Category Archives: time

Heraclitus and Parmenides time joke

From Existential Commics

From Existential Comics; Parmenides believed that nothing changed, nor could it.

For those who don’t remember, Heraclitus believed that change was the essence of life, while  Parmenides believed that nothing ever changes. It’s a debate that exists to this day in physics, and also in religion (there is nothing new under the sun, etc.). In science, the view that no real change is possible is founded in Schrödinger’s wave view of quantum mechanics.

Schrödinger's wave equation, time dependent.

Schrödinger’s wave equation, time dependent.

In Schrödinger’s wave description of reality, every object or particle is considered a wave of probability. What appears to us as motion is nothing more than the wave oscillating back and forth in its potential field. Nothing has a position or velocity, quite, only random interactions with other waves, and all of these are reversible. Because of the time reversibility of the equation, long-term, the system is conservative. The wave returns to where it was, and no entropy is created, long-term. Anything that happens will happen again, in reverse. See here for more on Schrödinger waves.

Thermodynamics is in stark contradiction to this quantum view. To thermodynamics, and to common observation, entropy goes ever upward, and nothing is reversible without outside intervention. Things break but don’t fix themselves. It’s this entropy increase that tells you that you are going forward in time. You know that time is going forward if you can, at will, drop an ice-cube into hot tea to produce lukewarm, diluted tea. If you can do the reverse, time is going backward. It’s a problem that besets Dr. Who, but few others.

One way that I’ve seen to get out of the general problem of quantum time is to assume the observed universe is a black hole or some other closed system, and take it as an issue of reference frame. As seen from the outside of a black hole (or a closed system without observation) time stops and nothing changes. Within a black hole or closed system, there is constant observation, and there is time and change. It’s not a great way out of the contradiction, but it’s the best I know of.

Predestination makes a certain physics and religious sense, it just doesn't match personal experience very well.

Predestination makes a certain physics and religious sense, it just doesn’t match personal experience very well.

The religion version of this problem is as follows: God, in most religions, has fore-knowledge. That is, He knows what will happen, and that presumes we have no free will. The problem with that is, without free-will, there can be no fair judgment, no right or wrong. There are a few ways out of this, and these lie behind many of the religious splits of the 1700s. A lot of the humor of Calvin and Hobbes comics comes because Calvin is a Calvinist, convinced of fatalistic predestination; Hobbes believes in free will. Most religions take a position somewhere in-between, but all have their problems.

Applying the black-hole model to God gives the following, alternative answer, one that isn’t very satisfying IMHO, but at least it matches physics. One might assume predestination for a God that is outside the universe — He sees only an unchanging system, while we, inside see time and change and free will. One of the problems with this is it posits a distant creator who cares little for us and sees none of the details. A more positive view of time appears in Dr. Who. For Dr. Who time is fluid, with some fixed points. Here’s my view of Dr. Who’s physics.  Unfortunately, Dr. Who is fiction: attractive, but without basis. Time, as it were, is an issue for the ages.

Robert Buxbaum, Philosophical musings, Friday afternoon, June 30, 2017.

Our expanding, black hole universe

In a previous post I showed a classical derivation of the mass-to-size relationship for black -holes and gave evidence to suggest that our universe (all the galaxies together) constitute a single, large black hole. Everything is inside the black hole and nothing outside but empty space — We can tell this because you can see outside from inside a black hole — it’s only others, outside who can not see in (Finkelstein, Phys Rev. 1958). Not that there appear to be others outside the universe, but if they were, they would not be able to see us.

In several ways having a private, black hole universe is a gratifying thought. It provides privacy and a nice answer to an easily proved conundrum: that the universe is not infinitely big. The black hole universe that ends as the math requires, but not with a brick wall, as i the Hitchhiker’s guide (one of badly-laid brick). There are one or two problems with this nice tidy solution. One is that the universe appears to be expanding, and black holes are not supposed to expand. Further, the universe appears to be bigger than it should be, suggesting that it expanded faster than the speed of light at some point. its radius now appears to be 40-46 billion light years despite the universe appearing to have started as a point some 14 billion years ago. That these are deeply disturbing questions does not stop NASA and Nova from publishing the picture below for use by teachers. This picture makes little sense, but it’s found in Wikipedia and most, newer books.

