9- Uncertainty versus duality
A wave of definite wave-length represents a particle with definite momentum. The shorter the wavelength l the higher the momentum h/l .Such a wave extends over the whole space. We already saw that if there is an attractive center the wave becomes a localized peak around this center like a cloud condensing to a droplet as it is the case in atomic orbitals .
However in reality an electron wave cannot extend over the whole space even if there is no attracting potential center. What we have in reality is a wave packet. The average spatial Extend of the wave packet represents the uncertainty of position of the electron.
It has been known for long that by taking an appropriate superposition of waves of different wave lengths one can construct any arbitrary function in space. This is a consequence of so called Fourier transformations. By appropriate I mean that one has to choose the weight of each particular wave-length so that the superposition gives the desired function.
It turns out that to construct a wave packet of a finite spatial extend with given wave-length l as in Figure 10a , one has to add small contributions from other wave lengths in close neighborhood of l (Figure 10b). The smaller one wants to make the spatial extend of the wave-packet the greater must be the contribution from other wave-lengths thus the wider the region dl becomes in figure 10b.
Figuere 10 a) wave packet in real space and b) its wave-length spectrum
Thus the sharper the position the more unsharp is the wave length and consequently the more unsharp the momentum p = h/
l and vice versa. This relation is formulated by Heisenberg in the following form known as Heisenberg uncertainty principle.dp dx >= h
However when he formulated the uncertainity relation originally he interpreted it in the following way. The momentum and the position of a particle cannot be measured simultaneously with arbitrarily high precision. He gave the following thought experiment to illustrate this principle:
To know the position of the particle you have to measure it. The simplest way to know the position is to look at it. To see something you must send light on it. There is however a momentum transfer from light to particle when light is reflected by the particle. Thus the particles momentum is disturbed during the measurement of the position. Classically the light intensity can be made so low that this momentum transfer can be made arbitrarily small. If the light is made of photons one cannot make the transferred momentum smaller then the momentum of single photon. To keep the momentum transfer small one needs light of longer wave length (momentum p= h/
l ). On the other hand To see an object the wave length of light must be shorter then the objects size. Otherwise the image of the object cannot be distinguished from the diffraction pattern of the wave. The sharper you measure its location (the shorter wave length , negligible refraction) the greater is the transferred momentum namely the greater the uncertainty in particle momentum.This interpretation of uncertainty principle shows how the minds could not free themselves from the concept of pointlike or billiard ball like entity. In truth One neither needs the concept of pointlike particle nor the discussion of momentum transfer in position measurement event to understand this principle if one uses the wave-only interpretation. It is the natural consequence of Fourier transformations.
The term “uncertainty of position” alone is misleading. It implies that there is a pointlike particle with a position but the position is merely uncertain. If there is only wave function it is obvious that we cannot speak of position at all. What is called uncertainty of position is the average spread of the wave function in space. The experimental revelation of the uncertainty principle namely the relation between the statistical variance in position and the statistical variance in momentum one obtains by the evaluation of of large number of position and momentum measurements with identically prepared wave functions is a consequence of 2 facts:
2. The fact that the squared magnitude of the wave function gives the probability of detection at a particular position or the of measuring a particular momentum.
The thought experiment of Heisenberg above implies that as if there is a pointlike particle with definite momentum and definite position (namely with a definite trajectory) and that it is only the inevitable momentum transfer that prevents us to determine these values simultaneously. Thus in this picture the uncertainty reflects OUR inability to determine these values because of inevitable experimental limitations. It doesn’t point to an intrinsic uncertainty. However we know that such an intrinsic uncertainty exist from the fact that we have to accept that even a single particle must go through both slits simultaneously if we want to explain the fact that the interference pattern in double slit interference experiment occurs even in the case when we lower the intensity of the electron beams up to a point so that the electrons go one by one through the slits. (see Chapter 4)
Heisenberg interpreted the double slit interference basing on his interpretation of uncertainty principle in the following way.
If we hit the electron with a photon in order to determine which slit the electron has passed through, namely if we narrow the uncertainty of its transversal position , there occurs a transfer of momentum with transversal components. This transversal kick changes the flight direction of the electron. This random change of direction for each electron acummulated leads to destruction of the interference pattern.
This is however an inadequate picture. That the uncertainity of transversal momentum increase doesnt mean that a momentum transfer has occured. A transversal momentum transfer would not change merely the uncertainity of momentum namely the average spread around the mean value of the momentum but the mean value (or the expectation value as it is called in quantum mechanics)of the transversal momentum itself .
