People are always shocked when they first begin programming because they quickly learn that even the slightest mistake is not tolerated. The slightest programming error can easily lead to catastrophic results. Usually, programs do not even compile on the very first few shots. Once you finally do get the program to compile and link into an executable file that can actually run on a computer, you quickly find that the program is now filled with all sorts of logical bugs that make it crash or perform improperly. Debugging software can be a very long and tedious process. Worse yet, even after the new code finally does successfully go through User Acceptance Testing and Production Assurance Testing, it can still have bugs that are only discovered after the new code goes into Production, and these bugs may not appear for several days and might be very difficult to repeat in a reproducible manner that allows you to troubleshoot the underlying problem. That is why, after all of that angst, new programmers are usually quite surprised to discover that perfectly debugged code always works. After all, it is quite amazing that a machine composed of many billions of transistors that are constantly switching on and off many billions of times each second could possibly yield the very same result each time it is run with the same set of input parameters no matter how many times you run the program. As we saw in The Fundamental Problem of Software it is the second law of thermodynamics operating in a nonlinear Universe that makes it very difficult to write and maintain software. This is largely due to the second law of thermodynamics introducing small bugs into software whenever software is written or changed and also to the nonlinear nature of software that allows the small software bugs to frequently produce catastrophic effects.
The reason that totally bug-free software always works is that, even though software is nonlinear and subject to chaotic behavior, it is still deterministic. That means that software always produces exactly the same results so long as the input data does not change. Physicists would say that software is deterministic because it always produces the same results so long as the initial conditions are not changed. For example, if your program will not compile, simply trying to recompile the exact same program again will not change the results. Unfortunately, if your program does compile, but has bugs that produce results that you do not like, simply rerunning the buggy program again will not produce different results.
Figure 1 - Given the initial conditions of a cannonball with velocity V0 and a cannon angle of inclination θ0 we can use Newton's laws of motion and gravity to calculate its path and the path will always be the same. Newton's laws of motion and gravity can be thought of as the software of the Universe, and it is deterministic in nature.
For example, given the initial position and velocity vector of a cannonball leaving a cannon, we can use Newton's equations of motion and Newton's theory of gravity to calculate how the cannonball will move with time. The cannonball will always follow the very same path through space and time because Newtonian physics is deterministic and always produces the same results for any given set of initial conditions or input data. Now because Newton's equations of motion can be expressed as linear differential equations, it means that we can use calculus to solve them and produce a mathematical formula that describes the motions we see in Figure 1. In fact, Issac Newton first invented calculus just so he could do so. However, the differential equations that describe most systems in our Universe are not linear differential equations and that means that we cannot use calculus to solve for a mathematical formula that describes how the system changes with time. Instead, we have to use a computer to simulate how nonlinear systems change with time.
Figure 2 - A double pendulum is a nonlinear system that can be described by nonlinear differential equations that cannot be solved using calculus. Instead, we need a computer to simulate how the nonlinear system changes with time. In the left simulation, angles θ1 and θ2 both start out at an angle of 0 radians to the horizontal. In the second simulation θ1 = 0 and θ2 = 0.1 radians which is 5.72958 degrees to the horizontal.
In Figure 2 above, we see a classic nonlinear system composed of two pendulums. If we start both pendulums out with nearly the same initial conditions, we see that at first, both systems behave in a very similar manner, but soon they depart and behave in entirely different manners. At first, it seems that nonlinear systems behave in an entirely random manner, but that is not so. Each time the two simulations start out, they both follow a deterministic path that is always the same. However, small changes to the initial conditions of the nonlinear systems can rapidly produce different paths. Physicists call that chaotic behavior. Since nearly all of the real systems in the Universe are nonlinear, much of the apparently random nature of the Universe stems from the fact that nearly all of the systems in the Universe are nonlinear. In fact, since all of our current theories in physics are deterministic in nature, except for quantum mechanics, most of the apparent randomness in the Universe must stem from its deterministic, but chaotic, nonlinear nature. That is why flipping a coin does not actually yield a truly random result. If you could exactly reproduce the initial conditions of a coin flip, you would always get the same result with each flip. But because coin flips are nonlinear, even the slightest change of initial conditions could yield just the opposite result. Since it is nearly impossible to exactly reproduce the initial conditions of a coin flip, they always seem to be random. In truth, coin flips are just deterministic nonlinear systems in a chaotic regime that makes them appear to be random.
Along Comes Einstein
In 1905, Einstein introduced his special theory of relativity, and soon after in 1908, the mathematician Hermann Minkowski reformulated Albert Einstein's special theory of relativity for a 4-dimensional spacetime.
Figure 3 - In 4-Dimensional spacetime, the Universe exists as an unchanging 4-Dimensional block of spacetime that describes the Present, Past and Future in an unchanging deterministic manner. If you know the conditions of Now, you can theoretically determine the Past and the Future in an unambiguous deterministic manner because all of the theories of physics are deterministic. Any seemingly random conditions in the Past or Future can be easily explained by the chaotic behavior of nonlinear, but nonetheless, deterministic systems interacting with each other.
With the introduction of the special theory of relativity in 1905, Einstein got rid of the concepts of absolute space and time and combined them into an eternal, absolute and unchanging block of spacetime that allowed the Universe to operate in a totally deterministic manner. But this did cause a problem for the concept of simultaneous events in the Universe that were separated in space. This is best explained by the two animations at:
http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/Special_relativity_rel_sim/index.html
In the first animation, the Einstein on the moving platform observes two photons arriving at the same time and concludes that both photons were emitted at the same time. In the second animation, the Einstein on the ground agrees that both photons hit the Einstein on the platform at the same time too, but concludes that photon A was emitted before photon B. Note that both Einsteins can claim to be standing still because neither one can detect any motion with experiments they perform because neither one is accelerating. What the special theory of relativity did for the Block Universe model of Figure 3 was to replace the concepts of absolute space and absolute time with the concept of an absolute 4-dimensional spacetime. With this approach, time was simply merged in with the three spatial dimensions (x, y, z) into a 4-dimensional spacetime of (x, y, z, t), and time just became a strange cousin of the other spatial dimensions. The distance between events in 4-dimensional spacetime is called the interval s and is defined just like the distance d between points in 3-dimensional space.
