Are Quantum Waves Tangible or Merely Mathematical Constructs?

[mysearch]

I guess I am trying to understand whether the description of a quantum waves aligns to any sort of physical wave description or should it only be interpreted as a mathematical construct? By way of an example and general reference, the following links make reference to http://phy240.ahepl.org/Chp5-MatterWaves-Serway.pdf" of a book called 'Modern Physics' by Serway, Moses and Moyer. Section 5.3 provides an introduction to ‘Wave Groups & Dispersion’, while section 6.2 addresses the ‘Wavefunction for Free Particle’ and the implications of a time evolving wave equation. However, this latter section is the primary focus of this post. In Chapter-6, on the page numbered 192, is the following quote:

“Notice that $\Psi$ itself is not a measurable quantity; however, $| \Psi|^2$ is measurable and is just the probability per unit length, or probability density P(x), for finding the particle at the point (x) at time (t).”​

Based on normal mechanical waves, $\Psi$ would correspond to some sort of amplitude, but the inference appears to be that this concept doesn’t apply to a quantum wave. However, based on deBroglie’s hypothesis, a matter wave is often described in terms of a wave packet that has both physical momentum $p=\hbar \kappa$ and energy $E=\hbar \omega$. However, in order to mathematically describe the wave packet many references use the idea of the superposition of continuous and infinite plane waves within a large spread of wavelengths, i.e. [k], as linked to Fourier theory. While Section 5.3 covers the basic idea, equation (6.5) in Chapter-6 states the basic dispersion relationship of deBroglie matter waves. This equation can also be cross referenced to the Wikipedia link ‘http://en.wikipedia.org/wiki/Dispersion_relation#De_Broglie_dispersion_relations"where the relationship is linked to kinetic energy, not total energy:


$$E= \frac {p^2}{2m}$$
$$\omega = \frac {\hbar \kappa^2}{2m}$$

However, if you read the sub-text to Figure 6.3, in Chapter-6, it appears to suggest that this equation is a source of dispersion of the composite matter wave packet, because each plane wave component, required to ‘construct’ the wave packet, travels at a different phase velocity based on the logic:

$$v_p = \frac {\omega}{\kappa} = \frac {\hbar \kappa}{2m}$$

However, it is not clear whether these superposition waves, in isolation, have any tangible physical existence to which physical mass [m] could be attributed, as required by the equation above. Equally, alternative approaches, which are also said to align to deBroglie’s hypothesis, appear to forward a different suggestion for the phase velocity, e.g.

$$E=mc^2=hf= \frac {hv}{\lambda}$$
$$\lambda = \frac {h}{m} \frac {v}{c^2}$$
$$v_p= \frac {E}{p} = \frac {mc^2}{mv} = \frac {c^2}{v}$$

So while we appear to have physical attributes linked to some sort of quantum wave description, it is unclear whether the wave description is representative of a quantum particle’s physical structure. So my first question is:

What velocity do the plane wave components of a wave packet propagate and do they have any physical inference?

In Chapter-6, there is an example (6.3) relating to the construction of a wave packet that is based on the assumption of a Gaussian distribution of an initial wave packet at (x=0) and the spread of its component wave numbers [k]. Without questioning either the assumptions or the mathematical derivation, at this stage, we arrive at an equation that appears to allow the dispersion time of a matter wave packet to be estimated, see Example 6.4, for an electron and a 1g marble, i.e.

Electron = $1.7 * 10^{-15}seconds$
1g Marble = $6 * 10^{16} years$

Under these examples is the quote:”

“The localization of an atomic electron is destroyed in a time that is very short, on a par with the time it takes the electron to complete one Bohr orbit.”​

Again, I would like to try to clarify what, if any, physical interpretation can be extrapolated from these figures and the quote above in contrast to the perception that a classical free particle will simply remain localised in space in the absence of any force:

Is the quantum description only saying that the mathematically probability of the location is dispersed or is there some inference of the physical dissipation of a particle as a waveform?

How does the structural composition of the marble that is made up of quantum particles, i.e. electrons, protons and neutrons, circumvent the dispersion process that appears to affect each quantum particle in isolation?

Can the idea of the dispersion of transverse superposition of waves be extended to 3-dimensions?

