Tunnelling mistake

In a previous post I wrote about quantum tunnelling. In that post I said that the energy of the different instances of the particle undergoing tunnelling were not larger than those of the barrier. I reached this conclusion by looking at the terms in an analytical solution of the Schrodinger equation and estimating their magnitudes. However, estimation can quite easily go wrong and I thought I should check by doing a simulation. I have performed a simulation and my conclusion was wrong. Some instances of the particle tunnelling through the barrier increase their energy. The program that performs the simulation and produces the figures shown below can be found at this link. The program is written in Python because it has sparse complex number matrix libraries, which were needed for the simulation. The link also includes some references to papers used in writing the code.

Tunnelling involves a wavepacket interacting with a potential that has higher energy than the wavepacket. After the interaction some instances of the particle are present on the other side of the potential. I made the potential have a height of 1.1 times the wavepacket’s mean energy. The wavepacket comes in from the left, some of it is reflected to the left and some continues to the right after the wavepacket interacts with the potential, as shown in the figures below. The red line represents the potential, the blue line is square amplitude of the wavepacket, which gives the probability of finding the particle in that location.


The energy of a wave with wavenumber $k$ is $k^2/2m$. I calculated the energy spectrum at each time by doing a fast fourier transform, finding the square amplitude at each wavenumber $k$, which is proportional to the probability of finding the particle to have that frequency when you measure it. The fourier transform includes both positive and negative frequencies: the first half of the spectrum is the positive frequencies, the second half is the negative frequencies. I calculated the square amplitudes for both halves of the FFT and added up the amplitudes that corresponded to the same energy (the $+k$ and $-k$ amplitudes). I then took all of the resulting energy spectra and divided them all by the amplitude of the initial energy spectrum and plotted the resulting normalised results on a semilog scale. This allowed me to see whether the components with energies above that of the initial wavepacket increased or decreased. The results are shown in the figures below at the same times as the figures shown above except that the last figure is omitted. The red lines are positioned so that if the spectrum is above the horizontal red line to the right of the vertical red line, then there is a higher probability of the particle having an energy above what it had before the potential.fftnorm0fftnorm400fftnorm500

It appears that the energy of some of the instances of the particle increased after interacting with the barrier and in particular the probability increased for energies above 1.1 times the wavepacket energy: the energy of the potential. The original post will be updated after I publish this post.


The Conservative Party’s Philosophy

In a recent article Daniel Hannan wrote:

Would you go an hour out of your way to get £100 discount on a £300 dishwasher? What about going an hour out of your way to get a £100 discount on a £12,000 car? If you’re typical, you’re much more likely to have answered yes to the first question than to the second. Which, logically, makes no sense at all. Either an hour of your time is worth £100 or it’s not. If the trade-off is in your interest – and, for most of us, it is – then you should make it both times. If you happen to be a hedge-fund owner or a gilded public-sector princeling, then you might rationally say no both times. But there is no sound basis for saying yes to one and not the other.

Hannan goes on to say that the Conservative Party’s leaders believe this kind of thing and base policy on it.

There are many differences between the decision about what dishwasher you should buy and what car you should buy. Many people drive their car to work, so their income depends on having a car that works. A dishwasher saves you some time when dealing with dishes, but if it breaks down you’ll still be able to pay for accommodation and food. So the cost of getting the car decision wrong may be a lot larger than the cost of getting the dishwasher decision wrong. So a person may stick with looking at cars from dealers he trusts rather than go looking for £100 off on a decision that could cost him thousands of pounds if he gets it wrong. So Hannan hasn’t given any reason to think actual decisions made by real human beings are irrational.

Hannan doesn’t mention the fact that according to his own theory politicians are irrational since they are human beings. So politicians should not be trusted to make decisions either according to this theory. And yet, somehow, politicians are fit to make decisions not only about their own lives, but also about the lives of their subjects.

The Conservative Party is run by technocrats who believe they know how you should live your life better than you do. The Conservative Party is opposed to individual liberty because its leaders think you are unfit to make your own decisions. That is part of why they are dragging their feet on Brexit. It also explains their contempt for freedom of speech, and their imposition of bad policy like plastic bag charges. And that’s why I won’t vote for the Conservative Party. I have respect for individual liberty, they have contempt for it.