Distance between points in space:
d² = ∆x² + ∆y² + ∆z²
Interval between events in spacetime:
s² = ∆x² + ∆y² + ∆z² + i²∆t²
where i² = -1
In 4-dimensional spacetime, the only distinguishing thing about time is that you have to multiply time by the imaginary number i in the equation for the interval between events. Otherwise, time is simply treated like the other spatial dimensions of x, y and z. Essentially, with the 4-dimensional spacetime Block Universe model of the Universe, the Universe became a static tube of salami that had seemingly always existed. Observers moving relative to each other simply sliced the static tube of salami at different angles. For example, in Figure 4 we see the Block Universe tube of salami cut at two different angles. In the first cut of the salami, two events happen at different positions in space and also different positions in time so the events are not simultaneous. This is the situation for our second Einstein on the ground who observed two photons being emitted at different places in space and also at different times. The second salami is cut at a different angle, which again has the two photons being emitted at different positions in space, but this time both photons are emitted at the same time on the time slice of salami. This is the situation for our first Einstein riding on the platform who observed two photons being emitted at different places in space, but at the same time in his reference frame.
Figure 4 – The special theory of relativity extended the Block Universe model by introducing the concept of an absolute spacetime salami that could be cut at different angles for observers moving relative to each other.
This new 4-dimensional spacetime Block Universe model, brought on by the special theory of relativity in 1905, really seemed to kill the concepts of absolute space and an absolute time because space and time got thoroughly blended together in the process of making the spacetime salami so that you could no longer tell them apart. If I am moving relative to you that means that my time can become your space and vice versa.
Spacetime Salami Cones
But you cannot cut the spacetime salami by more than 45o and that means the Universe has a speed limit equal to the speed of light. That means for each event A in the spacetime salami there is a light cone that defines what points in the history of the spacetime salami that could have affected event A and what future points in the spacetime salami that event A could affect. Physicists call these spacetime salami cones light cones.
Figure 5 - Above we see an event A within the spacetime salami. In the diagram, time T is measured in meters. A meter of time is the amount of time that it takes light to travel one meter. If we measure the spatial dimensions of X and Y in meters and also the dimension of time T in meters, then a beam of light will travel at a 45o angle because light travels one meter of distance in one meter of time. The event E in the above diagram is too far away from event A in the spacetime salami to have affected event A. Similarly, event A cannot affect event E. Physicists would say that events A and E are causally disconnected because they cannot cause each other to happen.
Einstein Squishes the Spacetime Salami with the General Theory of Relativity in 1915
The special theory of relativity does not work for accelerating reference frames like a SpaceX rocket blasting off. For that, we need Einstein’s general theory of relativity (1915). With the general theory of relativity Einstein extended the 4-dimensional spacetime Block Universe model to all forms of motion, including accelerating reference frames, and in the process demonstrated that accelerated motion was essentially the same thing as gravity. In the general theory of relativity, gravity is no longer a force between masses. Instead, gravity becomes another “fake force” like the centrifugal force. When you make a sharp turn in your car at high speed your body feels the “fake force” of centrifugal force pulling you away from the center of curvature of your turn. But that “fake force” is really just your body trying to move in a straight line through space according to Newton’s first law of motion. Similarly, in the general theory of relativity the gravitational force your body feels pulling you down to the Earth is simply your body trying to move in a straight line through spacetime and is also a “fake force”! If you jump off a cliff and find yourself in free fall, moving in a straight line through spacetime, gravity suddenly disappears, just as if you had been thrown from a car making a tight turn and had found yourself moving in a straight line through space with the “fake” centrifugal force suddenly disappearing too. In order to make that adjustment to the special theory of relativity that used flat time slices through the 4-dimensional spacetime salami, Einstein had to make the 4-dimensional spacetime salami internally curvy. In the general theory of relativity matter, energy and pressure can all cause spacetime to warp and become curvy, and it is the curvy 4-dimensional spacetime that creates the illusion of gravity. When there are no significant amounts of matter, energy or pressure present, the 4-dimensional spacetime salami is not internally distorted, so that slices through it are flat and we return again to the special case of flat spacetime that is covered by the special theory of relativity.
Figure 10 – In the general theory of relativity the 4-dimensional spacetime salami becomes internally curvy. Above we see a light cone within the distorted spacetime salami.
The Serene Block Universe of Albert Einstein in 1925
By 1925, Albert Einstein was the Einstein of his time and celebrated by all, including the general public, who probably at the time knew of no other physicist by name. Einstein was a Realist so Einstein's Block Universe consisted of real things, like electrons and protons that actually existed and moved about in the spacetime salami in a deterministic manner that could be measured in things like cloud chambers. His General Theory of Relativity eliminated the "spooky action at a distance" of Newton's theory of gravitation. Gravity was no longer magically created by masses at extreme distances that reached out across the vastness of space to pull on distant objects. Instead, gravity was now a local effect. The mass of a star distorts the spacetime around the star and a planet orbiting about the star was simply moving in a straight line through the locally distorted spacetime that the planet was traversing.
The electric and magnetic forces had been previously turned into local forces in the 19th century by people like Michael Farraday and James Clerk Maxwell. Instead of the electric and magnetic forces being mysterious forces arising from charged particles that could reach out into space and affect other charged particles, the electric and magnetic forces had been unified into a single electromagnetic force composed of electromagnetic fields. Charged particles created electromagnetic fields and then other charged particles experienced the electromagnetic fields when they moved through them. In fact, by 1925 people were already making money by broadcasting radio programs with ads using electromagnetic waves by vibrating these electromagnetic fields. So by 1925, all of the known forces of nature in Einstein's Block Universe were both deterministic and local.
There Was Only One Problem
Einstein's Block Universe did not work for small things like atoms. According to the most recent atomic theory of the day, atoms consisted of multiple electrons orbiting about the protons in a centralized nucleus. This meant that the electrons of an atom were constantly accelerating as they quickly raced about the nucleus. Maxwell's 19th-century equations for electromagnetic fields predicted that the accelerating electrons should give off electromagnetic waves and quickly lose energy and collapse into the nucleus. After all, people were making some serious money by accelerating electrons up and down radio towers to broadcast radio programs. Clearly, something was wrong with Einstein's Block Universe for small things.