Standard picture of the big bang theory. Expansions, but no contractions.

Standard picture of the big bang theory: A period of faster than light expansion (inflation) then light-speed, accelerating expansion. NASA, and Wikipedia.

We think the creation event occurred some 14 billion years ago because we observe that the majority of galaxies are expanding from us at a rate proportional to their distance from us. From this proportionality between the rate of motion and the distance from us, we conclude that we were all in one spot some 14 billion years ago. Unfortunately, some of the most distant galaxies are really dim — dimmer than they would be if they were only 14 billion light years away. The model “explains this” by a period of inflation, where the universe expanded faster than the speed of light. The current expansion then slowed, but is accelerating again; not slowing as would be expected if it were held back by gravity of the galaxies. Why hasn’t the speed of the galaxies slowed, and how does the faster-than-light part work? No one knows. Like Dr. Who’s Tardis, our universe is bigger on the inside than seems possible.

Einstein's preferred view of the black-hole universe is one that expands and contracts at some (large) frequency. It could explain why the universe is near-uniform.

Einstein’s oscillating universe: it expands and contracts at some (large) frequency. Oscillations would explain why the universe is near-uniform, but not why it’s so big or moving outward so fast.

Einstein’s preferred view was of an infinite space universe where the mass within expands and contracts. He joked that two things were infinite, the universe and stupidity… see my explanation... In theory, gravity could drive the regular contractions to an extent that would turn entropy backward. Einstein’s oscillating model would explain how the universe is reasonably stable and near-uniform in temperature, but it’s not clear how his universe could be bigger than 14 billion light years across, or how it could continue to expand as fast as it does. A new view, published this month suggests that there are two universes, one going forward in time the other backward. The backward in time part of the universe could be antimatter, or regular matter going anti entropy (that’s how I understand it — If it’s antimatter, we’d run into the it all the time). Random other ideas float through the physics literature: that we’re connected to other space through a black hole/worm hole, perhaps to many other universes by many worm holes in fractal chaos, see for example, Physics Reports, 1992.

The forward-in-time expansion part of the two universes model.

The forward-in-time expansion part of the two universes model. This drawing, like the first, is from NASA.

For all I know, there are these many black hole  tunnels to parallel universes. Perhaps the universal constant and all these black-hole tunnels are windows on quantum mechanics. At some point the logic of the universe seems as perverse as in the Hitchhiker guide.

Something I didn’t mention yet is the Higgs boson, the so-called God particle. As in the joke, it’s supposed to be responsible for mass. The idea is that all particles have mass only by interaction with these near-invisible Higgs particles. Strong interactions with the Higgs are what make these particles heavier, while weaker – interacting particles are perceived to have less gravity and inertia. But this seems to me to be the theory that Einstein’s relativity and the 1919 eclipse put to rest. There is no easy way for a particle model like this to explain relativistic warping of space-time. Without mass being able to warp space-time you’d see various degrees of light bending around the sun, and preferential gravity in the direction of our planet’s motion: things we do not see. We’re back in 1900, looking for some plausible explanation for the uniform speed of light and Lawrence contraction of space.As likely an explanation as any the_hitchhikers_guide_to_the_galaxy

Dr. r µ ßuxbaum. December 20, 2014. The  meaning of the universe could be 42 for all I know, or just pickles down the worm hole. No religion seems to accept the 14 billion year old universe, and for all I know the God of creation has a wicked sense of humor. Carry a towel and don’t think too much.

A simple, classical view of and into black holes

Black holes are regions of the universe where gravity is so strong that light can not emerge. And, since the motion of light is related to the fundamental structure of space and time, they must also be regions where space curves on itself, and where time appears to stop — at least as seen by us, from outside the black hole. But what does space-time look like inside the black hole.

NASA's semi-useless depiction of a black hole -- one they created for educators. I'm not sure what you're supposed to understand from this.

NASA’s semi-useless depiction of a black hole — one they created for educators. Though it’s sort of true, I’m not sure what you’re supposed to understand from this. I hope to present a better version.