Appreciate any knowledgeable insights or links to alternative references that might be thought to address these issues. Thanks.

 

[Black Integra]

Doesn't like classical mechanics, where we could interpret a wave in physical ways
for example, we can write the wave function of sound as the 'pressure' or 'displacement' function

$\Psi(x,t)=P_{max}sin(kx-\omega t+\phi_{p}) = S_{max}sin(kx-\omega t+\phi_{s})$

But in Quantum mechanics, there're NO way to describe the amplitude of wave function in a physical unit.
Sometimes we call the wave function as 'probability amplitude'

The wave function's amplitude is considered to be a probability that we will found the particle in that 'state'.
like

$\int^{t=-5}_{t=10}|\Psi(x=0,p=1)|^{2} = 0.75$

"The probability that a particle will be placed at position x=0 while its momentum is 1 during the time t=-5 to t=10 is 0.75"

So the wave velocity is the changing of wave's position in various time

sorry for my bad english

 

[mysearch]

Appreciate the feedback …

Doesn't like classical mechanics, where we could interpret a wave in physical ways…..The wave function's amplitude is considered to be a probability that we will found the particle in that 'state'.

Fair enough, I agree that aligning the square of the wave function amplitude to the probability density is pretty much the accepted position as I understand it. If so, it would appear that the wave function is a mathematical construct that does not necessarily say anything about physical wave structure of a particle. So are there any models of the ‘physical’ wave structure of a quantum particles?

…we can write the wave function of sound as the 'pressure' or 'displacement' function

As an aside, a sound wave is an interesting example of a 3-dimensional wave. Interesting in the physical sense because it propagates as a longitudinal pressure wave in 3-D. As such, it is not clear to me how a particle wave, if such a thing exists, could be described as a superposition of transverse waves, if restricted to 3-D space?

 

[Bill_K]

mysearch, Your difficulty is one that newbies to quantum mechanics often have, namely trying to apply an overly mechanistic interpretation. You're thinking of Ψ as "only a mathematical feature" of a particle that has "some other physical structure," whereas actually Ψ is the entire ballgame. There is no physical structure behind it. Particles in quantum mechanics are mathematical points. They have mass, possibly spin, possibly charge, but no other properties. They are not made of anything, so they cannot "disperse". Because there is nothing to be dispersed.

The probability amplitude Ψ(x,t) is a complex quantity that obeys the Schrodinger Equation. You can write down plane wave solutions to that equation, and wave packets which disperse. But Ψ is neither longitudinal like a sound wave nor transverse like an electromagnetic wave, because it is only a complex number, not a displacement.

 

[Maui]

"What is a Quantum Wave?" is a deeper, more sophisticated variation of the question "what is matter"? Sorry i cannot help, i don't think anyone else can really help you with this, other than offer an opinion.

I guess I am trying to understand whether the description of a quantum waves aligns to any sort of physical wave description or should it only be interpreted as a mathematical construct

In some situations it seems to have characteristics of both(as you surely know) but i am not a physicist(just an enthusiast looking to get a glipse at the deeper, poorly understood and misunderstood nature of matter)

 

[mysearch]

Thanks for the responses in posts #4 and #5, as both are interesting in different ways.

mysearch, Your difficulty is one that newbies to quantum mechanics often have, namely trying to apply an overly mechanistic interpretation. You're thinking of Ψ as "only a mathematical feature" of a particle that has "some other physical structure," whereas actually Ψ is the entire ballgame. There is no physical structure behind it.

Hi Bill, while I did study some quantum mechanics at university, some 40 years ago, I won’t argue with the label ‘newbie’ as I have pretty much started from scratch. However, I have some difficulty in accepting your position that ‘Ψ is the entire ballgame’ because it seems to deny the existence of any sort of physical reality. Replacing the concept of a particle with a wave structure makes sense, at least to me, when you can’t identify the ‘substance’ of a sub-atomic particle, but to say that the waveform is ‘only a mathematical feature’ while having an associated momentum and energy may suggest that quantum mechanics is still a limited description of physical reality, even though it can produce very accurate results.

"What is a Quantum Wave?" is a deeper, more sophisticated variation of the question "what is matter"? Sorry i cannot help, i don't think anyone else can really help you with this, other than offer an opinion.

