Widdekind

(Thanks for the replies) Can you create a "scalar photon", w/o using an electro-dynamic potential (A), but only a time-varying electro-static potential (V) ?

If [math]\vec{p}_{tot} = \gamma m \vec{v} + e \vec{A}[/math], what value of the momentum, [math]\vec{p}_{tot}[/math] or [math]\vec{p}_{conventional}[/math], is used in [math]E^2 = E_0^2 + (c p)^2[/math] ? And, what if there is also an electro-static potential, too ? For small electro-static & electro-dynamic potentials, the formula [math](E - V)^2 = E_0^2 + (\vec{p}_{tot} - e \vec{A})^2[/math] seems to correspond to the classical limit.

(Thanks for the clarification.) At room temperature, thermal energies are of order 0.04 eV. In-so-far, as this is (significantly) less, than the excitation / ionization energies of even valence electrons, thermal 'jostling' might not have enough energy to excite / ionize atoms at STP. What does, however, supply the energy for BB radiation? The KE / PE of the atoms, "sloshing around" in the potential wells, of their lattice sites, in some solid crystal (say) ?? Is it accidental, that BB radiation, being (in net) directed outward, from any parcel of matter, acts as a "compressive force", pushing the outer, radiating layers, of that chunk of matter, back inwards, towards their 'mates in the middle' (at least a little like some sort of surface tension) ?? It seems to me, that (1) photon non-conservation (creation/annihilation) plays a special role in (2) wave function collapse (measurement). And, photon non-conservation seems associated, with (3) the "totality" of wave function collapse, in photons, when they localize in absorption. And, finally, this "destructive wave function collapse", of photons, in absorption, seems associated with (4) photons having zero rest mass (rest mass provides an "irreducible kernel" of substance to quantum 'particles' ?). (Of course, this if for the atomic quantum scale. But, understanding these kinds of connections, at this scale, could help highlight the important physics, of corresponding processes, on the nuclear quantum scale.) How can neutral atoms BB radiate, at STP (<E> = 0.04 eV) ? What is the QM description, of accelerative radiation ? And, if two charges, in a plasma, interacted (normally, an Entanglement process), and then emitted BB photons (particle creation, not described by the SWE ??), would that constitute a wave function collapse (Type 1, "quantum jump") process ?

If Decoherence effects arise, from the coupling, of an ideal quantum system, to a "noisy" environment, full of phase-randomizing thermal interferences (phonons, photons); and if such an "environment", in thermal equilibrium, is, in practice, ultra-macroscopic; then doesn't Decoherence essentially, or at least ultimately, wind up appealing to Bohr's "microscopic / macroscopic" Copenhagen Interpretation, for quantum measurement ?

Wave functions spread, between measurements (from source to screen, say); and then the wave function 'collapses' upon measurement (into a single screen detector element, or 'micro-detector', like a phosphor or silver salt grain). Roughly related, this seems sort-of like a "dance like a butterfly, sting like a bee" behavior.

Doesn't BB radiation ultimately stem from collisional excitation / de-excitation processes ?

What would happen, if you performed a modified DC experiment, using a (switchable) Beam Splitter, instead of a (switchable) Mirror; and, then, lengthening the "switch line" (using Optical Delay Circuits), so that those portions of the photon's Wave Packet, which impinge upon the Interferometer, arrive "first", long before the "switch line" Wave Packet portion reaches its detector. QUESTION: Insofar, as wave function collapse is caused by contact, with the Detectors, would a longer "switch line" allow the Interference Pattern, observable at the intersection, of the "main lines", to be observed (in ~50% of the trials, i.e., at reduced intensity) ?? Fig. 2 -- Wave Packet portions in (transparent) red, showing "tenuous tendril tails" of probability amplitude connecting them. According to Greenstein & Zajonc's Quantum Challenge (pg. ~40), photon wave functions can be diffusively extended, across considerable spans of space -- even "billions & billions of miles", clear across the Solar System (in principle). Now, in John Wheeler's "quasar DC experiment", then, could we not interpret the photon's wave functions, as being de-localized, across the cosmos? Indeed, quantum entanglement functions (in principle) across the cosmos, too (from the "tenuous tendril tails", of probability amplitude, spanning the space between the wave packet portions). Evidently, the curvature of spacetime "in the background", causes the photon "spherical wave" to converge back upon itself. QUESTION: If the wave function, of a space-traveling photon, actually extends across the cosmos, then wouldn't the presence, of matter "in the middle" (the galaxy, in the figure), by virtue of being a potential absorber of photons "en route", actually reduce the signal intensity, received at Earth ??