From our outside perspective, an object tossed into a black hole will appear to move slower as it approaches the hole, and at the hole horizon it will appear to have stopped. From the inside of the hole, the object appears to just fall right in. Some claim that tidal force will rip it apart, but I think that’s a mistake. Here’s a simple, classical way to calculate the size of a black hole, and to understand why things look like they do and do what they do.

Lets begin with light, and accept, for now, that light travels in particle form. We call these particles photons; they have both an energy and a mass, and mostly move in straight lines. The energy of a photon is related to its frequency by way of Plank’s constant. E= hν, where h is Plank’s constant and ν is frequency. Their mass is related to their energy by way of the formula m=E/c2. This formulate is surprisingly easy to derive, and is often shown as E= mc2. In classical form, the gravitational force between a star, mass M, and this photon or any other object of mass m described as follows:

F = GMm/r2

where F is force, G is the gravitational constant, and r is the distance of the photon from the center of the star. The potential energy of a photon of the mass increases as it rises from the star surface, but the internal energy (proportional to frequency) decreases — the photon gets redder. The amount of internal energy lost to gravity as it rises from the surface is the integral of the force, and is thus related to the mass of the object and of the star.

∆E =  ∫Fdr = ∫GMm/r2 dr = -GMm/r

Lets consider a photon of original energy E° and original mass m°. Lets figure out the radius of the star r° such that all of the original energy, E° is lost in rising away from the star. That is let calculate the r for which ∆E = -E° as the photon rises to freedom. Lets assume, for now, that the photon mass remains constant at m°.

E° = GMm°/r° = GME°/c2r°.

We now eliminate E° from the equation and solve for this special radius, r°:

r° =  GM/c2.

This would be the radius of a black hole if space didn’t curve and if the mass of the photon didn’t decrease as it rose. While neither of these assumptions is true, the errors nearly cancel, and the true value for r° is double the size calculated this way.

r° = 2GM/c2

r° = 2.95 km (M/Msun).


Karl Schwarzschild 1873-1916.

The first person to do this calculation was Karl Schwarzschild and r° is called the Schwarzschild radius. This is the minimal radius for a star of mass M to produce closed space-time; a black hole. Msun is the mass of our sun, sol, 2 × 1030 kg.  To make a black hole one would have to compress the mass of our sun into a ball of 2.95 km radius, about the size of an SUV. Space-time would close around it, and light starting from the surface would not be able to escape.

As it happens, our sun is far bigger than a SUV and is not a black hole: we can see light from the sun’s surface with minimal space-time deformation. Still, if the mass were a lot bigger, the radius would be a lot bigger and the needed density less. Consider a black hole the same mass as our galaxy, about 1 x1012 solar masses (mostly dark matter), or 2 x 1042  kg. The Schwarzschild radius of a star with the mass of our galaxy would be 3 x 1012 km, or 0.3 light years, about 1/20 the distance to Alpha Centauri. This is far bigger than the six of our solar system, but far smaller than the actual size of the galaxy, 5 x 1017 km, or 50,000 light years. Still, the difference between 0.3 light years and 50,000 light years isn’t that great on the cosmic scale, and it’s worthwhile to consider a black hole comprising something 10 to 100 Billion times more massive than our galaxy — the universe as a whole.

The folks at Cornell estimate the sum of dark and luminous matter in the universe to be about 15 billion times the mass of our galaxy, or 3 x 1052 kg. This does not include the mass of the dark energy, but no one’s quite sure what dark energy is. Considering only this physical mass, the Schwarzschild radius for the universe would be about 4.5 billion light years, or about 1/3 the size of our universe based on its age. The universe appears to be 14 billion years old, so if it’s expanding at the speed of light, the radius should be 14 billion light years. The universe may be 2-3 times bigger than this on the inside (rather like Dr. Who’s Tardis it’s bigger on the inside) but in astronomical terms a factor of 3 or 10 is nothing: the size of the universe is remarkably similar to its Schwarzschild radius, and this is without considering the mass its dark energy must have.

Standard picture of the big bang theory. Dark energy causes the latter-stage expansion.

Standard picture of the big bang theory. Dark energy causes the latter-stage expansion.

