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I am trying to get a better intuitive understanding of the issues that surround Heisenberg’s uncertainty principle. Typically, most discussions centre on the equation:

[1] [tex] \Delta x \Delta p \geq \hbar /2 [/tex]

However, from a more classical perspective, there are said to only be 4 fundamental quantities in physics, i.e. length [x], time [t], charge [q] and mass [m]. Although I guess, in the context of quantum theory, it might be appropriate to ask whether mass [m] is an `effect` rather than a `cause`?

As such, equation [1] seems to masks these more fundamental quantities by using momentum [p=mv] given the ambiguity of mass [m] and composite nature of velocity [v] in terms of distance [x] and time [t]. Within the wider debate about the wave-particle duality, the Compton and deBroglie wavelengths seem key definitions:

[2] [tex] \lambda = h/mc = ht/mx_1 [/tex] :Compton

[3] [tex] \lambda = h/mv = ht/mx_2 [/tex] :deBroglie

As indicated, both these equations involve a velocity, which in-turn requires a measure of [x] with [t]. Of course, if mass [m] is an`effect` rather than a fundamental quantity, we might re-arrange [2] & [3] as follows:

[4] [tex] m_c = h/c \lambda = ht/x_1 \lambda [/tex] =equivalent photon `mass`

[5] [tex] m_v = h/v \lambda = ht/x_2 \lambda [/tex] =equivalent particle `mass`

As such, mass appears as a construct of [x] and [t], although there is an implicit reference to the energy of the wave with any given wavelength/frequency. I have only defined these relationships by way of clarification and reference, as I was primarily interested in trying to understand the scope of `uncertainty` in terms of these more fundamental units and from a number of perspectives:

1) Classical particle model:

If a classical mass [m] has a finite diameter, what is its exact location? I assume that the centre of mass would typically be used. However, if we define a unit volume of space that corresponded to the mass in question, be it an electron or planet, then I am assuming the peak mass distribution coincides when the unit volume aligns to the centre of the mass [v], which then effectively falls off as gaussian (?) distribution for all other locations. As such, it seems that there is a degree of `uncertainty` can be defined for even the location of a classical particle?

2) Measurement model:

If we assume an electron to be a particle, ascertaining its exact location seems problematic from the viewpoint of any practical measurement, if any interaction is required. For example, if a photon is used to measure the particle position, the photon will have energy [E=hf]. This implies that higher frequency/shorter wavelength photons will be able to more accurately locate a particle, but with the side-effect that a higher `collision` energy might actually affect the accuracy of the location we are trying to measure? Presumably, if we used lower energy photons, we would then lose resolution of the location due to the longer wavelength involved?

3) Classical wave model:

By definition, it would seem impossible to define a single fundamental wave as being located at a single point by virtue of its structure being defined by frequency and wavelength. Classically, such waves are normally described as propagating through some media with velocity [v], even if the media is actually empty space. However, while a standing wave can be described as stationary, isn’t it actually a composite superposition of 2 or more travelling waves? Again, the exact location of the standing would seem ambiguous.

4) Wave-packet model

Finally, if a particle is modelled as a wave-packet of finite length, doesn’t it still inherit all the ambiguity concerning position, as outlined in the previous cases? Equally, doesn’t a finite wave-packet have to be constructed as a Fourier series or superposition of component waves? If so, what are the component waves within the particle-wave packet, i.e. do they have any physical existence or is this purely a mathematical description?

Would appreciate any insights on offer. Thanks

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# Uncertainty with Hesienberg’s Uncertainty

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