Howson and Urbach vs Popper

Howson and Urbach claim to have refuted Popper’s ideas in “Scientific Reasoning: The Bayesian Approach”. In Section 4g they write:

Philosophers, such as Popper and Lakatos, who deny any inductive role for evidence, and who oppose, in particular, the Bayesian approach take note of the that scientists often do deal with particular instances of the Duhem problem by proposing alternative hypotheses; some of these philosophers have suggested certain normative rules that purport to say when such alternatives are acceptable and when they are not. Their idea is that a theory that was introduced ad hoc, that is, “for the sole purpose of saving a hypothesis seriously threatened by adverse evidence” (Hempel 1966, p. 29), is in some way inferior. The adhocness idea was largely inspired by certain types of scientific example, which appeared to endorse it, but in our view, the examples are misinterpreted and the idea badly flawed. The following are four such examples.

In each of these examples, the theory that was proposed in place of the refuted one seems highly unsatisfactory. It is not likely that any of them would have been advanced, save in response to particular anomalies and in order to evade the consequent difficulty, hence the label ‘ad hoc’. But philosophers who attach inductive significance to adhocness recognize that the mere fact that the theory was proposed under such circumstances is not by itself grounds for condemnation. For there are examples, like the following, where a theory that was proposed for the sole purpose of dealing with an anomaly was nevertheless very successful.

4 William Herschel, in 1781, discovered the planet Uranus. Astronomers quickly sought to describe the orbit of the new planet in Newtonian terms, taking account of the perturbing influence of the other known planets, and were able to deduce predictions concerning its future positions. But discrepancies between predicted and observed positions of Uranus substantially exceeded the accepted limits of experimental error, and grew year by year. A few astronomers mooted the possibility that the fault lay with Newton’s laws but the prevailing opinion was that there must be some unknown planet acting as an extra source of gravitational attraction on Uranus, which ought to be included in the Newtonian calculations. Two astronomers in particular, Adams and Le Verrier, working independently, were convinced of this and using all the known sightings of Uranus, they calculated in a mathematical tour de force where the hypothetical planet must be. The hypothesis was ad hoc, yet it was vindicated when careful telescopic observations as well as studies of old astronomical charts revealed in 1846 the presence of a planet with the anticipated characteristics. The planet was later called Neptune. Newton’s theory was saved, for the time being. (See Smart 1947.)

Their account of Popper’s position is wrong. In “The Logic of Scientific Discovery”, Chapter 4, Section 20, Popper writes:

As regards auxiliary hypotheses we propose to lay down the rule that only those are acceptable whose introduction does not diminish the degree of falsifiability or testability of the system in question, but, on the contrary, increases it. (How degrees of falsifiability are to be estimated will be explained in sections 31 to 40.) If the degree of falsifiability is increased, then introducing the hypothesis has actually strengthened the theory: the system now rules out more than it did previously: it prohibits more. The introduction of an auxiliary hypothesis should always be regarded as an attempt to construct a new system; and this new system should then always be judged on the issue of whether it would, if adopted, constitute a real advance in our knowledge of the world. An example of an auxiliary hypothesis which is eminently acceptable in this sense is Pauli’s exclusion principle (cf. section 38). An example of an unsatisfactory auxiliary hypothesis would be the contraction hypothesis of Fitzgerald and Lorentz which had no falsifiable consequences but merely served to restore the agreement between theory and experiment—mainly the findings of Michelson and Morley. An advance was here achieved only by the theory of relativity which predicted new consequences, new physical effects, and thereby opened up new possibilities for testing, and for falsifying, the theory. Our methodological rule may be qualified by the remark that we need not reject, as conventionalistic, every auxiliary hypothesis that fails to satisfy these standards. In particular, there are singular statements which do not really belong to the theoretical system at all. They are sometimes called ‘auxiliary hypotheses’, and although they are introduced to assist the theory, they are quite harmless.

So Popper does not say that every hypothesis introduced to save an existing theory is bad. Rather, he sez that if a new hypothesis is introduced to save an existing theory it should be a real advance in our understanding and should be testable. The Neptune theory passes this test since knowing there is a new planet is an advance in our understanding and can be tested. So the Neptune example doesn’t refute Popper’s actual position, as opposed to the position that Howson and Urbach invented and then attributed to Popper.

Communications Act 2003

Count Dankula was prosecuted under the Communications Act 2003, Section 127, which reads:

(1)A person is guilty of an offence if he—

(a)sends by means of a public electronic communications network a message or other matter that is grossly offensive or of an indecent, obscene or menacing character;

There is no objective definition of “grossly offensive” or “indecent” or “obscene” or “menacing”.

Almost anything you say will be grossly offensive to somebody. For example, any Jew who communicates online will offend Nazis by the fact of his existence. So all Jews are guilty under this act. Everyone who is in favour of people being able to deal with one another by voluntary transactions in capital markets will offend communists. Many left wing people will be offended by anyone criticising the welfare state. Some right wing people may be offended by people advocating the welfare state.