Many QM texts seem to use Spherical Waves (ei k r / r) to describe scattered Reflection Waves ([math]\Psi_R[/math]). Does this imply, that, in Compton Scattering, the photon's Wave Function is composed, of an incident plane-wave-like Wave Packet (ei k z), plus an outgoing, reflected, Spherical Wave, with an angular dependent Wave Vector ([math]\vec{k}( \theta )[/math]), which varies according to the Compton Formula ? Presumably, the electron would be, similarly, scattered out, into a spherically spreading Wave Packet (whose momentum distribution was Quantum Entangled, with the momentums of the reflected photon Wave Function, according to the conservation laws, for total linear momentum) ?? As a rule, the "Classical Distributions", found from high-intensity beams, which map out many individual wave functions, are the simple sum of all those individual wave functions, so that the "Classical Patterns" accurately map out the individual Wave Functions (Greenstein & Zajonc. Quantum Challenge, ch.2,8) ???

John von Neumann carefully distinguished between the two types of quantum evolution processes (Type 1 "quantum jumps" = "wave function collapse"; Type 2 SWE). Now, the probability amplitude, for a "quantum jump", is equal to the overlap integral ([math]< \Phi | \Psi >[/math]), between the quantum particle's current state ([math]\Psi[/math]), and the available end state ([math]\Phi[/math]). (The probability is the squared modulus of the probability amplitude.) QUESTION: Could you view "business as usual", Type 2, SWE-compliant, evolution, as a series of "self-collapses" of the wave function ([math]\Psi(t) \rightarrow \Psi(t+\delta t)[/math]) ? If: [math]\Psi(t + \delta t) \approx \Psi(t) + \frac{\partial \Psi(t)}{\partial t} \delta t[/math] [math] = \Psi(t) - \frac{i}{\hbar}\hat{E} \Psi(t) \delta t[/math] and if the wave function was normalized ([math]< \Psi | \Psi > = 1[/math]), then wouldn't the probability, for a "self jump" process be: [math]< \Psi(t+\delta t) | \Psi(t) > \; \approx \; 1 - \frac{i <E>}{\hbar} \delta t[/math] [math] \equiv 1 - i <\omega> \delta t[/math] ? In-so-far as such a "self-collapse" probability is [math]< 1[/math], what would happen, on those rare occasions, when the SWE "mis-fired" ([math]\propto - i <\omega> \delta t[/math]) ? Presumably, [math]\delta t \approx t_{Planck}[/math] ?? Could such "mis-fires", of the SWE, explain "spontaneous collapse" theories of quantum jumps ??? Comparison of "self-collapse" to "standard-collapse" probabilities (??) If the probability amplitude, of an "in-jump" process, is [math]P \left( \Psi(t) \rightarrow \Psi(t+\delta t) \right) \; \approx \; 1 - i <\omega> \delta t[/math], would the probability of an "out-jump" process grow as: [math]P \left( \Psi \rightarrow \Phi \right) = < \Phi | \Psi >[/math] [math] \approx < \Phi | \Psi(t) > - \frac{i < \Phi | \hat{E} | \Psi(t) >}{\hbar} \delta t[/math] [math]= a \left( 1 + \frac{i E_{\Phi}}{\hbar} \delta t \right)[/math] [math] \equiv a \left( 1 + i \omega_{\Phi} \delta t \right)[/math] where we have defined the transition probability amplitude [math]a \equiv < \Phi | \Psi >[/math], and then "acted to the left", with the Energy Operator... ?? Does this mean, accumulating said contributions over many time steps, that all transition probability amplitudes grow exponentially, as [math]e^{i \omega_{\Phi} t}[/math] ???

Photon non-conservation ([math]\hat{a}^{\pm}_{\vec{k}}[/math]) seems associated with "wave function collapse" (i.e., "quantum jump") processes.

