT}-1},$$

Rayleigh-Jeans law:


 * $$B_\nu(T) = \frac{2 \nu^2 k T}{c^2}.$$
 * $$B_\lambda(T) = \frac{2 c k T}{\lambda^4},$$

Planck explained further that the respective definite unit, ϵ, of energy should be proportional to the respective characteristic oscillation frequency $$\nu$$ of the hypothetical oscillator, and in 1901 he expressed this with the constant of proportionality h:
 * $$\epsilon=h\nu$$.

This is known as Planck's relation.

Another approach to how h would affect physics if it had other values is suggested by the fact that it is involved in Rydberg's constant:

We have therefore found the Rydberg constant for our hypothetical system of a nucleus with infinite mass, a +1 charge, and a single electron to be


 * $$ R_\infty = \frac{ m_{\mathrm{e}} e^4}{8 \varepsilon_0^2 h^3 c}. $$

As h increases, R decreases, As h decreases, R increases. As R decreases, the wavelengths increase. So the wavelengths increase as h increases. As R increases, the wavelengths decrease. So the wavelengths decrease as h decreases. As wavelengths decrease, the gaps between emission frequencies decrease. If R went to zero, then wavelengths become very long and gaps between frequencies become very small. If h went to zero, R would go infinite. Then the wavelengths would increase and the gaps between frequencies become very large Actually, R must stay below 1.097... E+300 or something will go wrong, maybe a division by O or something of that sort.

argument from Rayleigh-Jeans article
The Planck constant is not equal to zero in classical physics. The article is wrong at this point.

Classical physics had no idea of the quantum nature of phenomena, so it would have been strange indeed if it had included a term h in its equations, and even odder if it demanded that h = 0. In fact, the classical Rayleigh-Jeans equation produced predictions of radiance that diverged from empirical data more strongly the higher the frequencies of light emitted by a black body that were considered. The Wien approximation underestimated radiance at lower frequencies, and Planck's Law gave the correct predictions. Both of the latter two used equations that included h, and they did so to take account of an aspect of reality unheeded by classical physics. Since the latter two were derived from a classical basis, one would expect that if the modifications made to deal with the failings of classical physics were attenuated, then the predicted results would draw closer to the classical physics.

Such is indeed the case. The Rayleigh-Jeans equation predicts the lower end of the spectrum of a black body radiator fairly well, but the prediction of radiance gets progressively worse at higher frequencies. The Wien approximation is off at lower frequencies, but accurate at higher frequencies. Only Planck's law gives an accurate prediction. So, if we were to reverse history and eliminate the change Planck needed to make Planck's Law, then we would have the Wien approximation. It still retains h as an important factor, so it cannot be regarded as belonging to classical physics. So we can take the Wien Approximation and see what it is for cases in which "very high temperatures or long wavelengths" are involved, and we get the Rayleigh-Jeans equation.

Here is how I think the math is to be done, based on other Wikipedia articles previously cited by me:

We start with Planck's Law from the Rayleigh-Jeans law article:
 * $$B_\lambda(T) = \frac{2 c^2}{\lambda^5}~\frac{h}{e^\frac{hc}{\lambda kT}-1},$$

"In the limit of very high temperatures or long wavelengths, the term in the exponential becomes small, and so the exponential is well approximated by its first-order Taylor polynomial":


 * $$e^{\frac{hc}{\lambda kT}} \approx 1 + \frac{hc}{\lambda kT}.$$

Then
 * $$\frac{1}{e^\frac{hc}{\lambda kT}-1} \approx \frac{1}{\frac{hc}{\lambda kT}} = \frac{\lambda kT}{hc}.$$

By substitution:
 * $$B_\lambda(T) = \frac{2 c^2}{\lambda^5}~\frac{h\lambda kT}{hc},$$

And cancelling we get:
 * $$B_\lambda(T) = \frac{2 c}{\lambda^4}~\frac{kT}{1},$$



Note that removing a constant or a variable from an equation is not the equivalent of reducing it to zero. Since h/h=1, and in the case where temperatures are high or wavelengths are long, the non-classical terms drop out and we are back to the Rayleigh-Jeans law just quoted above.
 * $$B_\lambda(T) = \frac{2 c k T}{\lambda^4}.$$