There were some preliminary efforts to modify Einstein's Block Universe for small things like atoms, but the real breakthrough occurred in 1926 when Schrödinger and Heisenberg developed quantum mechanics. For more on that see Quantum Software. In 1926, Erwin Schrödinger was a 38-year-old physicist still struggling to make his mark in physics. The rule of thumb in physics is that most of the great advances come from people under the age of 30, probably because the inspiration/revelation step of the scientific method seems to come easier to the young at heart with less to lose. At this point, physicists had been working with the mathematics of waves for more than 100 years and were well aware that all known waves obeyed a differential equation of a particular form known as the wave equation. Schrödinger was convinced that for the de Broglie hypothesis about the wavelike behavior of matter to advance, a wave equation for particles was required. So in the winter of 1926, Schrödinger packed up his latest mistress and left his home and wife in Vienna for a couple of weeks of sabbatical in a Swiss chalet. There he worked out a famous wave equation for particles, now known as the Schrödinger equation. The story goes that he had two pearls, one for each ear, that allowed him to work undisturbed by his mistress. His wife back in Vienna certainly was not a distraction either. He truly was young at heart at the age of 38 and ripe for some inspiration/revelation.
To understand the significance of all this, we need to delve a little into the mathematics of differential equations. Imagine a very long taught guitar string stretched between two walls that are separated by a large distance. If we pluck the string near the left wall, a pulse will begin to travel to the right. Figure 11 below is a snapshot of a small section of a possible pulse at a particular time as it moves to the right.
Figure 11 (click to enlarge)
The motion of the pulse can be described by a scary looking differential equation, known as the wave equation for a stretched string, which describes how each small section of the string moves up and down as the pulse passes by.
The Wave Equation of a Stretched String
∂²y = μ ∂²y
── ── ───
∂x² T ∂t²
Now let’s spend a little time with the wave equation to show that it is really not so scary after all. The project plan for a large IT project can be pretty scary too if you look at the whole thing at once. However, if you break it down into its individual tasks, it looks much less formidable. It’s important to take life one 2x4 at a time. The first term in the equation on the left is called the second partial derivative of the pulse with respect to the x-axis (distance along the string). It is just the curvature of a small section of the string as the pulse passes by (see Figure 11):
∂²y
──
∂x²
When this term is a big number, it means that the curvature of the string is large and the string has a peak or valley. When this term is a smaller number, it means that the curvature is low and the string is kind of flat. The variable μ is the density of the string. When μ is large, it means that the string is heavy; when μ is small, it means that the string is light. The variable T is the tension in the string, or how tightly the string is stretched. So if we just look at what we have so far, we see that the curvature of the string pulse is equal to the density of the string μ divided by the tension T of the string times “something else”. This makes sense. When the string is stretched very tightly, the tension T is large, so the curvature of the pulse should get flatter (smaller). Also, when the density μ of the string gets bigger, the curvature of the pulse should get larger too, because a heavy string should be less flattened by a tension T, than a light flimsy string. Now we have to look at the “something else” term on the far right:
∂²y
──
∂t²
This term is called the second partial derivative with respect to time. It is just the curvature of the pulse in time – how fast a tiny section of the string accelerates up and down as the pulse passes by. A flat pulse will not accelerate up and down very quickly as it passes by, while a tightly peaked pulse will accelerate up and down quickly as the pulse passes by.
The wave equation for the stretched string now makes sense as a whole. It simply says that the curvature of the pulse along the x-axis gets bigger when the density μ of the string gets bigger, or when tension T gets smaller, or when the rate at which the string accelerates up and down as the pulse passes by gets bigger. When you work out the mathematics, the velocity of the pulse is given by:
____
v = √ T/μ
The hard part about differential equations is solving them. You have to find a curve that meets the above requirements. When you take a course in differential equations, they teach you all sorts of ingenious techniques using calculus to guess what curvy line fits the requirements. For the problem at hand, the solution to the wave equation for a stretched string fixed at both ends is a series of standing sine waves, which look something like Figure 12 below. The string can be excited into many standing waves, defined by a number n = 1, 2, 3,… which describes how many wavelengths of the standing wave just exactly fit between the two fixed points.
Schrödinger’s Equation
Classical 19th-century electrodynamics also had another problem. According to classical electrodynamics, the walls of the room in which you are currently located should be at a temperature of absolute zero, having converted all of the energy of the free electrons in the walls of the room into ultraviolet light and x-rays. This was known as the “Ultraviolet Catastrophe” at the time and is another example of an effective theory bumping up against the limitations of its effective range of reliable prediction. In 1900, Max Planck was able to resolve this dilemma by proposing that the energy of the oscillating electrons in the walls of your room was quantized into a set of discrete integer multiples of an elementary unit of energy E = hf.
Thus:
E = nhf
where
n = 1, 2, 3, ...
h = Planck’s constant = 4.136 x 10-15 eV sec
f = frequency of the electron oscillation
You can read a translation of his famous 1901 paper, that first introduced the concept of quantization at:
http://web.ihep.su/dbserv/compas/src/planck01/eng.pdf
Max Planck regarded his inspiration/revelation of the quantization of the oscillation energy of the free electrons and their radiated energy as a mathematical trick to overcome the Ultraviolet Catastrophe. But in 1905, the same year that he published the special theory of relativity, Einstein proposed that Planck’s discovery was not a mathematical trick at all. Einstein proposed that sometimes light, an electromagnetic wave in classical electrodynamics could also behave like a stream of quantized particles, that we now call photons, with energy:
E = hf
Similarly, in 1924 Louis de Broglie proposed in his doctoral thesis, that if light waves could sometimes behave as particles, then perhaps particles, like electrons, could also behave like waves with a wavelength λ of:
λ = h/mv
where
λ = wavelength
h = Planck's constant
m = mass of the particle
v = the velocity of the particle
Working with Einstein’s equation for the energy of a photon and de Broglie’s equation for the wavelength of a particle, Schrödinger had a problem. Unlike the velocity of a wave on a string, which only depended upon the density of the string μ and its tension T, the velocity of a particle’s wave depended upon the wavelength of the particle λ:
v = h
──
2mλ
Schrödinger believed that a particle must really be a wavepacket of many superimposed waves of different wavelengths that added up in phase near the location of the particle. Given the above equation, the waves in the wavepacket would tend to move at different velocities because they all had different wavelengths. The traditional wave equation, like the one for a wave pulse on a string, would not work under such conditions. Schrödinger overcame this problem with the following compromise equation, which sort of looks like a traditional wave equation. Note that the wavefunction Ψ is just a wiggly line, like the pulse on our string, and is pronounced like the word “sigh”, and m is the mass of the particle.
