Heisenberg’s Uncertainty Principle
The assertion of the Heisenberg uncertainty principle is that position [x] and momentum [p] of a particle cannot be simultaneously measured with absolute precision. The root of this assertion might be said to exist in the definition of the non-commutative nature of the position and momentum operator. This uncertainty can be expressed either as the product of the uncertainty in position [x] or momentum [p] or alternatively product of the uncertainty in energy [E] or time [t], although this latter definition is subject to certain caveats that will not be detailed at this stage.
So, in this context, it is suggested that Heisenberg's uncertainty principle is not making a general statement about the inaccuracy of any measurement of these quantities, but rather a more fundamental ‘uncertainty’ or 'indeterminacy’ that is inherent in the quantum description of nature.
Despite its fundamental implication, Heisenberg’s uncertainty principle is often described in terms of a practical measurement approach or conceptual wave approach. While, the practical measurement approach is somewhat easier to visualise, the more far reaching implications appear to stem from the conceptual wave approach, which possibly goes to the heart of the difference between quantum and classical mechanics. However, before outlining these two perspectives, it might be worth highlighting some general concerns regarding the nature of the uncertainty. As has been outlined, the non-commutative relationship between position [x] and moment [p] can be rationalised in terms of the following algebraic equation:
In this context [n,m] represent different points in spacetime, although the actual mathematics is usually presented in terms of a more abstract concept of linear vector space, known as Hilbert space, to be addressed in later discussions. However, at this stage, we might still question how any real observables of a system, i.e. [x] and [p], end up being associated with a complex component and how  is reconciled with the uncertainty principle in . It seems that one of the problems associated with understanding the uncertainty principle lies in the semantics stemming from the idea of the wave-particle duality. Again, it might be argued that it is the classical concept of a point-particle that is more difficult to reconcile, not only in terms of quantum mechanics, but the subsequent development of quantum field theory. In this latter context, it seems even more ‘logical’ to assume that the ‘substance’ of a particle may be a property of some form of wave process that is capable of transporting scalar energy in spacetime. If you simply accept this train of thought for the moment, a wave description implicitly has some uncertainty in its location, which we might label as [Δx] linked to the distribution of a ‘particle’ within its wavelength [λ]. Therefore, we might wish to use the basic idea associated with the Compton and deBroglie wavelengths as a baseline reference.
How we actually interpret  seems to depend on whether we apply it to the description of radiation in the form of a photon or as an electron particle in the form of a matter wave. Therefore, let us simply start by clarifying some of the variables in  in terms of a photon:
In this case, there would appear to be no ambiguity in terms of the frequency [f] of a given photon from which the wavelength [λ] can be calculated in vacuum using [c= fλ], such that the momentum [p] would appear equally certain. However, we seem to be able to calculate the momentum [p] without any knowledge of the location of a photon provided we know its frequency [f]. Therefore, in terms of the uncertainty principle in , we seem to have a situation where [Δp=0], while [Δx] might be said to be infinite, i.e. the momentum is known, while the location is totally ambiguous in normal spacetime.
Note: In practice, we only appear to know the position of a photon at a point of interaction with some other 'particle', but not on route, where even the interaction position might be subject to some degree of uncertainty given the ambiguity of the spatial distribution of the photon or particle. So, if we know [h] and know the frequency [f] of a photon is there any uncertainty about its energy or momentum?
But what about a matter-wave particle like an electron?
Note: In a similar fashion to the note above, we might question the position of a wave-matter particle if it is distributed across the wavelengths of its spatial wave structure. However, we might also question the accuracy of any measure of the energy or momentum of a wave-matter particle that is based on measuring its kinetic velocity [v], which is predicated on knowing the difference between its spatial positions at 2 points in time, possibly subject to relativistic effects,
While deBroglie's idea in  was applicable to matter waves, we have already highlighted a few anomalies in deBroglie's description, even though experiments appear to support the extension of  to matter waves. However, we might summarise the most salient issues in terms of the energy of a non-relativistic free electron:
However, the premise of  is rooted in the fact that the rest mass energy is much, much greater than the kinetic energy, when [v<<c]. However, if we now consider  in terms of kinetic energy only:
While there is some ambiguity in both  and  with respect to , the idea of a matter-wave packet that is localised in spacetime is often described in terms of a superposition of waves, each having a dispersive phase velocity [vp], which combine to form a wave packet with a group velocity [vg] that corresponds to the classical kinetic energy, as used in . However, it might be argued that there is an implicit uncertainty in the exact position [x] within the wave packet, as described, which we might generalise as follows:
What we might also realised from this description is that the kinetic velocity [v] of the wave packet is not a constant, as per a photon [c], and therefore a degree of uncertainty exists in both  and  regarding the exact location of any quantum particle, such that this uncertainty would also exist in the determination of momentum [p] based on position [x1,x2] and velocity [v=dx/dt]. Given that theory has to be ultimately verified by empirical measurements, it is unclear how any practical experiments can ever claim [Δx] to be zero, such that we might say that 'uncertainty' is only fundamental in the sense that there is no exact position within a wave-packet description. However, while the previous statements may not be accepted as mainstream, Heisenberg himself did not actually describe his principle in terms of uncertainty, but rather indeterminacy:
The term ‘uncertainty principle’ is an inaccurate translation of Heisenberg's description, as he actually describe this issue in terms of ‘indeterminacy’ rather than ‘uncertainty’. In this context, Heisenberg argued that all system observables, at the quantum level, simply have no definite value and, as such, there is no uncertainty, just an inability to assign an exact value of position [x] and momentum [p], when approaching the quantum scale.