I think this is a very honest answer and a position that modern physics does not always make clear to us ‘newbies’.

Particles in quantum mechanics are mathematical points. They have mass, possibly spin, possibly charge, but no other properties. They are not made of anything, so they cannot "disperse". Because there is nothing to be dispersed. The probability amplitude Ψ(x,t) is a complex quantity that obeys the Schrodinger Equation. You can write down plane wave solutions to that equation, and wave packets which disperse. But Ψ is neither longitudinal like a sound wave nor transverse like an electromagnetic wave, because it is only a complex number, not a displacement.

From an ‘ongoing’ learning perspective, I have approached quantum mechanics in some sort of chronological order, i.e. Planck, Einstein, Bohr, Compton, deBroglie, Schrodinger, Dirac, Born. As such, I am only up to about 1926, although it seems that most of the basic concepts are in place by this time? In post #1, I referenced http://phy240.ahepl.org/Serway-chp6-QM-in-1D.pdf" of a book on quantum mechanics, which focuses on Born’s interpretation of a quantum wave as a probability density. It seems that in order to align the ‘mathematical’ probability density to the perceived ‘physical’ location of the particle, the mathematical concept of Fourier superposition appears to be used to create a localised wave packet? Note, these are questions not statements on my part. As far as I can see, the plane wave components do not seem to have any physical reality, which then aligns to your position, although the reality of some sort of wave packet may still be open to debate. Anyway, I would really like to better understand the logic of Example 6.4: Dispersion of Matter Waves and in particular the derivation of the following equation:
$$\Delta x(t) = \sqrt{ \left[ \Delta x(0) \right]^2 + \left[ \frac{\hbar t}{2m \Delta x(0)} \right]^2 }$$
As far as I can see, this equation is linked to dispersion equation rooted in deBroglie’s hypothesis, i.e.
$$\omega = \frac {\hbar \kappa^2}{2m}$$
If ‘there is nothing to be dispersed’ what are these equation referring to? Would appreciate any insights you may be able to provide into any of the questions raised. Thanks

 

[jtbell]

I have some difficulty in accepting your position that ‘Ψ is the entire ballgame’ because it seems to deny the existence of any sort of physical reality. Replacing the concept of a particle with a wave structure makes sense, at least to me, when you can’t identify the ‘substance’ of a sub-atomic particle, but to say that the waveform is ‘only a mathematical feature’ while having an associated momentum and energy may suggest that quantum mechanics is still a limited description of physical reality, even though it can produce very accurate results.

The problem is that there is no general agreement about what might lie "underneath" standard quantum mechanics, or whether there is even anything at all "underneath". This is the subject of various interpretations of QM. The viable ones all make the same predictions for experiments that we can do, and as far as I know, they don't make any predictions that differ for experiments that can be done in principle; therefore they cannot be distinguished experimentally.

 

[PhilDSP]

Electron = $1.7 * 10^{-15}seconds$
1g Marble = $6 * 10^{16} years$

Under these examples is the quote:”

“The localization of an atomic electron is destroyed in a time that is very short, on a par with the time it takes the electron to complete one Bohr orbit.”​

How does the structural composition of the marble that is made up of quantum particles, i.e. electrons, protons and neutrons, circumvent the dispersion process that appears to affect each quantum particle in isolation?

One thing to keep in mind is the difference between frequency or spectrum dispersion (which de Broglie normally refers to) and wave amplitude dispersion (which you apparently are considering with the marble example)

Frequency dispersion:
$$\omega = \frac {\hbar \kappa^2}{2m}$$

Wave amplitude dispersion (dissipation):
$$\Delta x(t) = \sqrt{ \left[ \Delta x(0) \right]^2 + \left[ \frac{\hbar t}{2m \Delta x(0)} \right]^2 }$$

I believe de Broglie phrased it (at least in his later material) such that the particle is the singularity at the focal point of the wave packet. The particle would be expected to have little, if any, spatial extension so that an atom or molecule or large assembly of molecules would be associated with a mesh of waves - a separate one for each electron or nucleus in the system, rather than a single one for the entire assembly. But the application of the Schroedinger equation results in an interaction between particles so the mesh of waves would be at least be partially intertwined according to Schroedinger. Even then, it doesn't sound particularly valid to calculate the wave characteristics of a marble in that way.

