The author argues that, for consistency, the de Broglie wavelength of a moving material particle should take account of the medium in which the particle is moving. Louis de Broglie posited that a moving material particle should be considered in the same way as a photon. Now, the wavelength of a photon is related to the medium in which it propagates and it seems only logical that, for a moving material particle the nature of the medium should be captured by the inclusion of something which characterizes that medium. It is shown that this 'something' is the index of refraction.
The de Broglie wavelength is then shown to be given by: l = h/p.r , where h is Planck's constant, p, the momentum of the particle and r, the index of refraction. Further, it is demonstrated that the inclusion of the index of refraction in the de Broglie wavelength alters Heisenberg's position/momentum relationship (but not the energy/time relationship), and, in addition, modifies the Schrödinger equations. All of these return to their commonly recognized forms when the index of refraction is set equal to its vacuum value.
Table of Contents
Abstract
Introduction
Analysis – de Broglie
The Fidler diagram
Analysis-Heisenberg
Discussion
References
Abstract
It is argued in the following work that, for consistency, the expression for the de Broglie wavelength of a moving material particle should take account of the medium in which that particle is moving. Since de Broglie posited that a moving material particle should be considered in the same way as that of a photon, and the wavelength of a photon is related to the medium in which it propagates, then it would follow that, for a moving material particle the nature of the medium should be captured by the inclusion of something which characterises that medium; this is shown to be the index of refraction.
In this work we show that the expression for the de Broglie wavelength is given explicitly by: , where h is Planck’s constant , p, the momentum of the particle and r, the index of refraction.
Further, it is shown that this alters Heisenberg’s position/momentum relationship (but not the energy/time relationship) and, in addition, modifies the Schrodinger wave equations.
In all cases, the setting of the index of refraction to its vacuum value of unity restores all of these equations to their commonly-recognised forms.
Introduction
As is well known, de Broglie posited that all moving matter should have a wavelength in the same manner as a photon, and this was verified experimentally by Davisson and Germer when they showed that, in a version of the double slit experiment using electrons, the pattern on the ‘observing screen’ was the same as that observed when the initial beam was composed of photons, thus providing further evidence of the correctness of the concept of wave/particle duality.
The Fidler diagram 2 shows the behaviour of a photon in different media characterised by the index of refraction; it then follows that if the logic of the de Broglie hypothesis was pursued then the expression for the de Broglie wavelength of a moving particle should take account of the nature of the medium in which the particle was moving.
It was then shown that the expression for the de Broglie wavelength, was changed to:
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In many, if not all of the situations in which the de Broglie hypothesis is used, it is tacitly assumed that these take place in the vacuum where r = 1, thus giving the ‘standard ‘form of the expression for the de Broglie wavelength.
The inclusion of the refractive index in the de Broglie wavelength has implications for both Heisenberg’s position/ momentum inequality and Schrodinger’s wave equation, and this is shown explicitly to be indeed, the case.
Analysis – de Broglie
The fundamental concepts of Mechanics are mass, length and time. We now write these in the form of Planck units using parentheses to indicate the relevant concept.
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Moreover, we can define additional Planck quantities as follows:
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We now propose that the last four equations refer to electromagnetic energy at the ‘Planck condition’; hence, (L) is the wavelength, (T) the periodic time, and the Planck energy, (E) may be written in the form named by Wilczek 1 as the Planck-Einstein-Schrodinger equation, viz: ------------ (6), where is the Planck frequency. Further, dividing equation (2) by equation (3) gives:
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For a photon, equation (5) would be written:
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Multiplying equations (2) and (4) gives:
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The Fidler diagram
In the development of the Fidler diagram, 2 we borrowed a concept from Fluid Mechanics called the Strouhal number. This is an important dimensionless group associated with the oscillatory motion of a body suspended in a stream of fluid. We defined a radiation Strouhal number, where and l are the frequency and wavelength, respectively, of the radiation.
Now, the refractive index, r, of a substance is defined as: and, since the speed, v of propagation of a wave is given by the product then we see that the radiation Strouhal number is the inverse of the index of refraction. We chose the radiation Strouhal number in developing the equation from which the Fidler diagram was constructed for, being the inverse of the index of refraction, it lay within the range 0---1, unlike the index of refraction which can range between 1 and infinity. Such a latter magnitude was achieved by Hau et al, 3 where, switching rapidly a Bose-Einstein condensate from transparent to opaque to transparent succeeded in bringing light to rest, effectively producing a medium with an index of refraction of infinity.
