14:24
ok so after a more detailed analysis, in order to describe superposition and interference, what you end up with is basically a vector sum of the velocity fields that govern the object's momentum. This means the whole interpretation falls back into Bohm mechanics so there is nothing new here
although, it can still be argued that it enriched the ontology of Bohm mechanics somewhat. Here every bohmian velocity is interpreted as a complicated but periodic motion where the object's initial condition is uncertain (similar to ordinary Bohmian). Then the mismatch of the periodic motion of these objects with the test object (the measuring apparatus) give rise to nonzero probabilities
The classical limit is then recovered when the object's periodicity is much faster than the test object, thus causing the test object to interact as if the object's periodic trajectory is a solid object
The wavefunction then via the Bohm equation, prescribes the velocity field that the objects to be followed
I think I will need to check if I can derive Born's rule using this interpretation, starting with the simple case of a harmonic oscillator contacting with another test object which lowers to the oscillator periodically
We can set up the toy model as follows:
Let the test object to be at position $x_0$ when $t = Tn$ for some real period $T$ and $n \in \Bbb{N}$
Now the classical equation of motion of the harmonic oscillator is given by:
$x_h(t) = \frac{1}{2}x_0\cos (\omega t)$
Then, contact occurs whenever $x_h(t) = x_0(t)$
actually correction, the test object's equation of motion is given by $x_t(t) = x_0 \delta_{t,Tn}$
so contact occurs when $x_h(t)=x_t(t)$
$\frac{x_0}{2} \cos (\omega t) = x_0 \delta_{t,Tn}$
$\frac{1}{2} \cos (\omega t) = \delta_{t,Tn}$
$\frac{1}{2} \cos (\omega t) = 0, t \neq Tn$ or $\frac{1}{2} \cos (\omega Tn) = 1$
$t = \frac{\pi}{\omega}(\frac{1}{2} + n)$ or $\cos (\omega Tn) = 2$
looks like this actually is sensitive to initial conditions, thus we are no longer dealing with Bohm here
So that means, this is not an interpretation of quantum mechanics nor Bohmian mechanics
(Btw, I am assuming the motion of the test object and the object are orthogonal so I can simply its equation of motion using only deltas, since contact is only relevant in the x axis)
So if I want to check for all possible initial $x$ I should set $x_h(t)$ as:
$x_h(t) = x_0\cos (\omega t + t_0)$
Solving this for $t = Tn$ we have:
$x_h(Tn) = x_0 \cos (\omega Tn + t_0) = x_0$
$\omega Tn + t_0 = (\frac{1}{2}+n)\pi$
And solving for $t\neq Tn$ we have:
$x_n(t\neq Tn) = x_0 \cos (\omega Tn + t_0) = 0$
if $x_0 = 0$, then contact is certain, if $x_0 \neq 0$ we have:
Now let $f = \frac{1}{T}$ we thus have the equations:
$\frac{\omega}{f} + t_0 = (\frac{1}{2}+n)\pi$ or $\frac{\omega}{f} + t_0 = n\pi$
if the object periodic motion is faster than the test object, then we have $\omega >> f$, this gives:
$\frac{\omega}{f} \sim \omega$ and so we have:
$\omega + t_0$ on the LHS
so contact is entirely dependent on the motion of the object itself as expected intuitively
however, this contradicts the classical case because the fact that it need to satisfy those equations means contact is not guarantee for all initial conditions even as the frequency of oscillation of the object dominates the probing rate of the test object
What is this? Is this some sort of wizardry? Or is the server bad at counting?
Jokes apart! Why is my quarter rep more than my total rep?
It would be really helpful if you included the chemical elements involved (Chromium and Corundum) and their contributing properties that answer the question. Thank you!
