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Uncertainty with Hesienberg’s Uncertainty

  1. Apr 12, 2009 #1

    mysearch

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    Hi,
    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
     
  2. jcsd
  3. Apr 12, 2009 #2

    alxm

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    The Heisenberg uncertainty principle is a quantum-mechanical thing, so if you're going to try to elucidate/understand/criticize it, you should be doing so from the standpoint of quantum mechanics and nothing else.

    In short, every system can be described by a thing called the wave function, where observable properties of the system are eigenvalues of Hermetian operators acting on the wave function. Two observables cannot be simultaneously determined through measurements if the two corresponding operators do not commute, hence you have an uncertainty relationship for non-commuting operator pairs.

    Given a spatial wave function, the momentum operator is derived from the postulate that momentum is conserved, which is equivalent to the wave function being invariant under an infinitely small spatial displacement. Do a little math and you show that this means the momentum operator must be proportional to [tex]-\nabla[/tex]. Take the limit to the classical case and you find that the proportionality is [tex]-i\hbar\nabla[/tex].

    Momentum is thusly not a 'composite' of speed, distance, mass etc. Just a consequence of assuming invariance under a spatial displacement as the displacement becomes infinitely small.
     
  4. Apr 13, 2009 #3

    mysearch

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    I am only trying to understand. From this perspective, I am not sure that the suggestion to assimilate quantum mechanic purely from its perspective is realistic for many people coming to this subject for the first time. While I accept that `quantum maths` is an important aspect of this subject, it is not one that is easily mastered and can become increasingly abstracted from the physics if not anchored in some sort of physical interpretation. In part, this was why I outlined 4 models with different perspectives to try to understand whether the uncertainly existed in classical model, physical measurements or the description of particles as waves.

    Currently, I only have a limited understanding of eigenvalues, Hermetian and momentum operators et al, so some of your another suggestions will take some time to decode, but would appreciate if anybody could elucidate further on the following definition of momentum:
    Again, in part, I was trying to see if any physical clarification would come about by reducing composite quantities, such momentum and velocity, to more fundamental units, such as distance and time. I would also be interested how quantum mechanics defines mass within any wave description. The reason for asking is linked to the following statement from Wikipedia

    http://en.wikipedia.org/wiki/Uncertainty_principle
    “In quantum mechanics, a particle is described by a wave. The position is where the wave is concentrated and the momentum is the wavelength. The position is uncertain to the degree that the wave is spread out, and the momentum is uncertain to the degree that the wavelength is ill-defined.”
    Based on deBroglie’s wavelength definition, see equation [3] post-1, I would have thought it was more accurate to say that momentum was inversely proportional to wavelength? Also see equation [5] for the mass/wavelength correspondence. Equally, from the perspective of fundamental units, wavelength is a measure of distance, so is the uncertainty of both position and momentum rooted in determining position?

    Initially, when I searched the forum, I missed what appears to be the most extensive, if somewhat argumentative, discussion of the uncertainty principle, so will provide the link for anybody, like me, simply trying to understand the issues a little better:
    https://www.physicsforums.com/showthread.php?t=297557

    While I still need to review all the arguments in this thread in more detail, I would be interested in any further thoughts about the debate that seemed to be centred on whether uncertainty was purely a phenomenon of the quantum model or whether it also exists in the classical model and what are the fundamental differences between them. Thanks
     
  5. Apr 15, 2009 #4

    mysearch

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    As a beginner's attempt to try to understand the nature of `uncertainty` as being defined in a number of contexts, i.e. classical, wave and quantum physics, I have been looking at a number of references. While I would like to understand the quantum perspective better, I do not currently have the necessary maths background to fully interpret the physical meaning from a pure quantum mechanical/maths derivation based on Hermetian operators, eigenvalues etc, but was still hoping that I might gain some initial insight that was meaningful. One problem is that there are many descriptions of a type of `Uncertainty Principle` that appears to be based on an analogy that other texts then refute within the context of `Heisenberg’s Uncertainty Principle (HUP)`, e.g.

    http://en.wikipedia.org/wiki/Uncertainty_principle
    “The uncertainty principle is often explained as the statement that the measurement of position necessarily disturbs a particle's momentum, and vice versa—i.e., that the uncertainty principle is a manifestation of the observer effect. This common explanation is incorrect, because the uncertainty principle is not caused by observer-effect measurement disturbance.”

    The following link to a definition of the `uncertainty principle` is taken from the PF library, but it is unclear whether this outline derivation is not an example of the HUP being explained in terms of measurement inaccuracy?
    https://www.physicsforums.com/library.php?do=view_item&itemid=22

    In contrast, the next link appears to be discussing HUP from the general perspective of a wave model, but highlights the fact that quantum mechanics is only using aspects of wave physics to describe probability density in terms of a wave function:
    http://www.mtnmath.com/whatrh/node72.html

    With these caveats noted, the general wave model still seems to offer a general picture of why uncertainty is intrinsic to the quantum `wave` model and not just an issue of measurement accuracy. However, I am not sure that this statement is true, which is why I would like to get some clarification, if possible, of my following summary:

    A pure sine wave with just 1 single frequency can only be modelled as a Fourier superposition if it extends infinitely in space, i.e. it has an unambiguous wavelength [tex][ \lambda ][/tex], but an uncertain point of location. In contrast, an infinitely narrow wave-packet can conceptually have a defined location, but requires the superposition of an infinite number of waves with different wavelengths, such that the wavelength at any specific location is uncertain.​

    From deBroglie’s wavelength equation:

    [tex] \lambda = h/mv [/tex] therefore momentum [tex] p = h/ \lambda [/tex]

    This equation, in conjunction with the summary above, appears to suggest that the uncertainty in wavelength [tex][ \lambda ][/tex], when the location of the narrow wave-packet is well defined, corresponds to an uncertainty in momentum that seems to parallel Heisenberg’s Uncertainty Principle (HUP) albeit in only a descriptive manner rather than providing any rigorous mathematical proof.

    So the clarification being requested is whether this description is generally applicable to HUP?
     
    Last edited: Apr 15, 2009
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