In photoelectric effect, light waves cannot knock electrons out; and in a photons passing through many slit experiment, a photon cannot pass through many slits at the same time. Namely, the two physical processes, respectively, reflect one aspect of wave-particle duality of quantum particle. On the other hand, in photoelectric effect, photons can knock electrons out; in the many slit experiment, a photon light wave can pass through many slits at the same time. The two physical processes then are complementarily equivalent in wave-particle duality of quantum particle. That is, in wave-particle duality of quantum particle, the first and the second cases use the particle property and the wave property, respectively. Namely, a photon can show as either particle or wave, but cannot be observed as both at the same time for a physics process.
We now generally show them by exact deduction.
In 4-dimensional momentum representation of quantum theory, when considering wave function (phi (vec{p},E)) of momentum representation, one has25
$$ psi (vec{r},t) = frac{1}{{(2pi hbar )^{2} }}int_{ - infty }^{infty } {} phi (vec{p},E)e^{{i(vec{p} cdot vec{r} - tE)/hbar }} dvec{p}dE = frac{1}{{(2pi hbar )^{3/2} }}int_{ - infty }^{infty } {} varphi (vec{p},t)e^{{ivec{p} cdot vec{r}/hbar }} dvec{p} $$
(1)
Equation(1) is a general Fourier transformation of ( , phi (vec{p},E)) (about the plane wave energy E and momentum (vec{p})) from the four-dimensional momentum representation state vector ( , phi (vec{p},E)) to the projection of the plane wave basic vector (e^{{i(vec{p} cdot vec{r} - tE)/hbar }}) and making integration for getting ( , psi (vec{r},t)), which make ( , psi (vec{r},t)) have not only the characteristics of the probabilistic state vector of the particle but also the characteristics of the plane wave, i.e., make ( , psi (vec{r},t)) have the state vector characteristics of wave-particle duality.
Because the momentum representation state vector ( , phi (vec{p},E)) is nonlocal, it also reflects that the system has the global characteristics of momentum (vec{p}) and energy (E), this global property can be the integrity of the particle, e.g., even including different physics qualities, e.g., spin, since the different qualities are not related to space coordinates.
Therefore, the expression (1) exactly shows wave-particle dualitys origin which displays that the wave property is originating from the plane wave part of the general Fourier expansion, and the particle property is originating from the general Fourier expansion coefficients with the particles global property even including different spins.
Therefore, we discover, for arbitrary particle, on an aspect, it propagates with the plane wave of the four-dimensional momentum ((vec{p},E)) as the propagation amplitude of the plane wave; on another aspect, it moves as a particle with the four-dimensional momentum ((vec{p},E)). Especially, when the expanding coefficients have different spins, it moves as a particle with both the four-dimensional momentum ((vec{p},E)) and the different spins, which are the new true physics and the new physical pictures, and uncover the corresponding expressions contributions of both wave part and particle part of wave-particle duality origin. Namely, Eq.(1) is the function of unified expression of wave-particle duality.
A little bit of philosophical insight on what this work means that the unified expression of wave-particle duality is just the superposition state of wave-particle duality, and the superposition state of wave-particle duality is physically real.
Furthermore, the infinite big momenta and energy show their corresponding to infinite big velocity in Eq.(1), and then the infinite big velocity is included, i.e., the wave function (1) of coordinate representation has the contribution of infinite big momentum or speed, namely, the wave function at any spatial and time points has the contributions from negative to positive infinite big momenta or speeds. Similarly, when we do the inverse Fourier transformation of Eq.(1) about whole spacetime coordinates, we find that the wave function of 4-dimentional momentum representation has the contributions of the whole 4-dimentional spacetime, i.e., the wave function at any 4-dimentialal momentum spatial point has the contributions from the whole spacetime. Thus, the above both cases just the reasons that Feynman path integral can be done in whole 4-dimentional spacetime or momentum space.
Using Eq.(1), we have wave function of momentum representation at time t
$$ varphi (vec{p},t) = frac{1}{{(2pi hbar )^{1/2} }}int_{ - infty }^{infty } {} phi (vec{p},E)e^{ - itE/hbar } dE $$
(2)
On the other hand, using Huygens' Principle, one has the basic wave analysis:
Every point of a wave front may be considered the source of secondary wavelets that spread out in all directions with a speed equal to the speed of propagation of the waves. What this means is that when one has a wave, he can view the "edge" of the wave as actually creating a series of circular waves. These waves combine together in most cases to just continue the propagation, and in some cases there are significant observable effects. The wave front can be viewed as the line tangent to all of these circular waves26.
Further using Eq.(1) and Huygens principle above, we have N subwave functions through N slits
$$ psi (vec{r}_{j} ,t) = frac{1}{{(2pi hbar )^{2} }}int_{ - infty }^{infty } {} phi (vec{p},E)e^{{i(vec{p} cdot vec{r}_{j} - tE)/hbar }} dvec{p}dE = frac{1}{{(2pi hbar )^{3/2} }}int_{ - infty }^{infty } {} varphi (vec{p},t)e^{{ivec{p} cdot vec{r}_{j} /hbar }} dvec{p} $$
(3)
where j=1,2,,N. No losing generality and for simplicity, taking N=2 just shows the up slit and down slit, respectively, in Young's Double Slits in Fig.2.
Interference of a particle plane wave in Young's double slit experiment.
Therefore, Eqs.(1)(3) can also be seen as a kind of expressions of Huygens principle. Consequently, these Fourier expansions physically imply new physics, and are not only just the mathematical tools.
The superposition density function of two subwaves is just Eq.(5) in Section Solutions to Wheelers delayed choice puzzle and puzzle of a particles passing double slits simultaneously by the physics processes of the exact quantum physics expressions, the interference terms of the two subwaves in Fig.2 are just the third term and fourth term in Eq.(5).
These properties are exactly conforming to the plane wave properties of the single particle, thus a particle plane wave can simultaneously pass through N slits, for simplicity, Young's Double Slits in Fig.2, Eq.(3) just generally give the both subwave functions that simultaneously pass through N slits, for simplicity, two slits s1 and s2 in Young's Double Slits, respectively.
The N subwave functions have the same amplitude (phi (vec{p},E)) for some certain (vec{p},E), (e^{{i(vec{p} cdot vec{r}_{j} - tE)/hbar }}) (j=1, 2,, N) are just N plane subwave functions in Eq.(3), and the N probabilistic wave functions in Eq.(3) integrate for ((vec{p},E)) from negative infinite to positive infinite, i.e., having considered all possibility, which make the N expressions (3) exact.
The global property of a particle does not allow the single particle to simultaneously pass through N slits, for simplicity, Young's double slits, in reality, the interference of a particle wave is observed, which just show a particle wave simultaneously does pass the N slits, for simplicity, the double slits, but all theories up to now cannot solve the hard puzzle of a particles passing the N slits, e.g., Youngs double slits simultaneously.
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New quantum physics, solving puzzles of Wheeler's delayed choice and a particle's passing N slits simultaneously and quantum oscillator in experiments...