cleanup etc

Author: rcoreilly

content/complex-numbers.md

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+bibfile = "mechphys.json"
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+
+{id="figure_complex" style="height:20em"}
+![Complex numbers are just a way of representing two real values with one number, where these two values are aligned along two separate orthogonal dimensions. The imaginary number $i$, where $i^2 = -1$, is what keeps these two values orthogonal --- the first value $a$ is along the real axis, and the second value $b$ is along the imaginary axis. The complex conjugate, $c^*$, is simply subtracting the imaginary part instead of adding it (i.e., it represents a reflection along the imaginary dimension). Multiplying $c c^*$ gives the squared magnitude of the vector, which is a single real-valued scalar number. It is the (squared) length of the hypotenuse of the vector.](media/fig_complex_numbers.png)
+
+The symbol $\phi$ (another variant of the Greek symbol "phi", like $\varphi$) is used to represent a complex-valued state variable:
+
+- $\phi = a + i b $
+- $= \varphi_a + i \varphi_b $
+
+So, $\phi$ is composed of two separate real-valued numbers, designated $a$ and $b$ (or $\varphi_a$ and $\varphi_b$, to indicate that they are scalar state variables). A complex number is really just a way of representing two separate real valued numbers, aligned along orthogonal dimensions, in an efficient and compact manner ([[#figure_complex]]). It is essential to appreciate that, despite the presence of the imaginary number $i$ (where $i^2 = -1$ or $i = \sqrt{-1}$), _all you ever really have is two real-valued numbers._ There is nothing "imaginary" or mysterious or spooky about the second number in a complex number: all the $i$ does is keep these two values separate from each other.
+
+In the end, we will deconstruct all of our complex numbers into their real-valued components, and write purely real-valued expressions that determine their update rules. These expressions will be more complicated than the ones using complex numbers, but they are required for actually implementing the equations on the computer, and they also provide a more explicit and obvious indication of exactly what drives each value.
+
+Here's a few interesting facts about complex numbers:
+
+To do algebra on them, you just have to remember to _keep the real-values sorted separately from the imaginary ones,_ but otherwise treat them just like a pair of numbers:
+
+- $y = a + ib $
+- $z = c + id $
+- $y + z = (a + c) + i(b + d) $
+- $y z = a c + i a d + i b c - b d $
+- $= (ac - bd) + i (ad + bc) $
+
+Notice that this multiplication rule is the same as $(a + b)(c + d) = ac + ad + bc + bd$, where you just multiply everything through, except that you end up with these $i$ terms crossing over, and when you multiply $ib$ and $id$, you end up with $i^2bd$, at which point the $i^2$ disappears into a $-1$ (i.e., it crosses over from the imaginary world into the real one).
+
+As you should expect from [[#figure_complex]], adding two complex numbers is like adding two vectors, and multiplication is just like multiplying vectors. Complex numbers really are just a compact way of writing vectors!
+
+Multiplication by $i$: If you multiply a complex number by $i$, then you basically switch the real and imaginary parts: the real moves to the imaginary position, and the imaginary becomes real:
+
+- $i(a + ib) = ia - b = -b + ia $
+
+Geometrically, this is equivalent to rotating a vector by 90 degrees! If you do this four times, you'll end up back where you started (as you would expect by doing a 360).
+
+The **complex conjugate** of a complex number is just that number with imaginary dimension inverted:
+
+- $y^\* = a - i b $
+
+The primary use of such a thing is to find the magnitude of a complex number (i.e., the length of the vector that it represents), as:
+
+- $y y^\* = (a + i b)(a - i b) $
+- $= a^2 - iab + iab + b^2 $
+- $= a^2 + b^2 $
+
+This should be recognizable as simply the pythagorean theorem for the squared length of the hypotenuse of a right triangle ($a^2 + b^2 = c^2$). Again, complex numbers have no mystery: they just represent a two-valued vector.
+

content/dirac.md

@@ -4,8 +4,6 @@ bibfile = "mechphys.json"
 
 <!--- TODO: intro, also [[@MallickChandrashekar16]] -->
 
-# Complex Klein-Gordon Waves: Charge!
-
 We have mentioned that a major problem with the scalar KG equation developed so far is that it doesn't represent any kind of conserved value: you cannot compute some constant, unchanging value from the $\varphi$ state variables under this equation. Why is this a problem? If you want to develop a probabilistic interpretation of the wave, as in Schrodinger's equation, then you need this. But the conserved value that emerges naturally from the KG equation comes with two different signs, positive and negative, whereas Schrodinger's equation always produces a positive value. This is one of the major reasons why the KG equation is not widely discussed in quantum physics: it doesn't quite fit with the standard probabilistic framework.
 
