Two photon absorption spectroscopy is well known non-linear optic phenomena [1]. In this phenomena the virtual level is present which allows to absorb the second photon and thus excite the state with energy equal to two times the energy of the original quantum. The corresponding two-photon excited fluorescence is well known (and now three-photon excited fluorescence is known well). The most important observation connected with multiple quanta absorption is the non-linear dependence on power: this allows easily distinguish it from other phenomena.
The fundamental hypothesis outlined in the previous post [2] concerning the harmonics of de Broglie waves may be also stated for the photon itself. Indeed, the initial hypothesis of Plank concerning the photons [3] was the energy of photons itself is: E=n*h*ν [3]. From observation of photoelectric effect Einstein deduced the more commonly known rule: E=h*ν, which eventually lead to the development of the quantum electrodynamics and numerous discoveries.
However, any mathematical expression is only the approximation to the natural law, and the idea of the photons having energy of only E=h*ν may be very successful but not finally correct. Indeed, the double energy photon (with E=2*h*ν) may be so rare that virtually non-observable and thus making the Einstein idea so exceptionally great fit to the natural law that it looks absolute. The double energy photons may easily decay into two ordinary photons or mutate into the photon with double frequency.
Even the probability of the existence of such photons would be governed by the usual Boltzman rule (from [2]):
the population of double energy photons would be exp[-hν/(kT)] less compare to the usual photon. That value for relatively small energy infrared photons (1064 nm wavelength, 1.17 eV ordinary photon energy) would be at room temperature only 2.35*10exp(-20). This means that even such photons exist, they are so rare that virtually non-observable.
However, the two photon excited fluorescence is a convenient way to check theirs presence. Indeed, in addition to the quadratic in power term of such fluorescence (due to virtual levels creation [1]) an extremely small linear in power fluorescence is predicted. This fluorescence is so weak that it is necessary to consider the background created by usual thermal excitation (with some non-zero probability the same excited level may be reached by the usual thermal excitation according to the Boltzman formula P/Po=exp[-E/(kT)].
The trick to subtract background is that the two photon fluorescence is a resonant phenomena. It means that for the deviation of the wavelength from the resonant value it will quickly disappear. Since for the linear phenomena search the laser should not be powerful (to prevent observation of the more common quadratic in power two photon fluorescence) it may be with tuned frequency and thus allowing to observe the linear in power resonant phenomenon. When the frequency of the laser deviates from the frequency necessary for two quanta fluorescence (this frequency may be obtained from the quadratic term of the induced fluorescence) the observed linear term should quickly disappear (and the thermal background stay the same).
In summary: the observation of the linear term in the two photon fluorescence is predicted due to hypothetical existence of double energy photons (from Planks rule E=n*h*ν [3]), the phenomenon would be really weak (20 orders of magnitude weaker compare to one-photon fluorescence at least for infrared photons) but resonant in photon frequency.
References.
1. https://en.wikipedia.org/wiki/Two-photon_absorption
2. https://tipikin.blogspot.com/2020/05/quantization-rule-and-harmonics-of.html
3. http://web.phys.ntnu.no/~stovneng/TFY4165_2013/BlackbodyRadiation.pdf
Sunday, May 24, 2020
Wednesday, May 6, 2020
Quantization rule and harmonics of matter waves.
When modern scientist is recalling the quantization rule for photons, usually the famous E=h*ν is recalled. However, in the original Planck's derivation the more general rule of quantization was assumed: E=n*h*ν [1]. A similar rule of quantization was assumed by Nield Bohr concerning the orbital moment. The values n=2,3,4 .. are responsible for the excited states of the quantum system.
In quantum electrodynamics the quantization rule for electromagnetic field is similar to oscillator:
E=n*h*ν +0.5*h*ν
At the same time the wavelength of the de Broglie wave has only one value: λ=h/p=h/(m*v) not λ=n*h/p, where n is a number 1,2,3..., p is the pulse of the particle (applicable for any particle), m - rest mass and v - velocity of the particle (applicable only for non-relativistic case).
Since the matter waves are not really easy to investigate, it may happened that the de Broglie wavelength also follows the most general rule with the presence of the harmonics:
λ=n*h/p,
But they were simply overlooked and careful experiment would be necessary to discover them.
For de Broglie wave it is would not be easy to find presence of such harmonics, because during the interference experiment they would generate the maxima and minima, which coincide with maxima and minima of the main matter wave. If harmonic is present in the miniscule amount it will lead to some hardly observable effect.