If there really is both a particle and a wave associated with it as de Broglie stated, then it would stand to reason that the wave would dissipate in intensity or amplitude in space the further away from the particle you are.

 

[mysearch]

One thing to keep in mind is the difference between frequency or spectrum dispersion and wave amplitude dispersion.
Frequency dispersion:
$$\omega = \frac {\hbar \kappa^2}{2m}$$

From the description given in http://phy240.ahepl.org/Serway-chp6-QM-in-1D.pdf" , I thought it was suggesting that the wave packet is a construct of plane waves, which have a specific frequency/wavelength within a Gaussian distribution. However, because these plane wave components are assumed to have a different phase velocity, the superposition of these waves leads to an amplitude dispersion, which in the context of quantum mechanics is interpreted in terms of the square of the amplitude relating to the location probability density. Again, this is not a statement, but an issue for clarification. I was partly alluding to this issue in post #1 when asking about the interpretation of the phase velocity of the plane waves in superposition needed to create a localised wave packet. Here’s the question again:

What velocity do the plane wave components of a wave packet propagate and do they have any physical inference?

While the general consensus appears to be that quantum waves only have a mathematical inference, some sort of notional idea of velocity appears to be applied to them – see post #1 and below. However, as pointed out in post #6, there are ‘various interpretations of QM’, which might be segregated into ontological or epistemological differences, although my issue is far more mundane, as I simply cannot reconcile any concept of physical reality based solely on a mathematical construct – maybe I am too old school for this type of physics or have simply missed a key point – again clarifications welcomed.

Wave amplitude dispersion (dissipation):
$$\Delta x(t) = \sqrt{ \left[ \Delta x(0) \right]^2 + \left[ \frac{\hbar t}{2m \Delta x(0)} \right]^2 }$$

As also raised in post #6, I am not sure how the authors derived this equation, i.e. has it got anything to do with standard deviation? Here is a different variant that appears to leads to the same dispersion times given in Chapter-6, although its logic is suspect:
$$v_p = \frac {\Delta x(t)}{t} = \frac {\omega}{\kappa}= \frac {\hbar \kappa}{2m}; \ \ \ \ where \ \Delta \kappa = \Delta x(0)$$
$$t = \frac {\Delta x(t) \Delta x(0) (2m)}{h}$$

I believe de Broglie phrased it (at least in his later material) such that the particle is the singularity at the focal point of the wave packet. The particle would be expected to have little, if any, spatial extension so that an atom or molecule or large assembly of molecules would be associated with a mesh of waves - a separate one for each electron or nucleus in the system, rather than a single one for the entire assembly. But the application of the Schroedinger equation results in an interaction between particles so the mesh of waves would be at least be partially intertwined according to Schroedinger. Even then, it doesn't sound particularly valid to calculate the wave characteristics of a marble in that way. If there really is both a particle and a wave associated with it as de Broglie stated, then it would stand to reason that the wave would dissipate in intensity or amplitude in space the further away from the particle you are.

I am not sure how to interpret this in terms of the superposition of ‘continuous’ plane waves, which appear to have no obvious physical existence. I am assuming that a marble made up of billions of protons, neutrons and electrons might still be conceptually described as a superposition of waves, but also assume this would be as difficult as describing a thermodynamic system in terms of individual collisions rather than a statistical aggregate?

Anyway appreciate the feedback. Thanks

P.S. By way of a tangential issue, I was reading yet another article that floated the idea that a photon, having zero proper time when observed from any physical frame of reference, had no real existence in normal spacetime. As such, when measured, its wave function would collapse, but having no particle state, its energy has to be absorbed within some larger quantum system. I guess my first question is whether this makes any sense to those of you who understand quantum mechanics and second whether any work has discussed the idea of quantum matter waves 'existing' outside normal spacetime?

 

[PhilDSP]

What velocity do the plane wave components of a wave packet propagate and do they have any physical inference?

