Why does matter behave like particles when observed, and like waves when not?

Widdekind, on 22 January 2012 - 10:40 PM, said:

A wave-function has a momentum expectation value, at every point in space, which is proportional to the gradient of the wave-function, at that point:

$\propto \left( - \imath \hbar \frac{\partial}{\partial x} \right) \Psi(x)$

In analogy, a wave-function is a little like a flock of birds, flowing through space, with each "bird", at each point in space, having some momentum, at that point in space.

Yeah, wave functions change distance, but in order to "work backwards" from when a photon was measured as at a definite location, wouldn't that have to assume the photon was a point or in a defined location the whole way through?

This post has been edited by questionposter: 22 January 2012 - 10:51 PM

I understand, that quantum wave-packets have calculable "centroids" or "expectation values"; and, that those "centroids" evolve through time, according to the Classical equations of motion, whilst obeying Classical conservation laws. I.e. our macroscopic notions of "Classicality" arise from the "average" behavior, of quantum wave-packets. In analogy, wave-packets are like "schools of fish" or "flocks of birds", characterizable by an average "center of mass", and an average "overall momentum"; but which are also spread out through physical space (xyz) ("many many birds here & there"), and momentum space (p_xp_yp_z) ("some birds flying fast left, others flying slow right").

For example, for a fermion (F), composed of a "spin up" component (+), and a "spin down" component (-):

$\tilde{\Psi}_F(\vec{x}) = \Psi_{+}(\vec{x})|+\rangle + \Psi_{-}(\vec{x})|-\rangle$ ("flock of sea-gulls & flock of crows")

the "expected (linear) momentum":

$\langle \vec{p} \rangle \equiv \int d^3 x \tilde{\Psi}_F^{*}(\vec{x}) \left( -\imath \hbar \vec{\bigtriangledown} \right) \tilde{\Psi}_F(\vec{x})$

$= \left( \int d^3 x \Psi_{+}^{*}(\vec{x}) \left( -\imath \hbar \vec{\bigtriangledown} \right) \Psi_{+}(\vec{x}) \right) + \left( \int d^3 x \Psi_{-}^{*}(\vec{x}) \left( -\imath \hbar \vec{\bigtriangledown} \right) \Psi_{-}(\vec{x}) \right)$

$= \langle \vec{p}_{+} \rangle + \langle \vec{p}_{-} \rangle$ ("sea-gulls' momentum + crows' momentum")

and the "expected angular (spin) momentum":

$\langle s_z \rangle \equiv \int d^3 x \tilde{\Psi}_F^{*}(\vec{x}) \left( \hat{s}_z \right) \tilde{\Psi}_F(\vec{x})$

$= \frac{\hbar}{2} \left( \int d^3 x \Psi_{+}^{*}(\vec{x}) \Psi_{+}(\vec{x}) \right) - \frac{\hbar}{2} \left( \int d^3 x \Psi_{-}^{*}(\vec{x}) \Psi_{-}(\vec{x}) \right)$

$= \frac{\hbar}{2} \left( \langle _{+} \rangle - \langle _{-} \rangle \right)$ ("sea-gulls' whirling clockwise - crows' whirling counter-clockwise")

are well-defined quantities,

whose evolution through time obeys Classical equations of motion

Thus, via their expectation values, quantum wave-packets (of fermions) can be corresponded to Classical 'particles', having four degrees of freedom (p_xp_yp_z,s_z). Naively, a system of $N$ fermions, has $4N$ dof. And, such a system can undergo $\left( \begin{array}{c} N \\ 2 \end{array} \right) = \frac{N (N-1)}{2}$ interactions, between pairs of particles.

Now, each interaction, between wave-packets co-mingles, i.e. entangles, those wave-packets ("two flocks of birds collide, and get all mixed up"). And, each interaction imposes a conservation law, e.g. conservation of momentum $\left( \langle \vec{p}_1 \rangle + \langle \vec{p}_2 \rangle = \langle \vec{p}_1 \rangle' + \langle \vec{p}_1 \rangle' \right)$ ; and spin $\left( \langle s_{z,1} \rangle + \langle s_{z,2} \rangle = \langle s_{z,1} \rangle' + \langle s_{z,2} \rangle' \right)$ , from before to after the interaction (denoted by single quotes).

So, if the number of interactions, i.e. number of constraints, equals the number of dof; then the system is "uniquely determined"; and the system must adopt a unique set of values, for those dof ("number of equations = number of unknowns"). Naively, this occurs, for $N \ge 9$ , i.e. up to eight fermions can be mutually co-entangled, and still manifest quantum indeterminacy, e.g. at least four non-determined dof, corresponding to the linear momentum, and angular momentum, of the entangled "ensemble" of fermions ("the momentum & spin, of the 'super-flock', of all the now-co-mingled flocks").

I understand, that interactions "whittle away" remaining, non-determined, dof, until the system becomes uniquely specified, at which point quantum "decoherence" and wave-function "collapse" occur:

Quote

"a combination of measurements or encounters, which progressively limit the remaining degrees of freedom possessed by a quantum entity, will have the same effect, where the wavefunction collapses and a classical outcome is the result [i.e.] the conflicting interests of too many quantum mechanical effects happening at once, that leads to interference" (dickau)

This post has been edited by Widdekind: 25 January 2012 - 07:08 PM

LaTeX Code

Science Forums: Why does matter behave like particles when observed, and like waves when not? - Science Forums

Welcome to ScienceForums.Net!