Using equation 7 we may write:
In addition, , hence equation (10) may be written:
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And , , hence we may write equation (11) in its final form:
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It is immediately obvious that when r = 1 equation (12) becomes de Broglie’s equation.
De Broglie argued that what applied to a photon should also apply to all matter, in particular, that all moving particles had a wavelength. Equation (12) takes account of the fact that the speed of the photon is reduced below the luminal speed when the photon passes through anything other than the vacuum, and if, as de Broglie posited, all moving matter may be regarded as possessing wavelike qualities then we may thus interpret equation (12) as a modified version of de Broglie’s equation valid for all media. The experimental confirmation of equation (12) would, in all probability involve considerably more complication than the Davisson-Germer experiment which verified de Broglie’s hypothesis.
Analysis-Heisenberg
We state, without proof the General Uncertainty Principle applied to two Hermitian operators , and , i.e.
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Here, the are the standard deviations of their respective subscripts (although, a more informative title would be ‘the standardised deviations from the mean’), the vertical bars denote absolute value, the is the symbol for the expected value, and the square braces denote the commutator. It should be stressed that the General Uncertainty Principle (GUP) is totally unconnected with experimental measurements and is simply a mathematical process.
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Let us now consider the application of the GUP to the problem of the position and momentum of a particle in space.
The position operator is and the momentum operator in the x-direction , .
A moment’s reflection is sufficient to note that the substitution of the operators shown immediately above into equation (14) is a fruitless exercise for it will yield nothing more than a change of symbols. We require that the momentum operator be written in an analytical form.
Elsewhere, 4 we showed that Schrodinger’s wave function could be written
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Now, from equation (12), , hence we may write equation (15) as:
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It should be noted that Planck’s h has been replaced by Dirac’s .
Differentiating this wrt x gives: , hence, the operator, may be written:
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We now write the commutator acting on a dummy function, g, i.e . g) = .
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If it is assumed that the index of refraction is not a function of x, then the expectation value of the commutator is, and the modulus,
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Taking the lower limit of this equation we see that this represents an infinite number of rectangular hyperbolae with r as parameter.
If however, we replace by the product , from equation (12) there results:
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Taking the lower limit of this equation, it is seen that we have a single rectangular hyperbola with parameter and the dimensionless quantities as the variables.
Returning to equation (15) and differentiating it wrt time we get:
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This can be shown to be Hermitian, and so if we return to equation (14) and repeat the process there, but with the operators , and we get the Heisenberg relationship appropriate to the measurement of the energy of a system over time, i.e.
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It could be argued heuristically that the Heisenberg relationship relating to the measurement of the energy of a system over time should have no dependence upon the de Broglie wavelength, and the above analysis shows that this is indeed the case. The lower limit of this expression represents a rectangular hyperbola with as parameter.
If the system is an oscillator then if we divide equation (20) throughout by the product, T E, where, T is the periodic time then the LHS is: , and equation (20) then takes the same form as equation (19), i.e.
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At this juncture it is considered appropriate to dispel the general misconception that the Heisenberg relationships refer to experimental uncertainties in the quantities of interest. The relationships arise solely from the General Uncertainty Principle, which has nothing to do with experiments, and is a statement that the exact magnitude of any variable is fundamentally unknowable. This is widely confused with the Observer Effect, where the very act of measurement renders the result of the measurement imprecise.
Analysis-Schrodinger.
In all of the following the symbols have their usual connotation.
Schrodinger was concerned with the analysis of the motion of an electron in a potential field and from consideration of conservation of energy wrote the following simple equation:
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For clarity we confine ourselves to consideration of motion in one dimension, assume that the potential field is only a function of position, and, following Schrodinger we write:
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From the preceding section we have: and . Hence we may write equation (21) as :
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This is the Time-dependent Schrodinger equation incorporating the modification for the de Broglie wavelength.
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- William Fidler (Autor), 2020, Werner Heisenberg, Louis de Broglie and Erwin Schrödinger revisited, Múnich, GRIN Verlag, https://www.grin.com/document/915006
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