In Maxwell's theory, the nominal velocity of wave propagation is of course c. But it is known that the effective linear velocity is reduced in the presence of charge. See the Ewald-Oseen Extinction Theory or J. J. Thomson's monograph on the subject for rigorous descriptions. In “Quantum Theory”, Bohm provides a qualitative description of how that works for a hydrogen atom. The velocity of waves in the vicinity of the atom is highly dependent on the distance to the atom's center.

It seems de Broglie didn't think in those terms, yet they are implicit in the dispersion relations. What de Broglie was concerned with was the velocity component in the direction of the particle's trajectory, which is the group velocity for the wavefront as a whole in his proposal. Since Fourier components of the wave are evaluated at the same spatial point or region, they must share the same values of the dispersion relation (the uncertainty principle should affect the potential variation or uncertainty of the values in total but not the inter-component relationships). Therefore the common group velocity applies to each Fourier component also.

The physical reality of Fourier decomposition is very clearly shown by the fact that, using filters, we can remove components at will and the resulting wave will be exactly that described by Fourier analysis. Similarly, wave generators for separate components can be aligned spatially so that the resulting superposition equals exactly that of the wave described by Fourier composition.

 

[mysearch]

As always, appreciate the knowledgeable insights, but I am not entirely sure I understand the scope of some of the comments regarding Fourier component waves:

What de Broglie was concerned with was the velocity component in the direction of the particle's trajectory, which is the group velocity for the wavefront as a whole in his proposal. Since Fourier components of the wave are evaluated at the same spatial point or region, they must share the same values of the dispersion relation. Therefore the common group velocity applies to each Fourier component also.

Does the description above align to that given in http://phy240.ahepl.org/Serway-chp6-QM-in-1D.pdf" or is it expressing a different view? For example, the following quote is taken from the paragraph following equation 6.8:

“Subsequently, the packet develops according to the evolution of its plane wave constituents. Because each of these constituents moves with a different velocity vp=w/k (the phase velocity), the wave packet undergoes dispersion (see Section 5.3) and the packet changes its shape as it propagates (Fig. 6.3b). The speed of propagation of the wave packet as a whole is given by the group velocity dw/dk of the plane waves forming the packet.”​

Another thing that is not really clear to me in the discussion of quantum matter wave packets, constructed of Fourier component waves, is how this construct is extended to 3-dimensions. From a purely visual perspective, 3-dimensional mechanical waves are often described in terms of longitudinal pressure waves, e.g. sound waves, radiating outwards. In the specific case of quantum wave packets, is it valid to assume that the 3-D components of quantum matter waves are propagating towards and away, i.e. through, a point in space from all directions, irrespective whether this is viewed as mathematical or physical construct?

The physical reality of Fourier decomposition is very clearly shown by the fact that, using filters, we can remove components at will and the resulting wave will be exactly that described by Fourier analysis. Similarly, wave generators for separate components can be aligned spatially so that the resulting superposition equals exactly that of the wave described by Fourier composition.

When you say “the physical reality of Fourier decomposition” are you talking in general or specifically in terms of the components of a quantum wave packet. The reason I ask is that I got the impression that most people only see the Fourier component waves of a quantum matter wave packet as a mathematical construct. Thanks

 

[PhilDSP]

Does the description above align to that given in http://phy240.ahepl.org/Serway-chp6-QM-in-1D.pdf" or is it expressing a different view? For example, the following quote is taken from the paragraph following equation 6.8

The description I gave is really not aligned with the textbook. It assumed the perspective of traveling with the particle a specific distance away whereas it looks like the textbook considers a fixed point in space with time varying as the particle and wave travel past. In that case the velocities will change in time because different permittivity and permeability are being encountered.

It seems easier to understand and visualize things traveling with the particle (where the dispersion relation values are fixed) and consider how incident waves are affected, at least to be more aligned with what de Broglie was trying to convey (possibly). But in thinking about now, it seems you need both viewpoints because the fixed point in the textbook exposition is presumably in the same reference frame as the particle that is emitting radiation towards the particle that is traveling past.

When you say “the physical reality of Fourier decomposition” are you talking in general or specifically in terms of the components of a quantum wave packet. The reason I ask is that I got the impression that most people only see the Fourier component waves of a quantum matter wave packet as a mathematical construct. Thanks

I'd consider those statements as being general to any wave phenomenon. Is there really any reason to imagine de Broglie waves being any less physical than optical waves or radio waves or infrared waves for example?

