Saturday, October 27, 2012

Demand curve and de Broglie wavefunction

Demand curve and the de Broglie wave function
This entry has been hanging over me for sometime, and that I never published earlier was 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 maybe that they are not a real physical, tangible object, 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 has been developed from my work on the fractal, and culminates my discoveries and insights from the fractal.
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 the 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 verse.
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.


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 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. 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 reference to it yourself – but I will say it was not the theory, more than what it suggested, that caught my attention. 

I knew I was on to something significant early on in my fractal quantum thinking 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 maybe 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, I think with the fractal can trace it. It must at least have something to do with it, given my other discoveries. 

Some demand curve: background
The demand curve (shown below in fig, 2) is a downward sloping curve showing the relationship between 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 visa verse.

If there is any relationship between fig. 1 and fig. 2 then it maybe as simple as finding a connect 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 discussion. With this in mind, fractal theory would suggest from the outset that the two should be connected, universal, unbounded - irrespective of scale.

One at a time, I have been analysing these variables and other related patterns, but have found no direct relationship as 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 phenomena 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 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 quantity of triangles: it demonstrates the creation of a strait-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 (super position) of the triangles.  Amplitudes (A), wavelengths (λ) frequency (f) are derived from, and can be demonstrated in, every fractal development, and they all increase in their magnitudes in an exponential manner that corresponds to the quantum nature of electromagnetic spectrum.

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 amplitude of the wave, and not the wavelength. This should end the discussion, but for the fact that - and not withstanding 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 mathematician's and from readings, it is suggested that this it quite 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 not only in musical instruments, in 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 the 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 be derived between Qd and mv?
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 (by use of 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, but stemming from the wave nature insights of the fractal developed in this blog; the fractal does show concepts of velocity, and of frequency.
From classical physics: 

Wave velocity is equal to: 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 is the time taken for wave 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. Without time, mass, and distance, the calculation (in a classical sense) is impossible. In terms of the fractal their values are not apparent, not obvious, and any attempt to assign values to them are subjective or arbitrary. It maybe 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 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 interesting characteristic, and to the occurrence of this problem.

As I have pointed out, the fractal shows no concept of distance: so what to do, 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 becomes 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.
What have found is that the weaknesses of this analysis has exposed directly (for the first time) key issues of the fractal (and quantum) : time, distance and mass.
The strange thing is (for me) is that an understanding of the demand curve requires a great deal of 'faith', just like quantum.

I'll keep thinking. 

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








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