Let's consider for example the case of only one added harmonic. Let the first harmonic has maximum of 1 and decays as exp(-0.1*m), where m is the interference band number, m=0,1,2,3,4... In this case the amplitudes of the interference bands would be: Io=1, exp(-0.1), exp(-0.2), exp(-0.3), exp(-0.4)=1, 0.905, 0.819, 0.741, 0.670...
Second harmonic of de Broglie wave would have the amplitude of 0.01 and decays with distance away from the center according to the same law exp(-0.1*m), here m is the interference band for the second harmonic, which would coincide with a certain band for first harmonic (because wavelength is exactly 2 times larger). The amplitude would be I=0.01, 0.01*exp(-0.1), 0.01*exp(-0.2)… =0.01, 0.00905, 0.00819 ...
The sum of the amplitudes would be:
1.01; exp(-0.1); exp(-0.2)+0.01*exp(-0.1); exp(-0.3); exp(-0.4)+0.01*exp(-0.2); …..=
1.01; 0.905; 0.828; 0.741; 0.679; …..
In order to distinguish the case of the one and multiple harmonics the ratio of the amplitudes of the consecutive bands may be calculated: bamd1/band0; band2/band1, band3/band2.....
For exactly one de Broglie wavelength that would be monotonic function:
0.905, 0.905, 0.905 …. (constant in this example, because the chosen decay function was exponential)
For the sum of harmonics it would be: ratios are:
0.896; 0.915; 0.895; 0.916 …. - non-monotonic function, the superposition of monotonic function and ao*Cos(pi*m), where m is the interference band number.
The third and higher harmonics will add more "waviness" to the smooth function.
How to estimate the amplitude of the harmonics in matter wave? The idea of evaluation is inferred from the reciprocity principle: the particle is both matter and wave [2]. Assuming the matter wave is something real (similar to a photon, but permanently "attached" to the particle), the probability of the excitation of the second energy level would be similar to the idea proposed by Plank [1]:
population of each next level would follow Boltzmann rule [3]:
-log(Ni/N)~Ei/kT
But what is the expected energy of the initial de Broglie wave? Hypothesizing that the particle is both matter and wave it is possible to speculate about this value.
Since de Broglie wave is "attached" to the particle, the only velocity it may have is equals to the velocity of particle v. From the general rule connecting velocity, wavelength and frequency of the wave it follows:
v=λ*ν or λ=v/ν
here v is the velocity of the particle, λ is the wavelength, ν is the frequency of the wave.
Substituting λ into the formula for non-relativistic de Broglie wave:
λ=h/(m*v) and v/ν=h/(m*v)
which may be transformed as follows:
mv2=hν or mv2/2=hν/2
For non-relativistic particle the energy of de Broglie wave can not be larger than the full kinetic energy of the particles and quantized as oscillator (quite reasonable idea, because the zero energy of electromagnetic field has the same value). Assuming the next harmonic will have the energy according to the Planks rule (or quantum electrodynamic rule, similar to oscillator), the difference in energy between two levels for de Broglie wave would be double the kinetic energy of the non-relativistic particle (for relativistic particle the quantization rule is simpler and coincides with the quantization rule for photons).
Than the ratio of the amplitude of the second harmonic of de Broglie wave to the initial amplitude would be equal (from Boltzmann rule):
I/Io=exp(-2Ek/kT)
where Ek is the kinetic energy of the non-relativistic particle. For example for electron with possible to reach energy of 0.1 eV, observed at the room temperature (300 K, at this temperature de Broglie wave is still well resolved since 0.1 eV > kT), the ratio would be:
I/Io=exp(-2Ek/kT)=4.4*10exp(-4)
Which is small, but at numerous averaging is possible to reach and to discover.
References.
1. http://web.phys.ntnu.no/~stovneng/TFY4165_2013/BlackbodyRadiation.pdf
2. https://tipikin.blogspot.com/2019/09/the-possible-way-to-search-for-new.html
3. https://en.wikipedia.org/wiki/Maxwell–Boltzmann_distribution
In quantum electrodynamics the quantization rule for electromagnetic field is similar to oscillator:
E=n*h*ν +0.5*h*ν
At the same time the wavelength of the de Broglie wave has only one value: λ=h/p=h/(m*v) not λ=n*h/p, where n is a number 1,2,3..., p is the pulse of the particle (applicable for any particle), m - rest mass and v - velocity of the particle (applicable only for non-relativistic case).