-ħ² ∂²Ψ = iħ ∂Ψ
── ── ──
2m ∂x² ∂t
He had to make two modifications to the standard wave equation:
1. He used the first partial derivative with respect to time, instead of the second partial derivative with respect to time on the far right side of the “=” sign
2. The equation contained:
____
i = √ -1
(or in other words i² = -1 )
which meant that the Schrödinger equation was a complex differential equation, with an imaginary part containing the quantity “i”, the square root of -1. Now we all know that there is no “real” number, that when multiplied by itself (squared) produces a -1, but that does not scare off mathematicians! Several hundred years ago, mathematicians became comfortable with the idea of an “imaginary” number i, which they defined as the square root of -1. “Real” numbers, the kind of numbers that we are used to dealing with, are just numbers that do not have an imaginary part. A little later, physicists discovered that the Universe seemed to just love “imaginary” numbers. The imaginary number i started to pop up in all sorts of equations and was nearly as popular as π. Now the fact that Schrödinger’s equation contained an imaginary part meant that solutions to the equation, known as wavefunctions Ψ, would not be totally “real” either, because they would contain imaginary parts using the square root of -1. As we shall see, this implication created a problem for the interpretation of what exactly a wavefunction really was. All of the other waves we had experience with, like waves on a string, light waves, or water waves were “real” functions or curvy lines. What did a complex wavefunction, with both “real” and “imaginary” parts, mean?
Schrödinger’s Time Independent Equation for a Single Dimension
We can simplify Schrödinger’s equation by getting rid of the part that depends upon time for the cases where the energy E does not change with time. That is certainly true for the electrons in an atom, so long as they remain in their orbits and do not jump from one orbit to another and emit or absorb a photon.
-ħ² d²ψ(x) + V(x) ψ(x) = E ψ(x)
── ──────
2m dx²
In this equation, we use ψ for the wavefunction, instead of the full-blown Ψ, because it does not contain the part that varies with time. ψ is still just a wiggly line, like the pulse on our string, and is still pronounced like the word “sigh”. In addition, we added a term V(x) which is another function or wiggly line that describes how the potential energy of the particle varies as it moves back and forth along the x-axis. Imagine a straight road that runs across a hilly landscape and that you are riding a bicycle. If you ride your bicycle up a hill, you can tell that you are increasing your potential energy V(x) because it is hard to peddle up the hill as you convert some of your kinetic energy into potential energy. Similarly, it is easy to coast down a hill on a bicycle, because some of the potential energy V(x) that you have stored away, is converted back into kinetic energy. Another way of looking at this is that V(x) is also a way of describing a force. For example, the topography of the wiggly V(x) function seems to produce a force pulling you down the hill and another force impeding your progress up the hill. So the above formula allows you to calculate the wavefunction ψ for a particle subject to a force.
The Particle in a Box
Recall Figure 1 in my posting The Demon of Software, which depicted two containers full of molecules bouncing around. In 1872, Ludwig Boltzmann developed statistical mechanics by envisioning the molecules existing in a large number of microstates. This was many years before the arrival of quantum mechanics, so Boltzmann had to make up his microstates by arbitrarily force-fitting the molecules into little mathematical cubicles, like saying “I can tell by your energy that you are a perfect size 8”, at least approximately. Let us now apply Schrödinger’s equation to this problem to find the real microstates. To simplify the problem, imagine a single container of width “L” along the x-axis containing a single particle, such as an electron, in it. The electron is bouncing back and forth along the x-axis like a target in a shooting gallery. The electron only feels a force when it hits one of the walls on either the left or right side of the container at x = 0 or x = L, so the potential energy V(x) between the walls is zero. At the walls located at x = 0 and x = L, the electron feels an infinite force pushing it back into the container, so the potential energy hill V(x) at these two points is very steep and actually goes straight up to infinity.
When you solve Schrödinger’s equation for this problem, you get quantized solutions that are sine wavefunctions:
____
ψn(x) = √ 2/L sin(nπx/L)
n = 1, 2, 3, ...
En = n²h²
─────
8mL²
n = 1, 2, 3, ...
where
m = mass of the particle (electron in this case)
L = width of the box
h = Planck’s constant
n = quantum number
For example for the lowest energy level where n = 1:
____
ψ1(x) = √ 2/L sin(πx/L)
E1 = h²
─────
8mL²
What is happening here is that you can only fit sine waves into the box that have wavelengths that fit just right. The “n”s determine the number of wavelengths of the sine wavefunction that fits in the box properly and the energy En of each wavefunction. It is easier to see this as a plot rather than as an equation. Figure 12 shows the first three wavefunctions for n = 1, 2, and 3.
Figure 12
The above wavefunctions look a lot like the first three octaves of a vibrating guitar string, and indeed, are identical to what you get for a vibrating string.
Remember, I warned you about wavefunctions being strange. At first, nobody really knew what to do with them. Physicists are just like other people. They tend to take ideas that they are familiar with and project these ideas onto things that are new to them. Physicists have a few popular models. Grab a small pebble and keep making it smaller in your mind, until it is infinitely small with a dimension of zero. When you are finished, you have a particle. Now take the particle and throw it into a still pond. The resulting waves that propagate away are another good model. Take a bar magnet, as Michael Faraday did, and cover it with a piece of paper. Then sprinkle some iron filings over it. The “lines of force” that you see are a field. Faraday called it a field because it reminded him of a freshly plowed farmer’s field. So when Schrödinger came up with his equation, nobody really knew what to make of the wavefunction solutions to the equation. Schrödinger thought they might have something to do with the electric field of electrons, but he wasn’t quite sure.
In 1928 Max Born came up with another interpretation. Born proposed that the wavefunction was really a probability wave. Imagine that a gang of young thugs moves into your neighborhood and that they begin to knock off liquor stores late at night in the surrounding area. In order to avoid capture, the gang decides to only knock off each liquor store once in case the police are in hiding. If you plot the liquor stores that get knocked off versus time, you will see an expanding wave of crime. The odds of a particular liquor store getting knocked off increase as the wave passes by and diminishes as the wave moves on. Recall that usually, the wavefunctions that are solutions to Schrödinger’s equation are complex, meaning that they have both “real” and “imaginary” parts and that we are only used to dealing with solutions to wave equations that are real. Born knew that it was a mathematical fact that whenever you multiplied a complex number or function by its complex conjugate, you always obtained a real number because all of the imaginary parts disappeared. To obtain the complex conjugate of a wavefunction Ψ, all you have to do is change i to –i wherever you see it. The resulting function is denoted as Ψ*.
Born proposed that the probability of finding a particle at a position x was:
Ψ* Ψ = | Ψ|²
at the point x. So to plot the probability of finding the electron in our 1-dimensional box, we just have to square the absolute value of its wavefunction. Figure 3 shows the resulting plot.
Figure 13 (click to enlarge)
Now here comes the interesting part. Classical mechanics predicts that the probability of finding the electron at any point should be the same for each point along the x-axis, because the electron is just bouncing back and forth like a target in a shooting gallery, and that the electron could have any energy at all, since the energy is just defined by how fast the electron is bouncing back and forth. However, the wavefunctions that are solutions to Schrödinger’s equation predict that the energy of the electron is quantized and comes in discrete allowed values. They also predict that the probability of finding the electron along the x-axis varies according to the energy state of the electron defined by its quantum number n. For the lowest energy level, where n = 1, the electron is most likely to be found near the center of the box because ψ*ψ has a peak there. That is not too bothersome. However, for the second energy level, where n = 2, something really strange happens. There is a high probability of finding the electron on either the left or right side of the box, but never in the center! How can an electron move back and forth in the box without ever passing through the center? For the higher energy levels, where n = 3, 4, 5, …, there are even more dead spots where ψ*ψ = 0, and the electron will never be found! This is just another example of the quantum strangeness that is built into our Universe. I apologize for all of the math in this posting. If you feel a little confused, you are in good company. Nobody really understands any of this stuff.
Why Einstein Hated Quantum Mechanics
Most students soon grow to hate quantum mechanics because the assigned problems are so confusing and difficult, but Einstein had his own reasons. Because quantum mechanics only yielded probabilistic predictions of where electrons might be or how fast they might be moving, it was not a deterministic theory and it was not local either. According to quantum mechanics, electrons did not know where they were until they were measured, and when they were measured, their wavefunctions had to collapse to the point where they were found. This was known as the Copenhagen Interpretation of quantum mechanics and has been quite popular ever since it was first put forward in the 1920s. The Copenhagen Interpretation did not explain how measuring a particle collapsed its wavefunction, nor why the wavefunction that was spread out over the entire Universe was able to instantly collapse to a single point and do so much faster than the speed of light. That violated Einstein's theory of relativity and required a "spooky action at a distance" that made quantum mechanics nonlocal in nature. The Copenhagen Interpretation required that things that were separated by vast distances could instantly affect each other quantum mechanically and that made no sense to Einstein.
Because Einstein detested the Copenhagen Interpretation of quantum mechanics so much, he published a paper in 1935 with Boris Podolsky and Nathan Rosen which outlined what is now known as the EPR Paradox. But to understand the EPR Paradox we need a little background in experimental physics. Electrons have a quantum mechanical property called spin. You can think of an electron’s spin like the electron has a little built-in magnet. In fact, it is the spin of the little electron magnets that add up to make the real magnets that you put on your refrigerator. Now in quantum mechanics, the spin of a single electron can be both up and down at the same time because the single electron can be in a mixture of quantum states! But in the classical Universe that we are used to, macroscopic things like a child's top can only have a spin of up or down at any given time. The top can only spin in a clockwise or counterclockwise manner at one time - it cannot do both at the same time. Similarly, in quantum mechanics, a photon or electron can go through both slits of a double-slit experiment at the same time, so long as you do not put detectors at the slit locations.
Figure 14 – A macroscopic top can only spin clockwise or counterclockwise at one time.
Figure 15 – But electrons can be in a mixed quantum mechanical state in which they both spin up and spin down at the same time.
Figure 16 – Similarly, tennis balls can only go through one slit in a fence at a time. They cannot go through both slits of a fence at the same time.
Figure 17 – But at the smallest of scales in our quantum mechanical Universe, electrons and photons can go through both slits at the same time, producing an interference pattern.
Figure 18 – Again, you can see this interference pattern of photons if you look at a distant porch light through the mesh of a sheer window curtain or a pillowcase.
When you throw an electron through a distorted magnetic field that is pointing up the electron will pop out in one of two states. It will either be aligned with the magnetic field (called spin-up) or it will be pointing 180o in the opposite direction of the magnetic field (called spin-down). Both the spin-up and spin-down conditions are called an eigenstate. Prior to the observation of the electron’s spin, the electron is in a superposition of states and is not in an eigenstate. Now if the electron in the eigenstate of spin-up is sent through the same magnetic field again, it will be found to pop out in the eigenstate of spin-up again. Similarly, a spin-down electron that is sent through the magnetic field again will also pop out as a spin-down electron. Now here is the strange part. If you rotate the magnetic field by 90o and send spin-up electrons through it, 50% of the electrons will pop out with a spin pointing to the left, and 50% will pop out with a spin pointing to the right. And you cannot predict in advance which way a particular spin-up electron will pop out. It might spin to the left, or it might spin to the right. The same goes for the spin-down electrons – 50% will pop out spinning to the left and 50% will pop out spinning to the right.
Figure 19 - In the Stern-Gerlach experiment we shoot electrons through a distorted magnetic field. Classically, we would expect the electrons to be spinning in random directions and the magnetic field should deflect them in random directions, creating a smeared-out spot on the screen. Instead, we see that the act of measuring the spins of the electrons puts them into eigenstates with eigenvalues of spin-up or spin-down and the electrons are either deflected up or down. If we rotate the magnets by 90o, we find that the electrons are deflected to the right or to the left.