Equations sub 1
Here is what it seems to me the crux of the matter as developed in the Wien approximation article:

Max Planck empirically obtained Planck's law of black body radiation:


 * $$B_\lambda(T) = \frac{2 c^2}{\lambda^5}~\frac{h}{e^\frac{hc}{\lambda kT}-1}$$

In the limit of very high temperatures or long wavelengths, the term in the exponential becomes small, and so the exponential is well approximated by its first-order Taylor polynomial:


 * $$e^{\frac{hc}{\lambda kT}} \approx 1 + \frac{hc}{\lambda kT}$$

That being the case it is o.k. to write:


 * $$\frac{1}{e^\frac{hc}{\lambda kT}-1} \approx \frac{1}{\frac{hc}{\lambda kT}} = \frac{\lambda kT}{hc}$$

Therefore:

$$\frac{1}{e^\frac{hc}{\lambda kT}-1} \approx \frac{1}{\frac{hc}{\lambda kT}}$$

so


 * $$B_\lambda(T) = \frac{2 c^2}{\lambda^5}~\frac{h}{e^\frac{hc}{\lambda kT}-1}$$

And then we'll try


 * $$B_\lambda(T) = \frac{2 c^2}{\lambda^5}~\frac{h}{\frac{hc}{\lambda kT}}$$

This results in the two "h" factors canceling, and in Planck's blackbody formula reducing to
 * $$B_{\lambda}(T) = \frac{2ckT}{\lambda^4}$$

which is identical to the classically derived Rayleigh–Jeans expression as given in that article:


 * $$f(\lambda) = \frac{2\pi c k T}{\lambda^4}$$

except for the pi factor. Where did that come from? It must be that one equation is for blackbody radiation and the other is for something else. What is the "function of lambda?"

tables
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Note that "true" really means "not falsified," "not shown to be false."

Again, the fourth possibility describes a situation in which nothing has happened so the proposition has not been tested. Since it has not been tested it cannot have been falsified. So it is counted as "true."

Initials
In each cell below, the first line indicates IPA, the second indicates pinyin.


 * may phonetically be (a voiced retroflex fricative). This pronunciation varies among different speakers, and is not two different phonemes.

Finals
In each cell below, the first line indicates IPA, the second indicates pinyin for a standalone (no-initial) form, and the third indicates pinyin for a combination with an initial. Other than finals modified by an -r, which are omitted, the following is an exhaustive table of all possible finals. 1

It is of interest to point out that the only syllable-final consonants in standard Mandarin are -n and -ng, and -r which is attached as a grammatical suffix. If you see a Chinese syllable ending with any other consonant, it is either from a non-Mandarin language (southern Chinese languages such as Cantonese, or minority languages of China), or it indicates the use of a non-pinyin Romanization system (where final consonants may be used to indicate tones).

1 /ər/ (而, 二, etc.) is written as er. For other finals formed by the suffix -r, pinyin does not use special orthography; one simply appends -r to the final that it is added to, without regard for any sound changes that may take place along the way. For information on sound changes related to final -r, please see Standard Mandarin. 2 "ü" is written as "u" after j, q, or x. 3 "uo" is written as "o" after b, p, m, or f. 4 It is pronounced when it follows an initial, and pinyin reflects this difference.

In addition, ê is used to represent certain interjections.

Rules given in terms of English pronunciation
All rules given here in terms of English pronunciation are approximate, as several of these sounds do not correspond directly to sounds in English.

Pronunciation of finals
The following is an exhaustive list of all finals in Standard Mandarin. Those ending with a final -r are listed at the end.

To find a given final:
 * 1) Remove the initial consonant. For zh-, ch-, sh-, both letters should be removed, they are single consonants spelt with two letters.
 * 2) However, y- or w- are part of the final; do not remove those.
 * 3) Syllables beginning with y- and w- are simply standalone forms of finals beginning with i-, u-. and ü-.
 * 4) If the initial is j-, q-, and x-, and the final starts with -u-, then change the -u- to -ü-.

NPA
 中文      

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