superposition

Wave mechanics explains a lot of stuff, and helps us visualise things such as atomic and molecular orbitals, and even why electrons prefer to be a certain distance from the nucleus.

One cool thing you can try on holiday or if you have access to a swimming pool, or even a water basin is: Make sure the water is still. Take your two hands and put your index fingers about 15-25 cm apart (just a reasonable distance) and pointing at the water. Dab your fingers in and out of the water simultaneously and at regular intervals (say 2-3 times a second). Make sure light is cast on the water and pay attention to the shadows of the ripples on the bottom of the pool/sink and notice how in all that turbulence, as the waves "collide" and superpose, some parts of the water are absolutely still. Simple but just cool if you can get it right.
Plenty of people will be reading this thinking, "This guy is so basic, I can solve Schrodinger's Equation for a many electron system!" whereupon I reply, "no. you can't".

This helps tremendously if sources are placed in strange places and you can't really tell which direction current is flowing. Normally, node analysis is sufficient, but in some cases using superposition is necessary. Superposition is somewhat slow, because you have to redraw the circuit as many times as sources you have, unless you become very good at it.

This is a typical circuit you might choose to analyze with superposition:

                           R1                          
     -----------------/\/\/\/\/\----------------
     |             |                           |              
     |             |                           |
    /+\            \                          / \
   /   \           /               I Amp     /  ^\
  |  V  |V Volt    \ R2            Current  | I | |
   \   / Voltage   /               Source    \  |/
    \-/  Source    \                          \ /
     |             |                           |
     |             |                           |
     -------------------------------------------
                            |
                          -----  Ground
                           ---
                            -

So, we have to remove each source and analyze the circuit. Let's start by removing the voltage source. When we do that, we replace the voltage source with a short circuit. The circuit now looks like this:

                           R1                          
e2-->O----------------/\/\/\/\/\---------------O --> e1
     |             |                           |              
     |             |                           |
     |             \                          / \
     |             /                I Amp    /  ^\
     |             \ R2             Current | I | |
     |             /                Source   \  |/
     |             \                          \ /
     |             |                           |
     |             |                           |
     -------------------------------------------
                            |
                          -----  Ground
                           ---
                            -

That wasn't too painful. Now we replace the voltage source, and take out the current source. When you take out current sources, they must be replaced with open circuits. Here is that circuit:

                           R1                          
e2-->O----------------/\/\/\/\/\---------------O --> e1
     |             |                           |              
     |             |                           |
    /+\            \                         ----- +
   /   \           /               
  |  V  |V Volt    \ R2                            VOC
   \   / voltage   /                
    \-/  source    \                         ----- -
     |             |                           |
     |             |                           |
     -------------------------------------------
                            |
                          -----  Ground
                           ---
                            - 

Now we can find the total analysis by summing the results we have obtained. First, the voltage e2 turns out to be 0 + V = V. This makes sense because that node is hooked up directly to the voltage source. Next let's look at e1. e1 = (I * R) + V. The current going through the voltage source is I + (V/R). The current through R1 is I + 0 = I. The current through R2 is 0 + V/R = V/R.

This circuit can also be easily analyzed by node analysis. Node analysis gives you a system of equations, though. If there are many nodes and many sources, it will be difficult to solve this system. Superposition helps to alleviate this complexity, though it can be time consuming.

In Perl 6, a superposition is a simultaneous combination of various elements, generalized from quantum mechanics. For simplicity, Perl does not entangle itself with the details of entanglement.

Superpositions are constructed through the use of the any and all operators. These have infix equivalents: | and & (formerly bitwise or and and). These operators, incidentally, should be read "or" and "and."

Superpositions are a way to declaratively state what would previously have only been possible procedurally. So, instead of:

    if $a == 1 || $a == 2 || $a == 3 { ... }

You could write:

    if $a == 1 | 2 | 3 { ... }
# (which is)
    if $a == any(1, 2, 3) { ... }

Following the dreams of many early programmers. Of course, these statements really return a superposition of boolean: if $a was 2, then the former statement would be any(false, true, false). But a disjunctive superposition evaluates true if any of its arguments are true. Conversely, a conjunctive superposition evaluates true only if all of its arguments are true. This ends up DWIMming.