 Instead, it seems to make much more sense to interpret the KG waves as **waves of charge**, because charge is also strictly conserved, and it comes in both positive and negative varieties. Indeed, the authors that do write extensively about the KG equation adopt this interpretation (Greiner, 2000; Gingrich, 2004; Mandl and Shaw, 1984). Furthermore, we will see that this charge interpretation fits naturally with the coupling of this KG equation to the electromagnetic field, where the conserved charge value acts just like the electric charge in driving the field. Interestingly, this idea was pursued initially by Schrodinger in 1926, and has been pursued more recently in neoclassical self-coupled field theory (Jaynes & Cummings, 1963; Crisp & Jaynes, 1969; Barut & Van Huele, 1985; Barut & Dowling, 1990; Crisp, 1996; Finster, Smoller & Yau, 1999c; Radford, 2003; Masiello, Deumens & Ohrn, 2005). We discuss this approach, which is essentially an analytic version of our computational model, in more detail later.

content/hamiltonian.md

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-# Energy and Momentum Operators and the Hamiltonian
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+bibfile = "mechphys.json"
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-Using what we have just reviewed about 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 actually derive the KG 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.
+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).
 
-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](ch05_dirac.md) (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.
+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.
 
-You should be familiar with our computation of the total energy associated with a simple wave, which we calculated in the [Waves Chapter](ch02_waves.md). 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).
+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.
+
+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).
 
 Now, we're going to try to formulate the total energy associated with *the "particle" represented by the entire wave function*, instead of thinking in terms of each individual cell within the wave state. We'll see that we can compute the resulting total particle energy using local cell-level calculations, but the motivations and logic are different. There are still kinetic and potential contributions to this overall particle energy, but it is the overall velocity (actually momentum, which is just velocity times mass) of the particle, not the individual cell state, that we are concerned with in computing the kinetic energy.
 
-As discussed in the [Special Relativity](#special_relativity) section, the relativistic total energy of a particle moving with momentum $\vec{p}$ and having a rest mass $m_0$ is given by the following equation:
+As discussed in [[special relativity]], the relativistic total energy of a particle moving with momentum $\vec{p}$ and having a rest mass $m_0$ is given by the following equation:
 
 - **total relativistic energy:** $E^2 = \vec{p}^2 c^2 + (m_0 c^2)^2 $
 - $E = \sqrt{\vec{p}^2 c^2 + (m_0 c^2)^2} $
@@ -19,7 +23,7 @@ To do anything with this total energy equation, we need to be able to compute th
 
 - $\hat{p} = -i \hbar \vec{\nabla} $
 
-We highlight this because we'll keep using it again and again as we work our way up to the Dirac equation. Do not be alarmed by the presence of the $-i$ *imaginary number* at the start of this equation --- we'll get rid of it soon enough. See [Complex_Numbers](#complex_numbers) for more information if you want to brush up on your knowledge of these seemingly strange numbers at this point --- you'll need to really understand them in detail to understand the Schrodinger equation later in this chapter. They really are very simple once you get past all the imaginary business and recognize their actual practical application.
+We highlight this because we'll keep using it again and again as we work our way up to the Dirac equation. Do not be alarmed by the presence of the $-i$ *imaginary number* at the start of this equation --- we'll get rid of it soon enough. See [[complex numbers]] for more information if you want to brush up on your knowledge of these seemingly strange numbers at this point --- you'll need to really understand them in detail to understand the Schrodinger equation later in this chapter. They really are very simple once you get past all the imaginary business and recognize their actual practical application.
 
 The total energy operator for the wave state is computed in terms of the overall rate of change across all the wave cells, consistent with the notion that energy is a function of the velocity (kinetic energy) of the cell states: