maker progress on framing the issues, split out double-slit

Author: rcoreilly

content/configuration-space.md

@@ -8,7 +8,7 @@ Because it describes the _configuration_ of these elements, it is **exponential*
 
 The need for configuration space at a mathematically deep level arises because the equations being used are _linear_, so they cannot represent any kind of actual interaction among different particles. Without configuration space, every particle would fully superpose on every other particle --- they would just slip on past each other. This is in fact how _bosons_ (e.g., _photons_) behave, but not how _fermions_ like [[electrons]] behave.
 
-The [[pilot wave]] approach has been (perhaps unfairly) criticized for using configuration space, because it posits that the wave function is actually a "real" thing, thus exposing the implausibility of this otherwise purely [[tools vs models|calculational tool]]. See [[@NorsenMarianOriols15]] for an analysis of the contributions of configuration space to the pilot wave results.
+The [[pilot-wave]] approach has been (perhaps unfairly) criticized for using configuration space, because it posits that the wave function is actually a "real" thing, thus exposing the implausibility of this otherwise purely [[tools vs models|calculational tool]]. See [[@NorsenMarianOriols15]] for an analysis of the contributions of configuration space to the pilot-wave results.
 
 However, if the underlying dynamics of the system are _nonlinear_, and in particular involve interactions between [[stochastic particles]] and wave functions, then it is possible that these nonlinear interactions end up producing all of the relevant dynamics that are otherwise captured via the configuration space calculational tool. This is the approach taken here.
 

content/double-slit.md

@@ -0,0 +1,16 @@
++++
+bibfile = "mechphys.json"
++++
+
+{id="figure_double-slit" style="height:20em"}
+![The double-slit experiment. Narrow openings in the slits cause the wave to spread through diffraction, and because of the different distances traveled in the path from the two different slits to a given point on the far screen P, the waves will experience either constructive or destructive interference, resulting in the wavy bands of light and dark as shown. The quantum paradox here is that this pattern obtains even when a _single particle_ is emitted at a time. The particle only ever goes through _one_ slit or the other, but somehow the wave goes through both. Figure from wikimedia commons by Lacatosias.](media/fig_double_slit_expt.png)
+
+{id="figure_double-slit-elec" style="height:40em"}
+![Results of a double-slit experiment using electrons, with increasing numbers of electrons recorded (11, 200, 6,000, 40,000, and 140,000). The interference pattern emerges over time, even though only single electrons are detected on each trial. Figure from by Dr. Tonomura via wikimedia commons.](media/fig_double_slit_expt_electrons.jpg)
+
+The **double-slit** experiment (also known as Young's experiment) ([[#figure_double-slit]]), is said to illustrate the full mystery of quantum mechanics, and nicely demonstrates some puzzling aspects of wave-particle duality. Interestingly, the double slit experiment was around long before quantum mechanics, as a way of generating interference patterns with waves, but it "just got weird" when the intensity of the light, or beam of electrons or other particles, is reduced to the point where there is only a _single particle_ passing through the apparatus at a time.
+
+Surprisingly, one still observes the interference effect in this case ([[#figure_double-slit-elec]]). How can a single "hard little particle", all by itself, produce this wave-like interference effect?  There are _many_ other results that all add up to the strong conclusion that, somehow, elementary particles like electrons have _both_ wave and particle properties.
+
+The [[pilot-wave]] framework of de Broglie and Bohm provides the most natural, intuitive explanation of these effects: the wave goes through both slits, and the particle goes through one, but it is influenced by the wave.
+

content/epistemic-vs-ontic.md

@@ -10,7 +10,7 @@ If the quantum wave function is largely (or even partially) reflecting epistemic
 
 By contrast, the Copenhagen interpretation already takes a laissez-faire epistemic-level approach to the wave function in the first place: it is all just a big untouchable ball of mystery until you do a measurement anyway, so it might as well be epistemic or whatever! The Quantum Bayesianism (QBism) approach takes this to its logical extreme, with an entirely subjective epistemic treatment of the wave function ([[@FuchsMerminSchack14]]; [[@Mermin18]]).
 
-Critically, there is clear evidence from _within the pilot-wave approach itself_ that a not-insignificant portion of the pilot wave actually does represent epistemic uncertainty, because many different possible initial starting states must be modeled to capture our very real uncertainty about the precise starting state of any actual experimental configuration.
+Critically, there is clear evidence from _within the pilot-wave approach itself_ that a not-insignificant portion of the pilot-wave actually does represent epistemic uncertainty, because many different possible initial starting states must be modeled to capture our very real uncertainty about the precise starting state of any actual experimental configuration.
 
 The Heisenberg uncertainty principle dictates that there is a fundamental limit to which we can simultaneously determine all of the relevant degrees of freedom about a physical system, and in practice we almost certainly have well less certainty than this lower limit, because it is very difficult to make any kind of precise measurement of microscopic quantum-scale systems.
 

content/hamiltonian.md

@@ -2,11 +2,11 @@
 bibfile = "mechphys.json"
 +++
 
-Using [[special relativity]], plus the notion of **conservation of energy** --- i.e., that the **total energy** of the system is strictly conserved over time, we can derive the [[Klein-Gordon]] equation from first principles. In keeping with physicist's penchant for assigning people's names to concepts that would otherwise be very easy to understand if just spelled out, the total energy of the system is also called the **Hamiltonian** ($H$), and standard Newtonian physics can all be derived from the appropriate Hamiltonian (which is what W. R. Hamilton did).
+Using [[special relativity]], plus the notion of **conservation of energy** --- i.e., that the **total energy** of the system is strictly conserved over time, we can derive the [[Klein-Gordon]] (KG) equation from first principles. In keeping with physicist's penchant for assigning people's names to concepts that would otherwise be very easy to understand if just spelled out, the total energy of the system is also called the **Hamiltonian** ($H$), and standard Newtonian physics can all be derived from the appropriate Hamiltonian (which is what W. R. Hamilton did).
 
 This motif of using the total energy of the system to derive basic physical laws seems to work quite well in many cases, and is thus the primary way that such laws are derived for different definitions of the total energy. Essentially, the physical laws are latent in any given definition of total energy, and really just amount again to specifying the dynamics by which energy gets moved around in different ways, without ever gaining or losing any total energy.
 
-After we derive the KG equations from a relativistic total energy Hamiltonian here, we will then derive the Schrodinger equation from a different Hamiltonian at the end of this chapter, and then we'll extend the Hamiltonian to include spin and coupling to the EM field in the next chapter where we derive the [[Dirac]] equation (which is just a more complicated version of the KG equation). You will see that the total energy equation and the corresponding wave equation are very directly related mathematically, and thus this overall approach of using the total energy to derive the wave equation is a very powerful tool that is important to understand if you want to really understand what these wave equations are doing.
+After we derive the KG equations from a relativistic total energy Hamiltonian here, we will then derive the Schrodinger equation from a different Hamiltonian. The Hamiltonian can be extended to include spin and coupling to the EM field, to derive the [[Dirac]] equation (which is just a more complicated version of the KG equation). You will see that the total energy equation and the corresponding wave equation are very directly related mathematically, and thus this overall approach of using the total energy to derive the wave equation is a very powerful tool that is important to understand if you want to really understand what these wave equations are doing.
 
 You should be familiar with our computation of the total energy associated with a simple wave, which we calculated in [[waves]]. There we saw that for each cell element in our wave matrix, the total energy was the sum of the **kinetic** and **potential** energy, where kinetic energy is a function of how fast the state value is moving, and potential energy is a function of how much stress or tension there was between the state and its neighbors (i.e., the curvature of the space).
 

content/history.md

@@ -2,16 +2,6 @@
 bibfile = "mechphys.json"
 +++
 
-{id="figure_double-slit" style="height:20em"}
-![The double-slit experiment --- narrow openings in the slits cause the wave to spread through diffraction, and because of the different distances traveled in the path from the two different slits to a given point on the far screen P, the waves will experience either constructive or destructive interference, resulting in the wavy bands of light and dark as shown. The quantum paradox here is that this pattern obtains even when a _single particle_ is emitted at a time --- the particle only ever goes through one slit or the other, but somehow the wave goes through both. Figure from wikimedia commons by Lacatosias.](media/fig_double_slit_expt.png)
-
-{id="figure_double-slit-elec" style="height:40em"}
-![Results of a double-slit experiment using electrons, with increasing numbers of electrons recorded (11, 200, 6000, 40000, and 140000). The interference pattern emerges over time, even though only single electrons are detected on each trial. Figure from by Dr. Tonomura via wikimedia commons.](media/fig_double_slit_expt_electrons.jpg)
-
-The _double-slit_ experiment (also known as Young's experiment) ([[#figure_double-slit]]), is said to illustrate the full mystery of quantum mechanics, and nicely demonstrates some puzzling aspects of wave-particle duality. Interestingly, the double slit experiment was around long before quantum mechanics, as a way of generating interference patterns with waves, but it "just got weird" when the intensity of the light, or beam of electrons or other particles, is reduced to the point where there is only a _single particle_ passing through the apparatus at a time.
-
-Surprisingly, one still observes the interference effect in this case ([[#figure_double-slit-elec]]). How can a single "hard little particle", all by itself, produce this wave-like interference effect?  There are _many_ other results that all add up to the strong conclusion that, somehow, elementary particles like electrons have _both_ wave and particle properties.
-
 Historically, the early development of this wave-like property of particles was focused on understanding the nature of simple atoms like hydrogen, which has a single electron orbiting a nucleus. The dominant classical physical model of the atomic system in the early 1900's was the Rutherford model of 1911, with electrons as tiny points of charge and mass, orbiting a nucleus, much like planets orbiting the sun. This model had important failings, which the full development of quantum mechanics resolved, thus cementing the demise of the classical worldview, and solidly establishing quantum mechanics.
 
