Demand curve and de Broglie wavefunction

 

Demand curve and the de Broglie wave function

  I have since published in 2023: 

This entry has been hanging over me for some time, and I never published it earlier because I never thought I had it quite right or it never felt complete. Though those feelings have not changed, I have (now) decided to publish what I have, intending that my theory will kindle interest and discussion to further develop it. What really is a demand curve anyway? Do they really exist? The reality may be that they are not real physical, tangible objects but instead show the possibilities of goods and services in terms of price and quantity; as with quantum theory, to produce such a curve invokes the ‘measurement’ problem. It is as if the demand curve is a superposition of all the possible outcomes, just as in quantum mechanics.

My hypothesis:

The de Broglie wave function and the (consumer) demand function and corresponding curve are both different manifestations of the same thing.

This theory was developed from my work on the fractal and culminates my discoveries and insights into it.

Background.

Until now, in this blog, I have been attempting to decipher the fractal, and early on, suggested that the downward sloping demand curve is derived from an understanding of fractal development.  See my early entries:

Later I posited that the fractal when isolated, or in a state of ceteris paribus, resembles – what I think to be by no coincidence – the characteristics of quantum mechanics, super-position etc, and have suggested – based on fractal mechanics – that they are one and the same. As if to say: to understand quantum is to understand the fractal and visa versa.
It follows that if this is all so, then the demand curve is also (directly) related to quantum mechanics –

If so, how? It is not so clear, but here is what I have.

The de Broglie and the wave function

de Broglie suggested that ‘we’ are both waves and particles at the same time and that everything has a wavelength, from the smallest to the largest. The de Broglie diagram below illustrates the de Broglie equation:  

λ = h/p. 

where λ is the wavelength, h is the Planck constant, and p is the electron's momentum (mv, mass x velocity). Fig. 1 (below) is a diagram illustrating this de Broglie function—it is a downward-sloping log function.

Fig. 1 The de Broglie diagram

 Now, it is not my intent or my place to describe the de Broglie wave function more than this—I am really not the right person to do so, and besides, you can easily find references to it yourself—but I will say it was not the theory, more than what it suggested, that caught my attention. 

Early on in my fractal quantum thinking, I knew I was on to something significant when I identified that the fractal also demonstrates this wave-particle duality.

It was not until I saw the above diagram (Fig. 1) in an elementary physics textbook that I realised that there may be a direct relationship between demand and quantum - and something more significant to say about this most ‘unlikely’ of coincidences. The classical demand curve (fig. 2 below)  is also a downward-sloping log. log. function.

Fig.2 The demand curve

It may well just be a coincidence, but what if it is not? What if there is a connection? If there is, the fractal can trace it. Given my other discoveries, it must at least have something to do with it. 

Some demand curve: background

The demand curve (shown below in Figure 2) is a downward-sloping curve showing the relationship between the price of a good and the quantity demanded of the good. The curve illustrates the ‘law of demand’—all else being equal, if the price rises, the quantity demanded falls, and vice versa.

If there is any relationship between Fig. 1 and Fig. 2, then it may be as simple as finding a connection between the variables price and quantity demanded (from classic economics) and the variables wavelength and momentum (from quantum mechanics).

I am laying down a theory here. What I see. I expect there to be problems and discussions. With this in mind, fractal theory would suggest from the outset that the two should be connected, universal, and unbounded - irrespective of scale.

I have been analysing these variables and related patterns one at a time, but I have found no direct relationship yet.

1. Price and Wavelength

From classical economics, we have price(P) equal to marginal utility(mu):

P = mu. 

From the early work in this blog where I have suggested that utility, and thus price, is a fractal phenomenon and that the relationship between price and utility can be demonstrated in the (Koch snowflake) fractal development - where the marginal (or additional or extra) area after each iteration demonstrates the demand curve. See the diagram below.

Fig. 3 Please excuse the misleading label 'Fig. 1' - near the title of this diagram.

Fig. 4 below illustrates the logarithmic relationship between the Area and the number of triangles: it demonstrates the creation of a straight-line demand curve from an exponential function ( Fig. 3 above).