Since the matter waves are not really easy to investigate, it may happened that the de Broglie wavelength also follows the most general rule with the presence of the harmonics:
λ=n*h/p,
But they were simply overlooked and careful experiment would be necessary to discover them.
For de Broglie wave it is would not be easy to find presence of such harmonics, because during the interference experiment they would generate the maxima and minima, which coincide with maxima and minima of the main matter wave. If harmonic is present in the miniscule amount it will lead to some hardly observable effect.
Let's consider for example the case of only one added harmonic. Let the first harmonic has maximum of 1 and decays as exp(-0.1*m), where m is the interference band number, m=0,1,2,3,4... In this case the amplitudes of the interference bands would be: Io=1, exp(-0.1), exp(-0.2), exp(-0.3), exp(-0.4)=1, 0.905, 0.819, 0.741, 0.670...
Second harmonic of de Broglie wave would have the amplitude of 0.01 and decays with distance away from the center according to the same law exp(-0.1*m), here m is the interference band for the second harmonic, which would coincide with a certain band for first harmonic (because wavelength is exactly 2 times larger). The amplitude would be I=0.01, 0.01*exp(-0.1), 0.01*exp(-0.2)… =0.01, 0.00905, 0.00819 ...
The sum of the amplitudes would be:
1.01; exp(-0.1); exp(-0.2)+0.01*exp(-0.1); exp(-0.3); exp(-0.4)+0.01*exp(-0.2); …..=
1.01; 0.905; 0.828; 0.741; 0.679; …..
In order to distinguish the case of the one and multiple harmonics the ratio of the amplitudes of the consecutive bands may be calculated: bamd1/band0; band2/band1, band3/band2.....
For exactly one de Broglie wavelength that would be monotonic function:
0.905, 0.905, 0.905 …. (constant in this example, because the chosen decay function was exponential)
For the sum of harmonics it would be: ratios are:
0.896; 0.915; 0.895; 0.916 …. - non-monotonic function, the superposition of monotonic function and ao*Cos(pi*m), where m is the interference band number.
The third and higher harmonics will add more "waviness" to the smooth function.
How to estimate the amplitude of the harmonics in matter wave? The idea of evaluation is inferred from the reciprocity principle: the particle is both matter and wave [2]. Assuming the matter wave is something real (similar to a photon, but permanently "attached" to the particle), the probability of the excitation of the second energy level would be similar to the idea proposed by Plank [1]:
population of each next level would follow Boltzmann rule [3]:
-log(Ni/N)~Ei/kT
But what is the expected energy of the initial de Broglie wave? Hypothesizing that the particle is both matter and wave it is possible to speculate about this value.
Since de Broglie wave is "attached" to the particle, the only velocity it may have is equals to the velocity of particle v. From the general rule connecting velocity, wavelength and frequency of the wave it follows:
v=λ*ν or λ=v/ν
here v is the velocity of the particle, λ is the wavelength, ν is the frequency of the wave.
Substituting λ into the formula for non-relativistic de Broglie wave:
λ=h/(m*v) and v/ν=h/(m*v)
which may be transformed as follows:
mv2=hν or mv2/2=hν/2
For non-relativistic particle the energy of de Broglie wave can not be larger than the full kinetic energy of the particles and quantized as oscillator (quite reasonable idea, because the zero energy of electromagnetic field has the same value). Assuming the next harmonic will have the energy according to the Planks rule (or quantum electrodynamic rule, similar to oscillator), the difference in energy between two levels for de Broglie wave would be double the kinetic energy of the non-relativistic particle (for relativistic particle the quantization rule is simpler and coincides with the quantization rule for photons).
Than the ratio of the amplitude of the second harmonic of de Broglie wave to the initial amplitude would be equal (from Boltzmann rule):
I/Io=exp(-2Ek/kT)
where Ek is the kinetic energy of the non-relativistic particle. For example for electron with possible to reach energy of 0.1 eV, observed at the room temperature (300 K, at this temperature de Broglie wave is still well resolved since 0.1 eV > kT), the ratio would be:
I/Io=exp(-2Ek/kT)=4.4*10exp(-4)
Which is small, but at numerous averaging is possible to reach and to discover.
References.
1. http://web.phys.ntnu.no/~stovneng/TFY4165_2013/BlackbodyRadiation.pdf
2. https://tipikin.blogspot.com/2019/09/the-possible-way-to-search-for-new.html
3. https://en.wikipedia.org/wiki/Maxwell–Boltzmann_distribution
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