Quantum Wave Interpretation
In the story, so far, we have covered the initial idea that matter particles might also be described in terms of matter waves. However, the subsequent discussion of Schrodinger’s wave mechanics and Heisenberg’s uncertainty principle has possibly introduced some unsettling ideas, which may have thrown doubt on the actual nature of the waves in question; especially when described in terms of a wave function [Ψ]. As such, there is a question we cannot really continue to ignore:
What is the wave function [Ψ] actually describing?
There appears to be some, but not necessarily full, consensus that [Ψ] itself is not a measurable quantity, although |Ψ|2 is measurable, when interpreted as the probability per unit length, or probability density P(x) for finding the particle at the point (x) at time (t). As a consequence, it might be said that the wave function contains sufficient information about the particle to allow the probability of finding a particle in a given location in space. While it may be argued that this is a succinct definition, it does not seem to really address the spirit of the question raised above; so let us try to see if a more meaningful descriptions is possible. While a mechanical (SHM) wave would associated [Ψ] with some sort of amplitude, it is not clear that quantum mechanics is making any inference that can be anchored in physical reality. This said, based on deBroglie’s hypothesis, a matter wave packet is assumed to have the physical attributes of both momentum [p] and energy [E], which in terms of mechanical waves would be proportional to the wave amplitude.
In 1926, shortly after Schrödinger’s initial publication of wave mechanics, Max Born applied Schrödinger’s ideas to the issue of atomic scattering, which appeared to provide a method for determining the probability of a particle being scattered into some solid angle. However, the far more reaching aspect of this work was the interpretation of [Ψ] being linked to the probability density of a particle. In this context, the probability density simply helps to define whether a particle will be found in some interval [dx] about point [x], which we will quantified as follows:
So while it might be said that the square of [Ψ] is still somehow representative of the intensity of the matter wave, it is no longer a measurable quantity, such that [Ψ]2 is now said to only described the probability density P(x) for finding the particle at a point [x] at time [t]. In the context of mathematical probability, it is common practice to normalise the sum of all probabilities over all values of [x]:
We might therefore generalise the form of  to define the probability of finding a given particle in any finite interval [a<x< b]:
Of course, the form of  still appears to be a very abstract description, which does not seem to say too much about the actual form of wave the function [Ψ]. However, at this point, we need to be careful about specifying an actual wave function, as each system under consideration might take a different form. As such, the following example is simply illustrative of a wave function [Ψ] for which we want to determine the probability density:
The expression in  is representative of a generic Gaussian distribution curve, where [a] defines the peak height, [b] define the position of the central peak and [c] controls the width of the distribution. However, for the purposes of this discussion, we can simplify the form of  as follows:
In order to comply with the form of , we need to integrate the square of  over all space [x]. The square of the power term in  is just an additive process, while the infinite integral is a standard solution known as the Gaussian integral:
However, the form of  needs to be modified so that the total probability density equals unity.
Clearly, in this specific case, the coefficient [C] must be equal to the reciprocal of the square root of [π], such that total probability density, represented by the red shaded area in the diagram, equals unity. While the result in  is only intended to be illustrative of the mathematical process, it also suggests that the wave function Ψ(x,0) might represent some initial state of the wave-particle, which then propagates according to the general principles of Schrodinger’s wave equation. As such, we now need to consider the time evolution of the wave function.
Intuitively, we might initially assume that the description of a particle, as a wave, must be distributed over the length of the wave packet [Δx]. We might also assume, from a classical perspective, that the probability density, as previously described, would remain essentially constant as the particle continues to exist along some trajectory in space-time.
But does this assumption hold true in quantum theory?
In order to reflect on the question above in a little more detail, let us consider the case of a free particle, i.e. one that is subject to no force. Of course, in the current context, we want to quantify the particle in terms of its wave-like attributes. Therefore, we might define both the wave number [к] and the angular frequency [ω] in terms of the momentum [p] and energy [E] of a particle, based on the deBroglie relationships:
However, we have previously shown that deBroglie matter waves are predicated on the energy being linked to the kinetic energy of the particle, which leads to a dispersive relationship between the wave number [к] and the angular frequency [ω]:
The wave number [к] and the angular frequency [ω] can now be linked to the development of a wave function [Ψ] through either a standard trigonometric function or an equivalent complex form, e.g.
It is known that if the spatial variable [x] and time variable [t] only occur only within the form [кx-ωt], the oscillation normally represents a wave propagating in space and time, which might be be consistent with the idea of a free particle moving with a constant velocity, i.e. no force. However, the form of  is describing a continuous plane wave conceptually extending to [±] infinity. While this is a reasonable starting point, assuming that space-time puts no special restriction on the location of a particle, we need a wave mechanism that will produce a localised wave packet with a probability density similar in scope to that previously outlined in . In this context, we have already outlined how the superposition of plane waves can create a wave packet of any shape; although the animation below does not explicitly consider its evolution with time, because the issue of dispersion has not been taken into consideration:
For simplicity we could again assume a Gaussian distribution of plane wave numbers [к] within the superposition wave packet. As such, the wave function Ψ(x,0) woud initially describe a wave packet localised in some finite interval, as previous suggested by the animation above. So, in order to mathematically construct a localised wave packet, we need a linear superposition of plane waves, conforming to , but where each wave has a different wave number [к] conforming to a Gaussian distribution. As such, we will need to integrate the form of  over all values of [к]:
The coefficients, implied by a(k), specifies the amplitude of the plane wave with different values of the wave number [к] needed to produce the required shape of the wave packet. However, the dispersive relationship associated with matter wave, as given in , suggests that each plane wave would have to propagate at a different velocity [v]:
The implication in  is that the component waves in superposition cannot maintain a constant phase to one another and would result, in contradiction to the animation above, in a wave packet that changed shape as a function of time [t], as illustrated in terms of the following Gaussian distributions:
Based on Fourier theory, we might initially construct a localised wave packet using a Gaussian superposition of [n] plane waves. However, these waves would each propagate at a different velocity leading to an evolution of the wave packet with time:
- (a) is the initial localised wave packet Ψ(x, 0), which
represents a Gaussian distribution of [n] plane waves in
superposition at [t=0].