The evidence for dark energy is that the universe is expanding faster and faster instead of slowing. See figure. There is no visible reason for the acceleration, but it’s there. The significant amount of energy must have significant mass, E = mc2. If the mass of this energy is 3 to 10 times the physical mass, as seems possible, we are living inside a large black hole, something many physicists, including Einstein considered extremely likely. Einstein originally didn’t consider the possibility that the hole could be expanding. But now we know how to calculate the size of a black hole, and we know what a large black hole looks like from the inside. It looks just like home.

Wait for further posts on curved space-time. For some reason, no religion seems to embrace science’s 14 billion year old, black-hole universe (expanding or not). As for tidal forces, they are horrific only for the small black holes that most people write about. If the black hole is big enough, the tidal forces are small.

 Dr. µß Buxbaum Nov 17, 2014. You can drink to the Schwarzchild radius with my new R° cocktail.

On being a 16 year old girl

I’m not a teenage girl, in case you thought otherwise. I’m the father of a girl who just turned 16 though, and she asked me to write on the subject of what to expect from the next year or so. Here’s my sense of expectations.

You'll find yourself creeping up on adulthood, as a partner, not a kid; it's a scary and wonderful transition.

You’ll find yourself creeping up on adulthood, as a partner, not a kid; it’s a scary and wonderful transition.

In retrospect, you’re likely to say that 16 was among the best years of your life. These are the last glorious, innocent days with friends: days before competition means anything, before you really have to think of the world beyond your high school community. You are still hanging out, working together, and trying to feel your way towards a dim future as adults. Sorry to say, that’s in retrospect. While living through it, you’ll find this year fairly boring, and somewhat nerve-wracking. You’ll find your time filled with activities: school, home, hobbies. You’re likely to find these activities somewhat less stressful than before, because you’re more used to them and higher-up in the pecking order. Still, there are a lot of activities, and you’ll notice your day is pretty full. My advice: take time to enjoy your friends. Take pictures; they’ll be priceless.

At sixteen hobbies begin to be taken more seriously.

At sixteen, hobbies begin to be taken more seriously.

For both boys and girls, you are beginning the single most difficult, painful, and important transition of your life: the transition to adulthood. It’s not painful yet, but it will get worse in the next year or two. If all goes well, by age 22 or 23, you will be through it. Once through it, you will think of yourself as an independent moiety: someone who’s formed by us (your parents, family, and friends) but not defined by us. Once through it, if all goes well, you will be able to support yourself financially, and you will be able to live on your own. You are likely to want to live on your own too (teary smile) at least for a good portion of the year. At 16, this is a dimly seen, scary future, far off in the fog.

At this  point, you’re still tied to us, and I’m glad you don’t resent it. You’re happy to be a daughter, a sister to your siblings, a peer to your friends, and a student in your high school. Some teachers and classes you like, others less so. Your grades and hobbies are important to you, but your friends are more important. It’s nice to have high grades, but not so important as to disturb a friendship. You think of your hobbies as fun sidelights, and home as a place to relax with them. At home you write, read, draw, or cook for the fun of it. As the next few years wear on, this will change. You’ll think of yourself more as a writer, an artist, or plumber; as a private first class, or whatever. You’ll be good at something, not just generally bright. Some friends will fit better into this self-view; the friends who don’t fit will slowly drift away.

Sexuality rears its head in new ways at 16.

Sex rears its head unexpectedly.

At 16, I started thinking of myself as ‘an engineer’ or perhaps as a scientist or mathematician based on my hobbies and what I was good at in school. It became clear that I was not going to be an athlete, a historian, or a musician (though I retained an interest). Dropping options is a big, painful part of the transition. I recall almost hearing the doors closing behind me. You want to turn back, to catch the options. Know that, to not choose is to choose. As those doors close (and they should) new, better ones open that you didn’t realize existed. Losing friends and hobbies that are too high maintenance is good for you, and for them at this stage. Sex will rear its head, unexpectedly, and in new ways. Sexuality and homosexuality were words; for some they are becoming the dominant reality. For better or worse, you’ll be drawn in.