The EPR Paradox goes like this. Suppose we prepare many pairs of quantum mechanically “entangled” electrons that conserve angular momentum. Each pair consists of one spin-up electron and one spin-down electron, but we do not know which is which at the onset. Now let the pairs of electrons fly apart and let two observers measure their spins. If observer A measures an electron there will be a 50% probability that he will find a spin-up electron and a 50% chance that he will find a spin-down electron, and the same goes for observer B, 50% of observer’s B electrons will be found to have a spin-up, while 50% will be found with a spin-down. Now the paradox of the EPR paradox, from the perspective of the Copenhagen Interpretation, is that when observer A and observer B come together to compare notes, they find that each time observer A found a spin-up electron, observer B found a spin-down electron, even though the electrons did not know which way they were spinning before the measurements were performed. Somehow when observer A measured the spin of an electron, it instantaneously changed the spin of the electron that observer B measured. Einstein hated this “spooky action at a distance” feature of the Copenhagen Interpretation that made physics nonlocal, meaning that things that were separated by great distances could still instantaneously change each other. He thought that it violated the speed of light speed limit of his special theory of relativity that did not allow information to travel faster than the speed of light. Einstein thought that the EPR paradox was the final nail in the coffin of quantum mechanics. There had to be some “hidden variables” that allowed electrons to know if they “really” were a spin-up or spin-down electron. You see, for Einstein, absolute reality really existed. For Einstein, the apparent probabilistic nature of quantum mechanics was an illusion, like the random() function found in most computer languages. The random() function just points to a table of apparently random numbers that are totally predictable if you look at the table in advance. You normally initiate the random() function with a “seed” from the system clock of the computer you are running on to simulate randomness by starting at different points in the table.
However, in 1964 John S. Bell published a paper in which he proposed an experiment that could actually test the EPR Paradox. In the 1980s and 1990s, a series of experiments were indeed performed that showed that Einstein was actually wrong. Using photons and polarimeters, instead of the spin of electrons, these experiments showed that photons really do not know their quantum states in advance of being measured and that determining the polarization of a photon by observer A can immediately change the polarization of another photon 60 miles away. These experiments demonstrated that the physical Universe is non-local, meaning that Einstein’s "spooky action at a distance” is built into our Universe, at least for entangled quantum particles. This might sound like a violation of the special theory of relativity because it seems like we are sending an instantaneous message faster than the speed of light, but that is really not the case. Both observer A and observer B will measure photons with varying polarizations at their observing stations separated by 60 miles. Only when observer A and observer B come together to compare results will they realize that their observations were correlated, so it is impossible to send a message with real information using this experimental scheme. Clearly, our common-sense ideas about space and time are still lacking, and so are our current effective theories.
Why This is Important for Quantum Computers?
You might ask why we should care about these quantum mechanical subtleties of physics? What does this have to do with the real world of human affairs? The reason why is that quantum computers take advantage of these two things that have been bothering physicists ever since they invented quantum mechanics in 1926.
1. Superposition - A quantum bit, known as a qubit, can be both a 1 and a 0 at the same time. A classical bit can only be a 1 or a 0 at any given time.
2. Entanglement - If two qubits are entangled, reading one qubit over here can immediately let you know what a qubit over there is without even reading it.
Figure 20 – Superposition means that a qubit really does not know if it is a 1 or a 0 until it is measured. The qubit exists in a superposition of states meaning that it is both a 1 and a 0 at the same time.
Superposition is important because a classical computer with 127 bits of memory can be in only one of:
2127 = 1.701 x 1038 = 170,100,000,000,000,000,000,000,000,000,000,000,000 states.
But a quantum computer with 127 qubits of memory like the just-announced IBM Eagle processor can be in 170,100,000,000,000,000,000,000,000,000,000,000,000 different states all at the same time!
Entanglement is important because when two qubits are entangled, they can instantly affect each other no matter how far apart they are.
Figure 21 – When qubits are entangled, neither one knows if it is a 1 or a 0. But if you measure one qubit and find that it is a 1, the other qubit will immediately become a 0 no matter how far apart they are.
Superposition and Entanglement have both been experimentally verified many times even if they do not make much sense. In Quantum Computing and the Foundations of Quantum Mechanics and Quantum Computing and the Many-Worlds Interpretation of Quantum Mechanics, I covered two popular explanations for these phenomena known as the Copenhagen Interpretation and the Many-Worlds Interpretation of quantum mechanics. I also covered the Transactional Interpretation which behaves a bit like TCP/IP. The Copenhagen Interpretation maintains that when a quantum system is observed, it collapses into a single state so that a qubit that is in a superposition of being a 1 and a 0 at the same time collapses into either a 1 or a 0. Entangled qubits collapse in pairs. The Many-Worlds Interpretation maintains that a qubit in a superposition of being a 1 and a 0 at the same time is actually two qubits in two different universes. You are a being composed of quantum particles and when you measure the qubit, you are not really measuring the qubit, you actually are measuring in which universe your quantum particles are entangled with the qubit. In one universe you will find a 1 and in the other, you will find a 0. The same thing happens when you measure entangled qubits. In one universe the qubits are 1 and 0 and in the other universe, they are 0 and 1. The Many-Worlds Interpretation may sound pretty nutty, but it actually is a much simpler explanation and does not need anything beyond the Schrödinger equation that defines all of quantum mechanics. Plus, as David Deutsch has commented, if a quantum computer can perform the calculations of a million computers all at the same time, where exactly are all of those calculations being performed if not in Many-Worlds? For more on that see Quantum Computing and the Foundations of Quantum Mechanics.
So How Does a Quantum Computer Work?
The details are quite complex using quantum algorithms that use quantum gates for logical operations, but you should be able to get an intuitive feel just based on the ideas of Superposition and Entanglement. Remember, a quantum computer with 127 qubits of memory can be in 170,100,000,000,000,000,000,000,000,000,000,000,000 different states all at the same time and many of those qubits can be entangled together into networks of entangled qubits. This allows people to essentially write quantum algorithms that can process all possible logical paths of a given problem all at the same time!
Figure 22 – Imagine a large network of entangled qubits processing all possible logical paths at the same time producing massive parallel processing.
But What If Einstein Was Really Right All Along?