 The major problem with the classical atom was that it is fundamentally unstable: the electron should emit electromagnetic radiation as it orbits around the nucleus, and thus lose energy. As it loses energy, the orbit must get tighter, and eventually the electron should just collapse into the nucleus, just like one of those quarters you roll around in a gravity well at a science museum. Furthermore, as its orbit gets tighter, it should emit higher frequency radiation, predicting a continuous and increasingly high frequency emission spectrum. Instead, it was known that atoms emit consistent, discrete frequencies of radiation.
@@ -48,7 +38,7 @@ $$
 
 This wavelength is about .165 nanometers for the electrons in the Davisson-Germer experiment (very tiny, but enough to produce a measurable diffraction pattern through the crystal).
 
-Both de Broglie and Schrodinger thought that these [[matter waves]] were real physical things, like light waves. Furthermore, de Broglie suggested that the wave acted to _guide_ the point particle electron around, in his [[pilot wave]] theory. Schrodinger initially had an even more radical view, which abandoned the point electron entirely: he thought his wave equation described a wave of _charge density_ that _is_ the actual electron, without any need for a dual particle-like entity.
+Both de Broglie and Schrodinger thought that these [[matter waves]] were real physical things, like light waves. Furthermore, de Broglie suggested that the wave acted to _guide_ the point particle electron around, in his [[pilot-wave]] theory. Schrodinger initially had an even more radical view, which abandoned the point electron entirely: he thought his wave equation described a wave of _charge density_ that _is_ the actual electron, without any need for a dual particle-like entity.
 
 However, both of these attempts to provide a "physically realistic" perspective on the phenomena were quickly abandoned in the face of further evidence suggesting that the wave function fundamentally describes the _probability_ that a particle might appear at a given point in space when measured. As such, the wave is somehow "non physical",  and yet exerts physically-measurable effects.
 
@@ -70,7 +60,7 @@ This strong discretization of the laws of physics is at the root of many seeming
 However, in the 1950's, David Bohm reinvented the original _pilot-wave_ model of de Broglie, and showed that in fact it can fully explain the same phenomena as the standard Copenhagen QM framework. Critically, in this _de Broglie-Bohm pilot-wave_ framework, _particles always have a definite well-defined location_. There is no longer a complementary discretization of the world into measurement vs. wave-function evolution phases: the two are _always_ operating hand-in-glove, all the time.
 
 {id="figure_double-slit-deb" style="height:20em"}
-![Trajectories for particles in the double-slit experiment computed according to the de Broglie-Bohm pilot wave model. The interference effects can be seen as relatively localized bumps in the trajectories, corresponding to steep gradients in the Schrodinger wave equation. Critically, the underlying trajectories are considered to exist at all points even if you don't happen to observe them.](media/fig_double_slit_debroglie_bohm.png)
+![Trajectories for particles in the double-slit experiment computed according to the de Broglie-Bohm pilot-wave model. The interference effects can be seen as relatively localized bumps in the trajectories, corresponding to steep gradients in the Schrodinger wave equation. Critically, the underlying trajectories are considered to exist at all points even if you don't happen to observe them.](media/fig_double_slit_debroglie_bohm.png)
 
 [[#figure_double-slit-deb]] shows what the underlying trajectories of particles under the pilot-wave framework look like in a double-slit experiment, and [[#figure_double-slit-kocsis]] shows some recent data from an experiment where _weak measurements_ that minimally disturb the system allow one to infer particle trajectories, which look remarkably similar to those predicted by the pilot-wave model ([[@KocsisEtAl11]]).