[This follow-up has become rather long since eventually it also served as
exam preparation. Hope it helps, and CMIIW.]
Post by 4***@gmail.comWhat is ''quantum dualism'' ?
A term you invented.
BTW, in English one uses the quotation mark ("), not two straight
apostrophes, and one does not write space before a question mark (“?”).
No.
Post by 4***@gmail.comE=hf where ( h) is a ''quantum of action''- particle and (f) its frequency.
No. ℎ is NOT a particle, it is a _quantity_: the quantum of action
(“action” in physics does not mean the same as “activity” in colloquial
speech), or simply Planck(’s) constant.
The particle of which E = ℎ f is the total end kinetic energy is the photon
(formerly, Planck: „Lichtquant“ “quantum of light”), of monochrome
electromagnetic radiation of frequency f. Actually, from
E² = m²c⁴ + p²c²
with m = 0 it is
E = p c = (ℏ × |k⃗|) × c = ℎ/2π × 2π∕λ × c = ℎ∕λ × c = ℎ c∕λ = ℎ f
(Planck–Einstein relation).
The photon is also the quantum of the electromagnetic field.
It turned out to be necessary to introduce the idea that the energy of light
could only be available in portions (quanta) in order to avoid and solve the
*ultraviolet catastrophe*:
----------------------------------------------------------------------------
One can think of a closed, opaque box with a little hole in it as a good
approximation of a *black body*, i.e. an ideal, theoretical physical body
that absorbs all incoming radiation (and reflects none, therefore appears
black):
_____________
: _________ :
: : .`. : :
: : .' `._:
: : .' . `.
: :'. .' : :`.
: :__`.'____: : `.__ __
:_____________: :'. |PE
Such a black body is a perfect absorber, but therefore also a perfect
emitter: It emits absorbed radiation again as infrared (thermal) radiation.
But since there is only a little hole in the box, the emitted thermal
radiation is unlikely to get out of the box. So as long as radiation is
going into the box, the temperature of the box’s walls increases. When no
more radiation is going into the box, the box will develop into a state
where the walls of the box and its inside space have the same temperature
(thermal equilibrium).
But there is one major problem: Classically, the spectral energy density
(energy density for a certain frequency/wavelength) in the box is described by
u(f, T) = 8π/c³ k T f² Rayleigh–Jeans (radiation) Law,
where c is the speed of light, k is Boltzmann’s constant, T is the
temperature of the box, and f is the frequency of the radiation.
To determine the total energy density in the box, we would have to calculate
the integral over all frequencies that are absorbed by the box. But we have
defined that it is a perfect absorber, so it absorbs *all* frequencies, and
we would have to calculate the integral over *all* frequencies:
∞ ∞
u_tot = ∫ u(f, T) df = 8π k T/c³ ∫ f² df.
0 0
But this integral (the area under the curve of the function) is infinite as
the function value grows towards infinity; in particular, when frequencies
are high (as is the case with ultraviolet radiation, around 10¹⁵ Hz):
<https://www.wolframalpha.com/input/?i=plot+8*pi%2F(299792458)%5E3+*+1.38e-23+*+3000+*+f%5E2+for+f+%3D+0+to+1e15>
(plotted for T = 3000 K)
So *classically* (under the assumption that light is an electromagnetic
wave) the energy density in the box should be infinite. But such an energy
density is unphysical and it is NOT what is being observed. Thus, the
Rayleigh–Jeans Law fails to describe black body radiation completely
correctly (it is still a good approximation for long wavelengths).
Max Planck found in 1900 that, to solve this problem, one must assume that
the thermal radiation is the result of the oscillations of charge carriers
in the box’ walls, each one a little harmonic oscillator, that can only
have an energy, and therefore can only absorb energy, that is portioned
(*quantized*) into integer multiples of a constant,
ℎ ≈ 6.625 × 10⁻³⁴ J s
[now it occurs to me that maybe he called it “h” for „harmonischer
Oszillator“ – German for “harmonic oscillator”; I have read one
other account, I do not remember where, that claims it was “h”
for „Hilfsvariable“ “helper variable”].
If you do that, then the spectral energy density in the box is described
instead by
u(f, T) = 8π ℎ f³/c³ × 1/(exp(ℎ f/(k T)) − 1) Planck’s (radiation) law.