Now let us explore some interesting work by Sabine (pronounced Sabina) Hossenfelder that could restore an Einstein Block Universe that is deterministic, local, obey's Einstein's theory of relativity and is composed of real things like electrons that actually exist on their own even when nobody is looking at them. This will get pretty heavy and might even totally rewrite your current worldview, so be prepared. It might be best to watch a few YouTube videos first that describe the concept of a Superdeterministic Universe. A Superdeterministic Universe is simply Einstein's Block Universe with a little twist that accounts for all of the strange quantum behaviors of our Universe that have been experimentally verified.
What If We Live in a Superdeterministic Universe?
https://www.youtube.com/watch?v=JnKzt6Xq-w4&t=0s
Does Superdeterminism save Quantum Mechanics? Or does it kill free will and destroy science?
https://www.youtube.com/watch?v=ytyjgIyegDI&t=0s
Basically, Sabine Hossenfelder explains that if we give up the concept of Statistical Independence, we can restore Einstein's Block Universe which is both deterministic and local in nature. All of the quantum weirdness of quantum superposition and entanglement disappears by simply giving up the concept of Statistical Independence. At first, that seems like quite a bargain and one that Einstein could certainly buy into, but it does come with a price. The easiest way to give up Statistical Independence is to assume that Einstein's Block Universe, which is both deterministic and local, came about at the Big Bang as a static 4-dimensional spacetime salami that was allowed to rapidly expand by means of Einstein's General Theory of Relativity, but that all future events were predetermined at the time of the Big Bang in a deterministic and local manner. Since our entire Universe was all at the same spot at the time of the Big Bang and all the components of that early Universe could causally interact with each other and then evolve in time in a deterministic manner, there would be no violation of Einstein's theory of relativity if many billions of years later it was found that two detectors that were many billions of light years apart were still correlated when measuring the spins of electrons or the polarization of photons when a Bell-type experiment was performed.
Figure 23 - So long as the light cones of Detectors D1 and D2 and the event P that produced two entangled particles all exist within overlapping light cones, it would be possible for D1 and D2 and P to be correlated so that at the time of measurement at D1 and D2, the settings at D1 could seem to affect the settings at D2 in a manner that exceeds the speed of light. Such would not be the case if the two entangled particles were created at event X because D2 is causally disconnected from event X. Event X is not within the light cone of D2 and is therefore causally disconnected from D2.
The idea that the Einstein Block Universe of a spacetime salami that was initiated at the time of the Big Bang with a set of entangled initial conditions that then evolved in a deterministic and local manner like loading a program into a computer with some initial data parameters and letting the code then run in a deterministic manner to completion can be somewhat disturbing. Of course an AI Deep Learning neural network would not be bothered by this in the least. But living in a Universe that was completely superdetermined from the Big Bang does not sit well with most human beings. That is because most human beings do not think of themselves as part of the natural world. Instead, they think of themselves, and others, as immaterial spirits temporarily haunting a carbon-based body, and when that carbon-based body dies the immaterial spirit lives on. In this view, human beings are not part of the natural world. Instead, they are part of the supernatural with the capability of free will to change the future in a nondeterministic manner. But if consciousness and free will are just self-delusions even in a deterministic Universe, they certainly would be so in a Superdeterministic Universe. For more on that see The Ghost in the Machine the Grand Illusion of Consciousness and DishBrain - Cortical Labs Creates an AI Matrix for Pong With Living Neurons on a Silicon Chip. For Sabine Hossenfelder's take on the matter see:
You don't have free will, but don't worry.
https://www.youtube.com/watch?v=zpU_e3jh_FY&t=0s
To refute the concept of Superdeterminism, experimenters have done things to try to push Superdeterminism to the extreme. Instead of using random number generators that are close to Detector D1 and Detector D2 to set the angles of the detectors in a Bell experiment, they have used the colors of photons from very distant quasars to set the angles in an attempt to establish Statistical Independence of the detector settings. For example, in the paper below:
Cosmic Bell Test using Random Measurement Settings from High-Redshift Quasars
https://arxiv.org/abs/1808.05966
Abstract
In this paper, we present a cosmic Bell experiment with polarization-entangled photons, in which
measurement settings were determined based on real-time measurements of the wavelength of photons
from high-redshift quasars, whose light was emitted billions of years ago; the experiment simultaneously
ensures locality. Assuming fair sampling for all detected photons, and that the wavelength
of the quasar photons had not been selectively altered or previewed between emission and detection,
we observe statistically significant violation of Bell's inequality by 9.3 standard deviations,
corresponding to an estimated p value of ≲ 7.4 x 10-21. This experiment pushes back to at least
∼ 7.8 Gyr ago the most recent time by which any local-realist influences could have exploited the
"freedom-of-choice" loophole to engineer the observed Bell violation, excluding any such mechanism
from 96% of the space-time volume of the past light cone of our experiment, extending from the big
bang to today.
The researchers used one quasar that was 7.78 billion light years away and another quasar that was 12.21 billion light years away to determine the settings of Detectors D1 and D2. The light cones of these two distant quasars last intersected more than 7.78 billion years ago before the Earth even formed 4.56 billion years ago. Yet their experiment confirmed the seemingly "spooky actions at a distance" correlations that all Bell-type experiments have revealed.
Figure 24 - The above Bell Test experiment used the colors of photons from two distant quasars to try to set the polarimeters of their Detectors D1 and D2.
Figure 25 - Here is a close-up view of the experimental setup.
Figure 26 - The light cones of Detectors D1 and D2 last intersected more than 7.78 billion years ago.
Figure 27 - But that is not a problem if our current Universe was launched in a Superdeterministic manner at the Big Bang.
Superdeterminism Is Not an Interpretation of Standard Quantum Mechanics
It is important to realize that Superdeterminism is not just another interpretation of standard quantum mechanics. We covered the Copenhagen, the Many-Worlds and the Transaction Interpretations of quantum mechanics in Quantum Software, Quantum Computing and the Foundations of Quantum Mechanics, The Foundations of Quantum Computing and Quantum Computing and the Many-Worlds Interpretation of Quantum Mechanics. In all such Interpretations of quantum mechanics, quantum mechanics is considered to be a "complete" theory that uses wavefunctions to explain the strange behaviors of tiny particles. All such Interpretations consider quantum mechanics to be the real thing. The Interpretations of quantum mechanics just try to explain what the mathematics of quantum mechanics is trying to tell us.