As the year draws to a close, you’re likely to find our parental presence more and more annoying. This is a good thing; it’s what will get you out the door, and launch you as a person. We’re on your side here, but won’t be able to help as your old you will begin to fight the new one. You plan to go to college, perhaps away from home (both options are good), but some of your friends will want to stay at home and do on-line vocational courses, or get married as soon as possible. You’ll likely drift away from those friends. Some college-bound friends will pick schools far from yours, or will pick majors or activities that you’re not interested in. You’re likely to find yourself gravitating to those friends who’re going to your college or for majors that match yours. There is pain in realizing that you won’t be as close with the remaining friends. Know that doors are opening here in two ways. First, just as high school provided your current friends, college and pre-college will provide you a new group — ones you may keep longer since the relationship based on shared direction, not just shared experiences. Also, know that some of the friends who drift away now will come back later — perhaps when you and they are married.

You may come to realize that some of your closest friends are your competitors for college places, scholarships. This may seem bad (or disloyal) but it’s good. Competition will help you improve, and will increase your drive, and that’s what you need this year. Think of the relationship in “A separate peace.” Think of how the relationship between the young Harper Lee and Truman Capote likely shaped “to kill a mockingbird” and furthered both of their careers. Competition with your cousins is good too. Watch how they deal with competition and life choices. Are there family members that could be life models or coaches? A big reason we have the family reunions is so that you can have a choice of life models and coaches.

Teen jobs are rarely all that exciting, but are an important part of personal development.

Teen jobs are rarely all that exciting, but are an important part of personal development.

Money will become a lot more important in the next few years — for things, travel, school applications, and clothes. You’re likely to find it’s annoying to get your money from us, and you’re likely to start working more for money. This is the beginning of financial independence. As you do, you’ll find yourself becoming defined by the job you do, how much you make, and what you spend your money on. This is good, but includes a loss of Idyl. Your first jobs will not be great, and you may find leaches hanging on: financial and emotional. As annoying as it is to have a leech, it’s worse to be the leach. Try to avoid it; be a good friend or neighbor. You may want to buy stock, or start a company, or produce a product for sale: a book, album, or whatever. If it’s something you’re interested in and you try to make money at it, the experience will be worth the effort. Even if it turns out a financial failure, it will be an important part of the emotional and financial person you make of yourself. If you don’t go into business, you may get involved in politics or religion, moving right or left. That’s OK, and very normal — another part of self-development.

You may find yourself re-evaluating your thoughts on religion and government.

You’re already beginning to develop wonderful life-skills. We don’t compliment you enough on this. You’ve learned to cook eggs and noodles, and find you like the independence it gives. That’s the ticket to adulthood. You’ll need more life skills to give you real independence, but you’re on the right track. You’ll need to learn to do laundry, shopping, and cleaning. You’re likely to need to be better at driving, writing, and negotiating: all difficult things. You’re likely to go through emotional cycles or depression as you think of the stuff you can’t do, or don’t understand, the friends you’re losing, or the things you’d like to do, but can’t. Don’t stress; you’ve got 5 years, or more. We’re very proud of you, and will try to help by tutoring, hugs, more-freedom, and the assurance that independence is worth the struggle. All beginnings are difficult, and this is a big beginning.

You can define yourself by your hobbies or by your man, but try not to define yourself by your man's hobbies.

You can define yourself by your hobbies or by your man, but try not to define yourself by your man’s hobbies.

Switching schools includes the opportunity to reinvent yourself as something completely new. Most people do this to a greater or lesser extent. Embrace your inner weirdo, but not your inner crook. Try to invent yourself as something fun and active, not sinful or destructive. Try to be the young scholar, mechanic, artist, or athlete, not the young goth, gangster, or drug addict. High schools try to help here by exposing you to books and movies about alienated 16-20 year olds. Popular in my day were Great Expectations, The Outsiders, Catcher in the Rye, The Dead Poets Society, To Kill a Mockingbird, Slaughterhouse 5, and A Separate Peace. Take what comfort you can. School assignments will include essays on law, government, and God. Write honestly, with conviction. These assignments can help develop your life views and personality. We’ll try not to stifle you here, even when your opinions differ significantly from ours. Of course, if you come up with something truly stupid or awesome, we’ll tell you.