You can combine superpositions on both sides of a comparison:

    if 3 < 2 | 4 | 6 { ... }  # True
    if 3 < 2 & 4 & 6 { ... }  # False
    if 3 & 4 < 5     { ... }  # True
    if 3 & 5 < 5     { ... }  # False
    if 2 | 3 < 3 & 7 { ... }  # True
    if 6 & 7 < 4 | 6 { ... }  # False

Scalar arithmetic on a superposition works just like that on a vector:

    2 * any(1, 2, 3) == any(2, 4, 6)

You can combine superpositions:

    if all(1,2,3)*any(5,6) < 21 { print "No Alchohol!\n"}
# (Equivalent to:)
    if any(all(5,10,15),all(6,12,18)) < 21 { ... }
# (Or:)
    if all(any(5,6),any(10,12),any(15,18)) < 21 { ... }

As you can see, they are stored and evaluated recursively. This is very handy.

Perl 6 will allow extraction of the eigenstates of a superposition, with the states function. This along with the isa function allows you to decompose a superposition completely (despite the theoretical impossibility from quantum mechanics):

    sub printsuper($sup) {
        given $sup {
            when Superposition::Conjunction {
                print "all(";
                printsuper($_) for states;
                print ")";
            }
            when Superposition::Disjunction {
                print "any(";
                printsuper($_) for states;
                print ")";
            }
            default {
                print;
            }
        }
        print ",";
    }

(given..when is the Perl6 equivalent of C's switch..case. The exact class names that all and any produce have yet to be decided)

Note that the return value of a boolean comparison is another superposition of all the states that matched:

    (3 | 4 | 5 < 4)  == 3
    (3 | 4 | 5 < 5)  == 3 | 4
    (3 & 4 & 5 < 6)  == 3 & 4 & 5

This turns out to be a lot more useful than a bare boolean.

It is now decided that these will be called junctions, where all is a conjunction, and any is a disjunction. This was decided to help beginners grok the concept better, with a more familiar word.

The Superposition Principle

In Physics, the principle of superposition says roughly that if two particular sorts of behavior are allowed in a physical system separately, then if you try to cause both simultaneously, the result will be the sum of the two individual behaviors (where sum is defined in a sensible way, mathematically) called the superposition of the two. This principle is not true in all (or perhaps even most) physical systems, but it is often at least approximately true¹. The "principle" as I've just stated it undoubtedly sounds vague. That is because it is a very general property that exhibits itself in many different sorts of systems, which are then said to "obey the superposition principle". We can make the definition more exact by relating it to the mathematical idea of linearity, which I discuss in the last section.

When it applies, the superposition principle is extremely useful, because it allows us to take a complex situation and think of it as the sum of many, simpler parts. By being able to break down a complex problem into many simple problems, we are able to understand a lot more than we otherwise would be able to. In fact, most of our understanding of the physical world concerns situations where the superposition principle is true, and we are only recently starting to understand systems without the superposition, in the fields of nonlinear dynamics and chaos.

Now let's talk about a specific example of superposition to make this idea more concrete.

An Example of Superposition

The best example I can think of to illustrate the idea of superposition here is that of waves traveling on an idealized string. If this string obeys the superposition principle, and you can make two different wave packets (meaning two waves of certain shapes), A and B, on the string, then the principle tells you that when the wave packets overlap the result will just be as though one wave packet was stacked on top of the other one; the height of the result is the sum of the heights of the individual wave forms, in other words. At that point the two wave packets are said to be "superposed"². Furthermore, once they no longer overlap, each wave packet will continue on as if the other were never there. Here is a crude picture of what this would look like using a series of "freeze frames" at successive times:

  A  ______                                       B
    |      | --->                        <---  /\
    |      |                                  /  \
____|      |_________________________________/    \_____



                      _____/|
                     |      | /\
                     |      |/  \
_____________________|           \_______________________



                            /|
                           / |
                       ___/  |
                      |       \
                      |        \
______________________|         \________________________



                          /\
                         /  \
                        /    \ 
                       |      |
                       |      |
_______________________|      |__________________________



                          |\
                          | \ 
                          |  \____
                         /       |
                        /        |
_______________________/         |_______________________



                           |\_____
                        /\ |      |
                       /  \|      |
______________________/           |______________________



    B                                           ______  A
 <--- /\                                       |      | --->
     /  \                                      |      |
____/    \_____________________________________|      |____

I hope that diagram is clear. Please /msg me if it isn't and try to be as specific as possible about what confuses you. The first and last diagrams are supposed to be long before and after the two wave packets cross.