Roughly speaking, an Electro-dynamic potential, [math]\vec{A}[/math], can generate both [math]\vec{E}[/math] & [math]\vec{B}[/math], from [math]\vec{E} \approx \frac{\partial \vec{A}}{\partial t}[/math] & [math]\vec{B} = \nabla \times \vec{A}[/math]. Thus, an EDP, oscillating up & down along a single axis, would generate both the electric & magnetic fields, of a light wave. Is, then, the direction of [math]\vec{A}[/math] the polarization axis of light ?

That seems completely consistent with quantum theory. The de Broglie wavelength, of a 14 eV electron, is roughly: [math]\frac{p^2}{2 m_e} = 14 \, eV[/math] [math]p = \frac{h}{\lambda}[/math] [math]\therefore \lambda = \sqrt{\frac{h^2}{2 m_e E_0}} = 3.3 \AA[/math] That's comparable, to the atomic distance scales, of the available electron orbitals. Thus, electrons with really high incident kinetic energies, would start having de Broglie wavelengths smaller than the size of atoms. If the incident electron's wave function oscillated quickly, across the nuclear positive potential well, the overlap integral would start to self-cancel, leading to lower & lower transition probabilities. (An electron, whose de Broglie wavelength was aB, would have a KE of ~535 eV (1/2 keV).) EDIT: Technically, wouldn't the overlap integral involve the time-frequency phase factors, [math]e^{i \omega t}[/math]? Would beat frequencies, [math]e^{i (\omega - \omega_0) t}[/math], play any important part in the process ?? Or, would those phase factors cancel out, when you took the square modulus, in going from the overlap integral, [math]< \Psi_e | \Phi_{1S} >[/math], to the transition probability, [math]| < \Psi_e | \Phi_{1S} > |^2[/math] ? Edit 2: How would you compute the overlap integral, of an incident plane wave, [math]e^{i \vec{k} \vec{r}}[/math], with the ground state, [math]\Phi_{1S} \propto e^{- \alpha r} / r[/math] ? With only 1 power of r in the denominator, does the integral still converge ? Edit to Edit 2: On second thought, I calculate that the overlap integral, between an incident plane wave, and the 1S ground state, is [math]\propto \frac{1}{k^2 + \alpha^2}[/math], so that the overlap dramatically declines when [math]k > \alpha[/math], to wit, when the de Broglie wavelength is less than aB.

Insofar, as any interaction, can be described, as "beginning" ([math]P \approx 0[/math]), "peaking" ([math]P = P_{max}[/math]), and "ending" ([math]P \approx 0[/math]), then isn't the "peak probability" (Pmax) what quantum theory calculates, for the transition probability? If so, could [math]\partial P / \partial t = 0[/math]) "trigger" the "dice roll" of w.f. 'collapse' ??

Whenever Wave Functions collide with potential barriers, they bifurcate, into Reflected & Transmitted waves. Now, electrically charged particles create potential barriers. Thus, the contact collision, of (say) two electrons' wave packets, would amount to a "double barrier" Reflection & Transmission event, which could account for the ensuing Entanglement. Is this what happens*? * Entanglement, of two different types of particle (e.g. electron + photon) would result in scattered wave packets which would overlap, accounting for the phase entanglement, but which would propagate with different (group) velocities.

Thanks for the clarification & confirmation.

Thanks, that's an important point. Ultimately, it seems to me, that Einstein's explanation, of the photo-electric effect, in precise technicality, only proves that photons are absorbed in "lumps" (on wave function 'collapse'), not that they necessarily propagate through space as such localized "lumps" (cf. electrons' w.f.'s spread out, through double-slit apparati, before "collapsing" into a single micro-detector (phosphor grain), on the irregular array of the same, comprising the macro-detector (detector screen)).

(thanks for the responses) If a photon has an infinite lifetime ([math]\Delta t = \infty[/math]), then how can it have a non-zero spread in energy ([math]\Delta E \neq 0[/math]), and still obey the HUP ?