Thus the ultraviolet catastrophe is avoided: If the frequency becomes large,
then the first factor is still large, but the second factor is small. So
the function value approaches zero for large frequencies, and the integral
over the function over all frequencies f never becomes infinite:
<https://www.wolframalpha.com/input/?i=plot+8*pi+*+6.625e-34+*+f%5E3%2F(299792458)%5E3+*+1%2Fexp(6.625e-34+*+f%2F(1.38e-23+*+3000)+-+1)+for+f+%3D+0+to+1e15>
(again plotted for T = 3000 K)
And this is what is actually being observed.
There are several applications for this law. In astrophysics, where the
effective temperature of a star corresponds to its color, because depending
on that temperature its light has a maximum intensity at a certain
frequency/wavelength and it can be modeled as a black body.
A planet or moon can be modeled as a black body as well, and its surface
temperature can be estimated when its distance from its star is known.
This leads to the concept of a habitable zone around a star, important for
finding habitable exoplanets.
For example, one can use it to show that if there were no natural greenhouse
effect, the mean surface temperature on Terra (Earth) would be −18.5 °C
instead of 14 °C (liquid water unlikely, like on Mars), and that it is human
influence that increased that to 15 °C within the past century. (More than
16 °C mean surface temperature will be catastrophic, hence the globally
agreed “2-degrees-target” until 2100; less would be better, of course.)
1. Terra reflects 30 % of incoming radiation (albedo; so absorbs only
70 %);
2. the irradiated area is equivalent to that of a circle with the radius
of Terra (only the day side);
3. the emitting area is the surface area of a sphere with the radius
of Terra (day and night side)
0.7 F☉ π r² = 4π r² T⁴ σ
T = ∜(0.7 F☉/(4 σ)) ≈ 254.6 K ≈ −18.58 °C.
F☉ – solar constant: flux density of solar radiation at 1 AU
(solar luminosity L☉ = 4π R☉² T☉⁴ σ = 4π (1 AU)² F☉)
r – radius of Terra
T – mean surface temperature of Terra (if there were no atmosphere)
σ – Stefan–Boltzmann constant
The concept is carried over to *color temperature* in everyday life, e.g.
for light bulbs and computer displays. Presently, it is way past midnight
here and the Redshift software has automatically gradually adjusted the
color temperature of my laptop’s display to 3700 K because our biorhythm is
tuned to the apparent color of our star, which is more reddish closer to
sunset (due to Rayleigh scattering), to which this color temperature
corresponds; it will let me sleep better after writing this than if I had
been exposed to the bluish light of 6500 K that corresponds to sunlight in a
blue day sky (to which it will automatically revert if I use my laptop
during the day). In ambient lighting, color temperature makes the
difference between “soft”, “warm”, “natural”/“daylight”, and “cool” lights.
----------------------------------------------------------------------------
In 1905, Einstein showed that Planck’s assumption that the energy of
electromagnetic radiation/light is quantized in this way can explain also
the classically inexplicable photoelectric effect; therefore, that Planck’s
“quantum of light”, later called “photon”, and quantization of energy, was
more than the result of a mathematical trick to avoid infinities:
<http://hyperphysics.phy-astr.gsu.edu/hbase/mod1.html#c2>
This realization spawned the field of quantum mechanics (QM), which is at
the core of all modern physics and technology (even computers, and I am not
even talking about quantum computers).
Post by 4***@gmail.com(wave / particle duality - simultaneously )
Yes. But later in the development of QM it was realized that there is
actually no duality: all objects, including those who were previously
thought to be point-like particles (e.g., electrons), exhibit wave-like
behavior: they are properly described by a wave function that solves the
Schrödinger equation. It just does not show on larger-than-microscopic scales.
Post by 4***@gmail.comUhlenbeck and Goudsmit described how this action is possible: E=h*f
That is not a description of anything.
Yes. So what? (You have not used hbar. I did.)
--
PointedEars
Git: <https://github.com/PointedEars> | SVN: <http://PointedEars.de/wsvn/>
Twitter: @PointedEars2
Please do not cc me. / Bitte keine Kopien per E-Mail.
--
PointedEars
Twitter: @PointedEars2
Please do not cc me. / Bitte keine Kopien per E-Mail.