However, Superdeterminism is not such an Interpretation of quantum mechanics. Superdeterminism is a red flag trying to tell us that there is something wrong with standard quantum mechanics. Superdeterminism is trying to tell us that quantum mechanics is not a "complete" theory because something is missing. Superdeterminism tells us that quantum mechanics needs to be tweaked or replaced by a more fundamental theory because current quantum mechanical theory is just a very useful approximation of the true natural world.
For example, Einstein was a Realist and thought that tiny particles like electrons were real things with real properties that existed even if nobody was around to observe them. Einstein wondered,"Is it enough that a mouse observes that the Moon exists?". He did not like quantum mechanics because it was not deterministic and not local in nature. Instead, it was a probabilistic theory describing the behaviors of small particles in probabilistic terms. Because of that, Einstein thought that standard quantum mechanics must be an "incomplete" theory that was just an approximation of the actual behavior of tiny particles. Einstein thought that there must be some "hidden variables" or parameters that were missing from the wavefunction defined by quantum mechanics that really explained what happened in a deterministic manner when a detector interacted with a particle. These "hidden variables" or hidden parameters might be carried along by the "real" particle or they might come from somewhere else.
The idea of "hidden variables" might sound rather strange but it reminds me very much of a very perplexing troubleshooting problem that I had many years ago while working at United Airlines with Tuxedo C++ code. Tuxedo was developed back in the 1980s as a Transaction Processing Monitor like IBM's CICS that could run under Unix. Nobody knew it at the time, but Tuxedo was actually an early form of modern container-based Cloud computing. A Tuxedo Domain was really a Cloud Container that could crank up multiple Tuxedo Services written in C++ in a dynamic manner. Since then, Tuxedo was purchased by Oracle and is still available today for Cloud Computing on a Hybrid or external Cloud. Anyway, back in about 2001, my development group had a problem with an intermittent bug in one of our C++ Tuxedo Services. We had a very difficult time trying to troubleshoot this bug because it had a "hidden variable"! The problem was that a Tuxedo Domain would crank up a new instance of a C++ Tuxedo Service when the load increased and sometimes that would trip the "hidden variable" bug! Below is a description of the C++ code that caused the "hidden variable" bug.
int i;
More code;
More code;
More code;
More code;
if ( i == 1 )
{
Do this code;
Do this code;
Do this code;
Do this code;
Do this code;
Do this code;
}
The problem was that the original programmer did not initialize the variable i to zero with the code:
int i = 0;
So the bug happened when a new instance of the C++ Tuxedo Service was cranked up by the Tuxedo Domain and the section of memory allocated for the new instance of the C++ Tuxedo Service just happened to contain a binary "1" for the memory location of the "hidden variable" i. If the variable i was not set to a different value by some other code in the C++ Tuxedo Service before it reached the "if" statement above, then the "hidden variable" i contained a binary "1" in it and the "Do this code;" block of code would execute by mistake. This did not happen very often, so this was a very intermittent bug that was very hard to debug. The interesting thing to note is that the declaration of the "hidden variable" i was many thousands of lines of code distant from the "Do this code;" block of code that caused the damage. If our Universe is similarly the Superdeterministic result of a highly entangled and correlated set of particles and their descendants arising from the Big Bang, then such would also be the case.
For More on Sabine Hossenfelder's Work
You can read some accessible papers by Sabine Hossenfelder in the Cornell University arXiv at:
Rethinking Superdeterminism
https://arxiv.org/abs/1912.06462
Abstract
Quantum mechanics has irked physicists ever since its conception more than 100 years
ago. While some of the misgivings, such as it being unintuitive, are merely aesthetic,
quantum mechanics has one serious shortcoming: it lacks a physical description of the
measurement process. This “measurement problem” indicates that quantum mechanics
is at least an incomplete theory – good as far as it goes, but missing a piece – or, more
radically, is in need of complete overhaul.
Here we describe an approach which may provide this sought-for completion or replacement:
Superdeterminism. A superdeterministic theory is one which violates the assumption
of Statistical Independence (that distributions of hidden variables are independent
of measurement settings). Intuition suggests that Statistical Independence is an essential
ingredient of any theory of science (never mind physics), and for this reason Superdeterminism
is typically discarded swiftly in any discussion of quantum foundations.
The purpose of this paper is to explain why the existing objections to Superdeterminism
are based on experience with classical physics and linear systems, but that this experience
misleads us. Superdeterminism is a promising approach not only to solve the measurement
problem, but also to understand the apparent nonlocality of quantum physics. Most importantly,
we will discuss how it may be possible to test this hypothesis in an (almost) model
independent way.
Superdeterminism: A Guide for the Perplexed
https://arxiv.org/abs/2010.01324
Abstract
Superdeterminism is presently the only known consistent description of nature that is
local, deterministic, and can give rise to the observed correlations of quantum mechanics. I
here want to explain what makes this approach promising and offer the reader some advice
for how to avoid common pitfalls. In particular, I explain why superdeterminism is not a
threat to science, is not necessarily finetuned, what the relevance of future input is, and
what the open problems are.
For the more mathematically gifted, you can use the Search feature in the upper right corner of the arXiv webpages to search for more mathematically challenging papers by Sabine Hossenfelder. Her blog is at:
BackReaction
http://backreaction.blogspot.com/2019/07/the-forgotten-solution-superdeterminism.html
and her YouTube channel is at:
Sabine Hossenfelder
https://www.youtube.com/@SabineHossenfelder
Conclusion
Thus, if we really do live in a Superdeterministic Universe, it means some more work needs to be done to produce a replacement theory for standard quantum mechanics that incorporates realism, determinism, relativity and is local in nature. Like the General Theory of Relativity, such a more-fundamental theory would likely be described by nonlinear differential equations that induce chaotic behaviors in tiny particles. The seemingly random behaviors of particles in standard quantum mechanics would then just be the result of particles existing in the chaotic regimes of such nonlinear differential equations.
Comments are welcome at
scj333@sbcglobal.net
To see all posts on softwarephysics in reverse order go to:
https://softwarephysics.blogspot.com/
Regards,
Steve Johnston