Other examples of de-localized atomic wave function states:

In 1973, Edward Tryon calculated, that the sum of matter's Rest-Mass Energy & Gravitational Potential Energy is E0 + Eg ~ 0. This can be seen, from the above equations, by replacing the Bohr Radius w/ the classical electron radius (aB --> r0), which one can do b/c the correspondences, while quite close comparatively, are still rather rough: [math]\frac{G M_V m}{R_H} \approx \frac{e^2}{4 \pi \epsilon_0 \, r_0} \equiv m c^2[/math] And, thus, [math]R_H \approx \frac{G M_V}{c^2} \approx R_S[/math] To wit, the Hubble Radius (RH), of the visible cosmos, is quite close to the Schwarschild Radius (RS), of all the mass making up the visible cosmos. This again seemingly suggests, that the cosmos is quite close to "critical" & "closed" (M.Wolff. Schrodinger's Universe, pp. 136-137).

I understand, that the SWE, on its own, also does not predict the electron's spin projections -- it must be augmented by the Pauli Spin Matrices. Could you not do essentially the same, w/ the KGWE as well ?

(thanks for the replies) If w.h.t. (1) F = dp/dt; (2) F = -dV/dx; then could you consider, that the "electro-static potential (V)", which an electron "emits", and whose gradient defines the electro-static force-field emanating from said electron, as a "cloud" of virtual photons, of varying spatial density, so that their "electro-magnetic pressure" (P ~ dp/dt) generates the "push" associated with the electro-static potential (V) ?? In symbols, can you equate -dV/dx = dp/dt (of the virtual photon "cloud") ??

Can virtual photons be described, as having positive momentum (by which "(EM) forces" are exerted), but imaginary rest-mass-energy (being "off-mass-shell"), so that their total energy (& energy density, of the associated virtual photon "cloud" shadowing quantum particles) is zero: E2 = E02 + (pc)2 = 0 ? Imaginary rest-mass-energy, would translate into imaginary frequency (E = hf), which would make virtual photons fizzle away, exponentially, in time (eift --> e-ft), explaining why they never venture far from the particles producing them. (This might mean, that the lowest frequency virtual photons, would last the longest, and have the "longest reach" (??).)

In practice, wouldn't the electrostatic potential be "strong" only w/in a few nm of the nucleus? Atomic electron orbitals can be decomposed, as the sum, of inflowing & outgoing spherical waves (eikr/r), which, taken together in opposing pairs, produce Standing Waves (M.Wolff. Schrodinger's Universe). For the Hydrogen 1S state, ~e-ar/r, the overlap integral: [math]< k_r | \Phi_{1S} > \; \propto \int_0^{\infty} 4 \pi r^2 \frac{e^{i k r}}{r} \frac{e^{-\alpha r}}{r} dr \propto \frac{\alpha + i k}{\alpha^2 + k^2}[/math] where, in the last step, the imaginary exponential has been decomposed, into sines & cosines, and those integrals computed "by parts". This makes the fractional contribution: [math]\left| < k_r | \Phi_{1S} > \right|^2 \propto \frac{1}{\alpha^2 + k^2}[/math] Since [math]\alpha \approx a_B^{-1}[/math], we see that the Heisenberg Uncertainty Principle is obeyed. Since plane waves, comprising (electron) wave packets, naturally scatter, off of nuclei, into spherical waves, one can start to see, how a simple scattering interaction, could "generate" or "set up" such "spherical standing waves" (which would then, presumably, be "augmented" & "exaggerated" during the "quantum jump", from free particle, to bound orbital, even as the "tails" fizzled away).

When a free electron is incident upon a bare proton, how does one compute the "recombination" probability, for the formation of a neutral atom? If the electron "wave packet" is moving at even thermal velocities, 100-1000 m/s, the initial interaction, between the two particles, is surely fairly fleeting -- perhaps a trillionth of a second (??). When the electron "wave packet" is far away from the proton, the Overlap Integral (and, hence, Transition Matrix Element) is ~0. Then, a trillionth of a second later, the scattered "wave packet" ricochets away, and the Overlap Integral returns to ~0. In between, briefly, the Overlap Integral increases, to some maximum value, and then declines, again, back down to negligibility. Would the effective "recombination" probability, be computed, from the maximum Overlap Integral ??

Thanks again again for the link, it is very informative. Gary Zukav says essentially the same thing, albeit qualitatively, w/o the mathematical rigor: The author goes on to describe the Strong Force, as an exchange of virtual pions between nucleons, in essentially similar terms.