- (b) represents the same wave packet, at some later time [t].
The apparent distortion of the wave packet is caused by the fact that
plane waves with smaller wave number [к] propagate at a slower
velocity [v], as defined in .
- (c) shows that the probability density function must also spread out, such that probability of locating the particle ultimately extends over all space.
It is not obvious as to how we should physically interpret the suggestions outlined above. At one level, the wave function Ψ(x, 0) in (a) and (b) appear to be linked to the conceptual amplitude of the wave packet, i.e. the superposition wave; while the probability density function |Ψ(x, 0)|2 in (c) appears to suggest that the location of the matter particle is somehow dispersed over an increasingly large region of space, as illustrated by the spread of the blue pulse with respect to the original black pulse in the following animation:
However, if the wave function is only a mathematical concept, is there any physical meaning to the dispersion being described?
At one level, the superluminal phase velocities of the component waves might be questioned in terms of physicality, such that we might also question the physical meaning of the spread of the wave packet. However, the concept of momentum and energy is still linked to the group velocity of the wave packet, which remains unchanged, even though the animation above implies the wave packet to be spreading in space. Of course, this 'logic' seems to beg the basic question:
If wave packet is an effect and the cause is the superposition of phase waves; how can the the effect be real, if the cause is not?
However, in the absence of any immediate answer to this question, at this point, let us simply return to the idea of Heisenberg’s uncertainty in which a narrow wave packet [Δx] requires a broad spectral content [Δк] and vice versa. As such, we might quantify this relationship in the form [ΔxΔк≈1]. Therefore, if we assume that the superposition of each plane wave within the wave packet propagates independently, the wave packet at any time [t] is defined by the wave function that accounts for the dispersion expression given in .
So, while we might establish an initial condition that conforms to the first diagram above, each [к] component wave moves with a different velocity such that the wave packet would appear to disperse as a function of time [t]. However, if we are to actually solve the wave function originally presented in , we need to rationalise an expression for the function a(к). Again, we might simply assume that the distribution of the wave numbers [Δк], required by the wave packet, follows some basic Gaussian distribution.
As such, we can now substitute  into  to give:
However, before we can evaluate the integral we need to complete the square in the exponent as follows:
While  may not look any simpler, the second term on the right is now constant with respect to [к], such that it can be moved outside the integral and the remaining expression reduced to a variable [z]:
We can now present the integral in  as follows:
We can now see that  has the same form as  and, as such, the integral has the standard solution known as the Gaussian integral. Therefore, the wave function can now be reduced to the form:
If we compare  with , we see that we still have a Gaussian function, which has a single maximum at [x=0] and decays smoothly to zero on either side of this point, as illustrated in the first diagram below:
The Gaussian wave function in  is representative of a particle initially localized around [x=0] with an amplitude [C]. At [x±2α], the amplitude falls from its maximum value by a factor [1/e], such that we might associate [α] with the spatial variable [Δx]. Of course, the implication of the diagram, which will be interpreted as the time evolution of the wave function Ψ(0,t), such that we might present  in the form:
Again, we might substitute for a(k), based on , and solve the integral, as per  using the standard Gaussian integral:
Again, we might question how quantum mechanics, which appears to be predicated on the principles of normal wave propagation, leads to such a fundamentally different interpretation. In terms of mechanical waves, the square of the amplitude of the wave is proportional to the energy. However, if we tried to extend this idea into quantum mechanics, the implication would be that the rest mass energy of a matter wave would also be dispersed. Clearly, there is an important difference in the interpretaion of quantum waves, such that it may be useful to consider some actual examples of quantum dispersion.
What we might recognise in  is that the wave function retains its Gaussian distribution, while reflecting some aspect of the dispersion relationship given in . However, at this point, we might try to quantify the dispersion of a matter wave through an example using an approximation based on .
So, by way of an example, let us assume a free electron is initially localized to a region of space 0.10 nm wide and then try to approximate how much time will elapse before the localization of the electron probability amplitude is lost to dispersion. However, in order to continue with the approximation, we will need to approximate a value for Δx(t), which we might simply define as some reasonably large multiple of Δx(0), e.g. 10:
We might also try to compare the dispersion time of the electron above to that of a 1g marble localised within 0.1mm.
Based on the approximation in , the localization of the electron probability density is destroyed in about the same time it takes the electron to complete one Bohr orbit. In comparison, the localisation of the 1g marble, given in , can be converted to about 9.55*1015 years. So, while sub-atomic particles are constrained within the composite mass of the marble, the quantum effects of dispersion disappear on any measurable time scale. Of course,  was only a crude approximation of a Gaussian wave function; however, it should also be noted that the while Gaussian wave packets are often used to represent the initial system state, this is itself only a simplifying mathematical assumption, which may in-turn only be a crude approximation of any quantum reality, whatever that may mean. While we shall leave the discussion at this point, we might wish to table some questions:
Do quantum waves reflect any form of physical reality or are they
just a mathematical construct?
If only a mathematical construct, what can be inferred from the description of dispersion?
Through what process does the quantum composition of the marble circumvent the dispersion process that appears to affect each quantum particle in isolation?