 

[mysearch]

Is there really any reason to imagine de Broglie waves being any less physical than optical waves or radio waves or infrared waves for example?

Personally, I think this is one of the central questions that confuses many people when starting to review quantum mechanics, i.e. are we only talking about a mathematical model that simply gives answers that align to experimental data or a more profound description of physical reality? Reading around this subject is not always helpful, as there appears to be so many interpretations, both scientific and philosophical. For example, at the Solvay Congress of 1927, Heisenberg and Born were reported to have stated:

“We regard quantum mechanics as a complete theory for which the fundamental physical and mathematical hypotheses are no longer susceptible of modification.” ​

However, it is not totally clear whether they meant ‘complete’ in terms that it is as far as we can go into the quantum realm or ‘complete’ in the sense that QM is both a mathematical and physical description of the quantum realm?

…actually Ψ is the entire ballgame. There is no physical structure behind it. Particles in quantum mechanics are mathematical points. They have mass, possibly spin, possibly charge, but no other properties. They are not made of anything, so they cannot "disperse". Because there is nothing to be dispersed.

While not wishing to misrepresent Bill_K words, my interpretation of his comment above was that he considered the quantum function, and presumably quantum waves, to only be a mathematical construct. However, I was particularly struck by the range of opinion in another PF thread entitled ‘https://www.physicsforums.com/showthread.php?t=541962" from which the following quote is taken:

“So, what is quantum mechanics? Even though it was discovered by physicists, it's not a physical theory in the same sense as electromagnetism or general relativity. In the usual ‘hierarchy of sciences’ with biology at the top, then chemistry, then physics, then math, quantum mechanics sits at a level between math and physics that I don't know a good name for. Basically, quantum mechanics is the operating system that other physical theories run on as application software (with the exception of general relativity, which hasn't yet been successfully ported to this particular OS). There's even a word for taking a physical theory and porting it to this OS: "to quantize. But if quantum mechanics isn't physics in the usual sense, if it's not about matter, or energy, or waves, or particles, then what is it about? From my perspective, it's about information and probabilities and observables, and how they relate to each other.”​

While Aaronson is clear in his own views about what quantum mechanics is, it is unclear whether he addresses the question as to what physicists, not mathematicians, would really like quantum mechanics to be. From my personal perspective, I would like physics to help me understand the fundamental nature of reality and, in this general context, some form of wave theory appears to be the most likely as it is the only mechanism I know that can transport energy, if you reject the idea of particles at some point within the quantum realm. Of course, this would again return to the question:

What is a quantum wave?

 

[fizzle]

The physical reality of Fourier decomposition is very clearly shown by the fact that, using filters, we can remove components at will and the resulting wave will be exactly that described by Fourier analysis. Similarly, wave generators for separate components can be aligned spatially so that the resulting superposition equals exactly that of the wave described by Fourier composition.

A filter doesn't really "extract" a Fourier component, it simply responds to the overall input excitation and produces output based on the filter's characteristics (with phase and amplitude changes, etc.).

Also, isn't superposition mainly a mathematical/accounting concept? If I add two waves together, are there really two waves there or just a single value? I'd say there's just a single value. The best example is where you have two electromagnetic waves whose E and B completely cancel -- superposition says that there are two waves at a point but reality says that there's nothing there.

 

[PhilDSP]

A filter doesn't really "extract" a Fourier component, it simply responds to the overall input excitation and produces output based on the filter's characteristics (with phase and amplitude changes, etc.).

You're talking about implementation issues (which is probably off-topic). A filter can be casual (as you apparently are thinking) or acausal. Causal filters work as you describe but acausal filters are not so constrained and may operate with prior knowledge of the signal attributes.

Also, isn't superposition mainly a mathematical/accounting concept? If I add two waves together, are there really two waves there or just a single value? I'd say there's just a single value. The best example is where you have two electromagnetic waves whose E and B completely cancel -- superposition says that there are two waves at a point but reality says that there's nothing there.

Superposition is just the assumption of additive linearity within each dimension for each attribute under consideration.