Your friends will start dating, or discussing boys, and you are almost certain to start looking at boys differently: first as exciting possibilities, and then as potential mates. Part of the attraction involves the ability to define yourself by the boy you choose. This is a comfort and a curse. The comfort is that it avoids you having to define yourself, or grow up quite. The curse is that the boy doesn’t know who he is either. You’ll find that some boys are nice and some are grounded, others are not. And some are really messed up. With the right kind, you’ll find you can do more as a pair than as a single. Eventually you’re likely to pair off; in our community that happens at about 21-24. When it happens, I hope it’s with a nice, grounded fellow. It works best if you first know who you are, but even otherwise, it can work. Couples sometimes discover who they are together. And, at that point the transition will be over. You’ll be a married adult; you’ll introduce yourself as Mrs Shnicklefritz, or as Dr. and Mr Schniklefritz, or whatever, and we’ll prepare ourselves to spoil our grandkids.

Dr. Robert E. Buxbaum, proud father of you and your two older siblings. October 4, 2014. Though further along in their life paths, I can hope that the older siblings will enjoy these thoughts too. I’ve previously mused about US education, and whether ADHD was a real disease. For my older daughter’s 21st birthday, I invented a new mixed drink.

Dr. Who’s Quantum reality viewed as diffusion

It’s very hard to get the meaning of life from science because reality is very strange, Further, science is mathematical, and the math relations for reality can be re-arranged. One arrangement of the terms will suggest a version of causality, while another will suggest a different causality. As Dr. Who points out, in non-linear, non-objective terms, there’s no causality, but rather a wibbly-wobbely ball of timey-wimey stuff.

Time as a ball of wibblely wobbly timey wimey stuff.

Reality is a ball of  timey wimpy stuff, Dr. Who.

To this end, I’ll provide my favorite way of looking at the timey-wimey way of the world by rearranging the equations of quantum mechanics into a sort of diffusion. It’s not the diffusion of something you’re quite familiar with, but rather a timey-wimey wave-stuff referred to as Ψ. It’s part real and part imaginary, and the only relationship between ψ and life is that the chance of finding something somewhere is proportional Ψ*|Ψ. The diffusion of this half-imaginary stuff is the underpinning of reality — if viewed in a certain way.

First let’s consider the steady diffusion of a normal (un-quantum) material. If there is a lot of it, like when there’s perfume off of a prima donna, you can say that N = -D dc/dx where N is the flux of perfume (molecules per minute per area), dc/dx is a concentration gradient (there’s more perfume near her than near you), and D is a diffusivity, a number related to the mobility of those perfume molecules. 

We can further generalize the diffusion of an ordinary material for a case where concentration varies with time because of reaction or a difference between the in-rate and the out rate, with reaction added as a secondary accumulator, we can write: dc/dt = reaction + dN/dx = reaction + D d2c/dx2. For a first order reaction, for example radioactive decay, reaction = -ßc, and 

dc/dt = -ßc + D d2c/dx2               (1)

where ß is the radioactive decay constant of the material whose concentration is c.

Viewed in a certain way, the most relevant equation for reality, the time-dependent Schrödinger wave equation (semi-derived here), fits into the same diffusion-reaction form:

dΨ/dt = – 2iπV/h Ψ + hi/4πm d2Ψ/dx               (2)

Instead of reality involving the motion of a real material (perfume, radioactive radon, etc.) with a real concentration, c, in this relation, the material can not be sensed directly, and the concentration, Ψ, is semi -imaginary. Here, h is plank’s constant, i is the imaginary number, √-1, m is the mass of the real material, and V is potential energy. When dealing with reactions or charged materials, it’s relevant that V will vary with position (e.g. electrons’ energy is lower when they are near protons). The diffusivity term here is imaginary, hi/4πm, but that’s OK, Ψ is part imaginary, and we’d expect that potential energy is something of a destroyer of Ψ: the likelihood of finding something at a spot goes down where the energy is high.