From a wave only viewpoint it is only the destruction of the phase coherence between the two wave parts (coming from left and right slit) and not the transversal momentum transfer that leads to destruction of interference pattern.
As we will discuss in the next chapter in quantum nondemolition experiments the phase difference between the wave parts coming from slit-A and slit-B is affected and the interference pattern is lost although there is a negligibly small momentum/energy transfer.
The uncertainty relation applies not only to momentum and position but it can be formulated between any two physical quantities, product of which has dimensions of Planck constant. Such variable pairs are called complementary variables. As plane wave is a wave-function form where the momentum has a definite sharp value, for each physical observable (angular momentum, energy etc.) there is a set of corresponding wave-function-forms where this observable has definite sharp values. For angular momentum for example these functions are the so called spherical harmonics that were already known to mathematicians before the quantum mechanics emerged.
These lead us to following conclusions:
1. The term collapse or reduction of the wave function is somewhat misleading. These terms are leftovers from early discussion on double slit experiments where the collapse refers to the “sudden(?)” transition of wave function from an spread form to a very localized form in a position measurement. We know however while the wave function becomes very localized in real space during a position measurement it becomes very spread in momentum space namely while dx becomes small dp increases accordingly. There is no reduction but merely a transition from one form to another form.
2. The wave particle duality is nothing but the uncertainty relation between the complementary variables momentum and position. Thus the wave particle duality that inspired also the pilotwave theories looses its originally central role. Wave (A wavefunction with a sharp wave length but extending over the whole space) and pointlike-particle (an infinitely sharp peak in space) are two extreme forms of the same wave function. The physical reality can come from time to time depending on external conditions close to either side but it newer matches exactly one of them. This but frees us from the concept of a pointlike entity complementary to wave and takes us back to the wave function only viewpoint.
This is the third meaningshift in the concept of wave particle duality. Remember now the earlier phase of the theory(chapter 7 second meaningshift in duality) where the pointlike particle was considered to be real physical entity although with uncertain position and the wave function was interpreted as a mathematical construction, a probability wave or an uncertainty wave. Now it seems the particle in the sense pointlike entity has disappeared completely and we are left only with the wave function. That there are no need for pointlike entities doesn’t mean however that there are no countable entities. That Particles in the sense countable entities exist is known since Millikan’s experiments and his discovery of elementary charge. What quantum mechanics teaches us is that the imagination of these countable entities (particles) as pointlike entities is a misconception
For long (even since demokritos) it has been regarded as self evident, as very natural fact that matter must be made up of small indivisable identical particles of limited number of types . Why do these countable entities exist in nature is a nontrivial justified question. We will come back to it when we discuss the second quantization in chapter 10.
In modern quantum mechanics there are no pointlike particles and therefore no problem of wave particle duality in the sense pointlike-extended but there is only the wave function. . In this new picture the particle number n in a system determines merely the dimensions of configurationsspace namely the number of variables the wave function depends on(3 spatial coordinates for each particle giving 3n variables) .
But if we get rid of pointlike particle and end up again with a wave-function only viewpoint what can be said about the objections discussed in chapter 6
Remember now the earlier phase of the Copenhagen interpretation (chapter 7second meaningshift in duality) where the pointlike particle was considered to be real physical entity although with uncertain position and where the wave function was interpreted as a mathematical construction, a probability wave or an uncertainty wave because of conclusion 3 at the end of chapter 6 . In this viewpoint the position measurement was interpreted as a real event where the physically real particle reveals itself in one of the probable positions. Thus the probability wave interpretation had a meaning in connection with physically real entities(pointlike particles) and physically real events(position measurement) namely predicting the probability of a physically real event that happens to a physically real entity. But if the measurement is only an indeterministic transition from one form of the probability wave (spatially extended form), to another form (a sharply localized form although never pointlike) we don’t have a physically real entities and physically real events at all . What happens to something that is itself not physically real (wave-function) is not a real physical event may it be an U process or an R processes.
It is fashionable nowadays to believe that this is what the nature teaches us and we have to show the mental maturity to accept it .
In a paper David mermin states :1
Correlations have physical reality that which they correlate does not.
Thus what exist is only this information wave and not the objects the information is about.
This is a weird redefinition of reality.
There are fortunately physicists like P.R. Wallace and E.T. Jaynes who think different and criticize the Cophenagen interpretation or its variations. 2,3
E.T. Jaynes says 3 :
It is not science but mediaval necromancy.
I agree with jaynes and Wallace . I will try to show that the nature doesn’t force us to accept such a weird definition of physical reality.
1 Mermin 1998 p.753
2 Wallace 1996
3 Barut 1980 p.42 (Contribution of E.T.Jaynes)