Fig. 4

1.2 Fractal wave and superposition
More recent discoveries from the fractal (in this blog) have suggested that there is a wave-like nature to fractal development and that the MA curve above shows all the (infinite) positions (superposition) of the triangles.  Amplitudes (A) and wavelengths (λ) frequency (f) are derived from and can be demonstrated in every fractal development. They all increase in their magnitudes exponentially, corresponding to the electromagnetic spectrum's quantum nature.

1. 3 Is price (derived from the marginal area in the fractal) really the same as the wavelength in the de Broglie diagram? 
Directly, no. The price (as far as understood) is the wave's amplitude, not the wavelength. This should end the discussion but for the fact that - and notwithstanding the understanding that both Wavelength (λ) and the Amplitude (A) are independent of each other in classical wave theory - that the two (in the case of the fractal) are both inextricably linked as having an exponential nature, and both share infinity in their scale. It is as if λ is equal to or related to f. In discussions with mathematicians and from readings, it is suggested that this is entirely possible.
In Ian Stewart's Taming The Infinite, Quercus Publishing 2008, page 124: on Bassel functions,' The amplitudes of these waves still vary sinusoidally with time, but their spatial structure is more complicated.' 'The wave equation is exceedingly important. Waves arise in musical instruments and the physics of light and sound. Euler found a three-dimensional version of the wave equation... Clark Maxwell extracted the same mathematical expression from his (Euler's) equations for electromagnetism and predicted the existence of radio waves.'
And, from Wikipedia: '... and based on the theory of Fourier decomposition, a real wave must consist of the superposition of an infinite set of sinusoidal frequencies.' on the subject of electromagnetic wave equations.

2. Quantity Demanded (Qd) and Momentum (mv)

Can any relationship between Qd and mv be derived?

Qd

• Qd: is the quantity demanded for a good in a specific time period - so is a rate or frequency.
• Qd is said to be exponential in nature and is shown and demonstrated to be exponential (using the fractal) in this blog.

mv 

Momentum is equal to mass(m) times velocity(v).

There is no obvious concept of mass in the fractal, so mass may be assumed to be 0. However, stemming from the wave nature insights of the fractal developed in this blog, the fractal does show concepts of velocity and frequency.

From classical physics: 

Wave velocity is equal to the frequency(f) multiplied by the wavelength (λ) 

v = fλ

f: is equal to 1 divided by the Period(T)  

f=1/T

T: is equal to the time a wave takes to produce one cycle.

time:

λ: is equal to the distance* between the crest to crest or trough to trough

*It is here, with the fractal, that two issues of position (time and distance) present themselves, or as it would be, don't present themselves. The calculation (in a classical sense) is impossible without time, mass, and distance. In terms of the fractal, their values are not apparent or obvious, and any attempt to assign them to them is subjective or arbitrary. It may be concluded that a fractal in isolation has no concept of time nor distance - something I shall address in a separate entry due to this interesting characteristic and to the occurrence of this problem.

The one (real) reference to the time that the fractal may offer may be found in the iteration: the change from one iteration to another. Again, these are concepts I aim to develop, but for now:

It shall be assumed that T = to the iteration number (i).

T = (i)

So, T (the period from crest to crest in the Koch snowflake wavelength) is equal to 6 iterations: 

T = 6

f = 1/6

To calculate the velocity, we need λ, the wavelength, and to calculate this, we need a distance.

Distance: As with time, I shall address the fractal issues of distance in a separate entry due to this exciting characteristic and the occurrence of this problem.

As I have pointed out, the fractal shows no concept of distance, so what should we do? Should we leave it?

I am currently working on ways to measure this distance, but until then I'll:

Assume that the wavelength behaves as a sinusoid and that the frequency and the wavelength become exponentially smaller. 

  

Final word.

I have nothing to conclude, no absolute finding, but like all things I'm looking at, (I suggest) they cannot be overlooked.

We have found that this analysis's weaknesses have directly (for the first time) exposed key issues of the fractal (and quantum): time, distance and mass.

The strange thing (for me) is that understanding the demand curve requires a great deal of 'faith', just like quantum physics.

I'll keep thinking. 

Wouldn't it be a hoot to think that 'economics' and 'physics' are connected?