Applicability of Superposition

Perhaps one of the most important examples of where the principle of superposition is exactly true is the theory of electromagnetic fields in vacuum (the classical theory, anyway³). If two different electric and magnetic fields can exist in a certain region, then the sum of those fields is also a possible behavior. Also, if a certain configuration of electric charges and electrical currents cause one field, and a second set of charges and currents cause a different field, then if both sets of charges and currents are present it will cause the sum of the two fields. It is this last property that makes the superposition principle particularly useful in studying electromagnetism. When you have a complicated source (some set of currents and charges), you can think of that source as the sum of many simpler sources. You can then figure out the effect of each of those sources separately. Afterward, you can add up the resulting fields to get the total field that the total source will cause. The simpler sources can be point charges, sheets of charge, or other simple charge distributions, and you can think of many similarly simple current sources. This technique can make the situation much easier to figure out. Another approach using the same idea is using a multipole expansion. Even if the source can't be described exactly by a few simple sources, this will often give a good approximate description.

In many other cases, the superposition principle holds for a good approximate description of the system in question. Most systems⁴ that have a stable equilibrium configuration will behave linearly for small deviations from that equilibrium[2], and disturbances from equilibrium should stay small (since it's a stable equilibrium), so you can describe them approximately with a theory that obeys the superposition principle as long as the motion does not start out too large. What is "too large" is defined by how far the system can go from equilibrium before nonlinear behavior becomes important, which depends on the details of the system in question. This sort of approximate validity is usually the case for electromagnetic fields inside of a medium, like glass or water. The superposition principle will be true for small field strengths, but for strong fields it will not hold true.

One example of a situation in which the superposition principle holds approximately is an electromagnet with an iron core. Putting electrical current into the electromagnet causes an electromagnetic field in the core. The core, which is often a ferromagnetic material, is there because the field caused in it by the current is stronger than the field that would be caused in empty space. If you add two small currents in the electromagnet, the resulting field will be approximately the sum of the fields that would have been caused by each current alone, obeying the superposition principle; however, if you add a large current the increase in magnetic field within the core will not be as large as you would expect from the superposition principle, because eventually the iron core begins reach "saturation". This is the point where nonlinearities in the behavior of the core have become important.⁵

The most common context in which to discuss superposition is the study of waves. A linear wave theory is one that describes waves in a system in which the superposition principle holds exactly or at least approximately. Systems described by a linear wave theory include electromagnetic waves (a.k.a. light), gravitational waves, sound waves, and mechanical vibrations. Waves occur in a system that is close to stable equilibrium, so as I discussed before this often means that the superposition principle will hold true as long as the waves are small enough. When we have the superposition principle to rely on, even the motion of complicated wave packets can be understood, because we can understand them as nothing more than the sum of many simple, periodic waves. As long as we can understand how the simple waves behave, then we can add up the results to understand how the complex wave packet behaves. When two different waves come together, the result of superposing the two waves is usually called interference. In systems that only approximately obey the superposition principle, when the waves get too large and we no longer have simple superpositions, we enter the realm of waves in nonlinear media, which are much more difficult to understand and predict. Most of the time at a high school or undergraduate level you will only ever talk about waves in systems that obey the superposition principle, because those ones can be understood so much more simply and completely.

The superposition principle is useful to understand a huge variety of systems. The reason it's so useful is that it allows us to break down a complex problem into many simple problems and solve it piece by piece, since we can get the result just by adding up the results of the individual pieces. Practically all widely studied, well understood systems obey the superposition principle; the concept of superpositon is central to most of the study of Physics. Systems that don't are said to be "nonlinear" and are studied in nonlinear dynamics and chaos, but usually you can't say nearly as much about them as you can about linear systems with superposition. Superposition is the consequence of a mathematical property called linearity, which I'll finish up by describing.

Mathematically Defining Superposition

I've said that systems that obey the superposition principle are called linear systems. One way this is often defined mathematically is to say that the quantities of the system you're interested in behave according to an equation of motion that is a linear differential equation:

a_n(t)*(dⁿy/dtⁿ) + … + a₂(t)*(d²y/dt²) + a₁(t)*(dy/dt) + a₀(t)*y = 0

where dⁿy/dtⁿ is the n^th derivative of y(t) written in Leibniz notation. The important feature of that equation is that if a function y_B(t) is a solution and a function y_C(t) is a solution then the function y_D(t) = y_B(t) + y_C(t) is also a solution. y_D is the superposition of y_B and y_C. While the above is an ODE, it could just as well have been a PDE. To put it more generally and formally, a linear system is one in which the solutions (the possible behaviors) form a vector space. This is true of the solutions of the differential equation above.

Sources:

Su`per*po*si"tion (?), n. [Cf. F. superposition. See Super-, and Position.]

The act of superposing, or the state of being superposed; as, the superposition of rocks; the superposition of one plane figure on another, in geometry.

© Webster 1913.