The form of this diffusion is linear, a mathematical term that refers to equations where solution that works for Ψ will also work for 2Ψ. Generally speaking linear solutions have exp() terms in them, and that’s especially likely here as the only place where you see a time term is on the left. For most cases we can say that

Ψ = ψ exp(-2iπE/h)t               (3)

where ψ is not a function of anything but x (space) and E is the energy of the thing whose behavior is described by Ψ. If you take the derivative of equation 3 this with respect to time, t, you get

dΨ/dt = ψ (-2iπE/h) exp(-2iπE/h)t = (-2iπE/h)Ψ.               (4)

If you insert this into equation 2, you’ll notice that the form of the first term is now identical to the second, with energy appearing identically in both terms. Divide now by exp(-2iπE/h)t, and you get the following equation:

(E-V) ψ =  -h2/8π2m d2ψ/dx2                      (5)

where ψ can be thought of as the physical concentration in space of the timey-wimey stuff. ψ is still wibbly-wobbley, but no longer timey-wimey. Now ψ- squared is the likelihood of finding the stuff somewhere at any time, and E, the energy of the thing. For most things in normal conditions, E is quantized and equals approximately kT. That is E of the thing equals, typically, a quantized energy state that’s nearly Boltzmann’s constant times temperature.

You now want to check that the approximation in equation 3-5 was legitimate. You do this by checking if the length-scale implicit in exp(-2iπE/h)t is small relative to the length-scales of the action. If it is (and it usually is), You are free to solve for ψ at any E and V using normal mathematics, by analytic or digital means, for example this way. ψ will be wibbely-wobbely but won’t be timey-wimey. That is, the space behavior of the thing will be peculiar with the item in forbidden locations, but there won’t be time reversal. For time reversal, you need small space features (like here) or entanglement.

Equation 5 can be considered a simple steady state diffusion equation. The stuff whose concentration is ψ is created wherever E is greater than V, and is destroyed wherever V is greater than E. The stuff then continuously diffuses from the former area to the latter establishing a time-independent concentration profile. E is quantized (can only be some specific values) since matter can never be created of destroyed, and it is only at specific values of E that this happens in Equation 5. For a particle in a flat box, E and ψ are found, typically, by realizing that the format of ψ must be a sin function (and ignoring an infinity). For more complex potential energy surfaces, it’s best to use a matrix solution for ψ along with non-continuous calculous. This avoids the infinity, and is a lot more flexible besides.

When you detect a material in some spot, you can imagine that the space- function ψ collapses, but even that isn’t clear as you can never know the position and velocity of a thing simultaneously, so doesn’t collapse all that much. And as for what the stuff is that diffuses and has concentration ψ, no-one knows, but it behaves like a stuff. And as to why it diffuses, perhaps it’s jiggled by unseen photons. I don’t know if this is what happens, but it’s a way I often choose to imagine reality — a moving, unseen material with real and imaginary (spiritual ?) parts, whose concentration, ψ, is related to experience, but not directly experienced.

This is not the only way the equations can be rearranged. Another way of thinking of things is as the sum of path integrals — an approach that appears to me as a many-world version, with fixed-points in time (another Dr Who feature). In this view, every object takes every path possible between these points, and reality as the sum of all the versions, including some that have time reversals. Richard Feynman explains this path integral approach here. If it doesn’t make more sense than my version, that’s OK. There is no version of the quantum equations that will make total, rational sense. All the true ones are mathematically equivalent — totally equal, but differ in the “meaning”. That is, if you were to impose meaning on the math terms, the meaning would be totally different. That’s not to say that all explanations are equally valid — most versions are totally wrong, but there are many, equally valid math version to fit many, equally valid religious or philosophic world views. The various religions, I think, are uncomfortable with having so many completely different views being totally equal because (as I understand it) each wants exclusive ownership of truth. Since this is never so for math, I claim religion is the opposite of science. Religion is trying to find The Meaning of life, and science is trying to match experiential truth — and ideally useful truth; knowing the meaning of life isn’t that useful in a knife fight.

Dr. Robert E. Buxbaum, July 9, 2014. If nothing else, you now perhaps understand Dr. Who more than you did previously. If you liked this, see here for a view of political happiness in terms of the thermodynamics of free-energy minimization.