It should also be pointed out that the wave packet, as outlined above, does not seem to have any 'physicality' within the accepted quantum description. This is possibly one of the hardest aspects of quantum mechanics to understand, as it seems to suggest that the mathematics of quantum mechanics exists independently of any obvious form of physical reality, at least, in as much as can be tangibly verified.
Note: One other issue might be raised at this point regarding indeterminacy versus uncertainty in respect to  related to the Planck constant, i.e. h=6.62607004*10-34joules.seconds and the Planck scale in general. Estimates of the size of the smallest particle within the general particle model, i.e. the electron, suggests that its size exists, at least, on a scale some 16 orders larger than the Planck scale, i.e. 10-18 metres. As such, it is difficult to understand the degree of precision being assumed about any physical measurement of position or momentum.
The basis of this approach can be linked to a famous thought experiment, also attributed to Heisenberg, in which the position of a particle is measured using a ‘microscope’. However, we might initially characterise the nature of this approach in relatively simple terms. Let us assume that we are trying to measure the position of a quantum particle using its physical interaction with photons of a known wavelength. In this context, the limit of accuracy associated with the measurement might be best achieved using very short wavelength photons, which also comes with the implication of higher frequency [f] and energy [E]:
In this context, the use of higher energy photons would also imply a greater disturbance of the position of the quantum particle during the measurement process itself. In Heisenberg’s thought-experiment, the quantum particle is illuminated by X-rays, i.e. photons, which are scattered back towards the lens of the microscope as illustrated in the diagram below:
As such, the measured position of the particle has an uncertainty given by the following approximation linked to the angular dispersion of the aperture:
Here [λ] is the wavelength of the scattered photons and [α] is the angle subtended by the particle to the microscope lens. However, the process of observation involves innumerable photon-particle collisions with some of the scattered photons entering the lens of the microscope. To enter the lens, a scattered photon with wavelength [λ] and momentum [p] must have also have a momentum component along the x-axis:
Therefore, this component of momentum must also be subject to a degree of uncertainty, because momentum is conserved when the photon scatters.
While we might expect to reduce the uncertainty in the measurement by reducing the frequency impact of the photons used to ‘illuminate’ the particle, this would only result in a corresponding reduction of spatial [x] resolution of the microscope, which will then cause an increase in the uncertainty in the position [x] of the particle. Indeed, by combining  and , we find that the uncertainty in the position and in the momentum of the observed particle are approximately related by:
This general result might be said to characterise Heisenberg's uncertainty principle, as it highlights the idea that greater accuracy in position is only possible at the expense of greater uncertainty in momentum, and vice versa. However, the precise statement of the uncertainty principle complies to the form shown in  below, which essentially represents the average uncertainty:
So through this thought-experiment Heisenberg suggested that a precise determination of position is only possible at the expense of total uncertainty in momentum, although it has to be recognised that this might only be a conceptual statement. In fact, analysis of the microscope experiment, which takes into account the Compton effect, shows that a precise determination of position is impossible. This simplified analysis of Heisenberg's microscope experiment has only attempted to illustrate the fundamental role of Planck's constant in any measurement:
We have already touched on the general issue of wave dispersion and matter waves in which a particle is described in terms of a superposition of waves. However, we have also introduced the general concept of a wave function, based on Schrodinger’s wave equation, which suggests that the wave function Ψ(x,t) will spread out spatially.
Note: The discussion entitled 'A Matter of Perspective' explores some additional issues surrounding the uncertainty principle, but where equations  and  further question the dispersive nature of matter waves.
As such, if the position [x] of a particle is measured at time [t], the resultant wave function would reflect the dispersion of the probability density |Ψ(x,t)|2 for any point in space. By the same token, the probabilistic nature of the wave function also suggests that if we repeatedly measure the position [x], even under the same conditions, each measurement would be different. Therefore, the formulation of a particle as a matter wave also appears to introduce some level of intrinsic uncertainty in comparison to the classical model of a particle. For the purposes of this discussion, we might wish to consider a wave function that can be described in terms of a Fourier superposition. By way of general introduction, we shall initial describe a wave packet localised in spacetime in terms of a 1-dimensional square pulse, which is constructed from the Fourier transform:
Fourier analysis allows any wave function to be constructed from a superposition of plane waves, which have integral wavelengths defined in terms of a wave number [k=2π/λ]:
In this case, we are simply assuming that the wave number distribution f(k) corresponds to the diagram on the left, which then has a corresponding probability density |Ψ(x)|2, as shown on the right. We can characterise this relationship by pointing out that when the wave number distribution is wider, i.e. lots of wavelength components, the probability density distribution is narrower. However, we shall proceed using a Gaussian wave packet as a general example, where the wave function is given by a Gaussian function:
We can characterise the wave number distribution and the wave function as per the two diagrams below. Here, the width [a] of the wave function becomes narrower, when the width of the wave number distribution is wider.
Based on deBroglie’s matter wave description, we can define a relationship between the wavenumber [k] and the momentum [p] of the matter wave-particle:
This wave approach suggests that when the position coordinate [x] and momentum [p] of a particle are measured, the dispersion of the momentum [Δp] will be larger in those states with a smaller dispersion [Δx] of position [x]. Therefore, in the example of the Gaussian wave packet, if we want to make the width of the wave function [a] smaller than [Δx], the dispersion of momentum [Δp] would have to be:
Again, the relationship in  turns out to only be an approximation, which when subject to more rigorous quantum analysis takes the final form of Heisenberg’s uncertainty principle, as shown in . As such, quantum mechanics seems to assert that if we measure the position [x] and the momentum [p] of a particle simultaneously, then there is an intrinsic degree of uncertainty in either the position [x] or the momentum [p]. However, as reflected in the initial section of the discussion, the indeterminacy of position [x] appears to the more fundamental issue, especially as momentum [p] of a matter wave packet is a composite value based on a change in position [x] with time [t].