The word “obscene” is in a similar position. What you consider “obscene” depends on your standards. It’s also not clear what “menacing” means. Making threats of physical violence was illegal before this law was passed. So presumably menacing means something else, but this hasn’t been spelled out.

The section continues:

(2) A person is guilty of an offence if, for the purpose of causing annoyance, inconvenience or needless anxiety to another, he—

(a)sends by means of a public electronic communications network, a message that he knows to be false,

(b)causes such a message to be sent; or

(c)persistently makes use of a public electronic communications network.

So if you try to annoy or inconvenience anybody using the internet “persistently”, then you’re guilty. People often have misunderstandings about whether an action was intended to be annoying. Also, people go out of their way to pretend not to be annoyed, so it’s often difficult to tell whether you’re annoying somebody.

So it is impossible to tell whether you’re obeying this law. And that means the government can prosecute anyone for anything they say online. I don’t think a mistake like this can be made innocently. Perhaps the government wanted to be able to destroy anybody they like. Or perhaps they are so sloppy that they passed this law without thinking about what it means. In either case, the government that passed this law, and all the governments since then that didn’t repeal it, are evil or negligent or both

The prosecution of Count Dankula

Today the British government convicted Count Dankula of being grossly offensive for making a YouTube video of a dog raising its paw in front of a television. The prosecutor, the judge and the law used to prosecute Count Dankula are anti-moral and anti-intellectual garbage.

Any new moral idea that is an improvement will contradict standard moral ideas in some ways. As a result, some people will find moral improvement offensive. Moral progress is often required for intellectual progress since morality is about what you should do, including how you should conduct discussion. The British government’s actions are an attack on practices required for intellectual and moral progress.

By prosecuting Count Dankula for speech, the British government have conceded that their policies and ideas are intellectually and morally indefensible:

Let us consider the effect that coercion produces upon the mind of him against whom it is employed. It cannot begin with convincing; it is no argument. It begins with producing the sensation of pain, and the sentiment of distaste. It begins with violently alienating the mind from the truth with which we wish it to be impressed. It includes in it a tacit confession of imbecility. If he who employs coercion against me could mould me to his purposes by argument, no doubt he would. He pretends to punish me because his argument is important, but he really punishes me because his argument is weak.

William Godwin, An Enquiry Concerning Political Justice Vol. II, Book VII, Chapter II

Teenager’s brains and their ideas

Teenage brains can’t tell what’s important and what isn’t claims an article in New Scientist. There are lots of articles and books like this one claiming that behaviour X is a result of a person’s brain. This kind of argument is primarily an excuse for denigrating people.

Any human brain is a universal computer – it is capable of computing anything that can be computed. And since the information processing done by a person’s brain can be computed, this means that any human brain is capable of computing anything that another human brain can compute. As such, the difference between the brains of different people is how they are programmed. It is not a hardware difference.

In addition, human beings are capable of creating new explanatory knowledge. Even a person learning stuff that other people know has to recreate that knowledge in his own mind by the same kind of process that somebody creating completely novel knowledge would use. He guesses and criticises those guesses. Saying that there is some knowledge teenagers can’t create implies that there is some limit to what sort of explanations they could guess and criticise short of what an adult could do. No such limit has been explained. Nor does any such limit make sense. A child learns language, which includes all of the explanatory knowledge that is common enough for people to coin words to refer to it. For example, the word “fall” involves the idea of motion, a distinction between higher and lower, the idea of interactions that can raise objects and oppose falling motion, the idea of objects that you can fall off and so on. People had to discover this knowledge and come up with new words to describe it. So there is no hard and fast distinction between knowledge that lots of people happen to know now and knowledge that is not common in terms of the process of creating that knowledge. If you can learn English well enough to communicate at all then you can learn anything. There are examples of young children knowing more than their teachers or other adults, e.g. – Gauss.

A person can look for an explanation of his current ideas and priorities and change them if that explanation has flaws. This is true of all human beings, including teenagers and young children. Both teenagers and children can and do change their ideas and priorities in the light of new information. So if you want to explain the behaviour of any human being, including a child or teenager, you need to understand what explanations that person had adopted.

So trying to explain a person’s behaviour by referring to his brain is irrelevant and unhelpful. It’s like trying to explain why a character in a computer game has particular dialogue by talking about electrons in circuits. The reason why a character has particular dialogue is that somebody wrote that dialogue. The dialogue writer chose particular words in light of his knowledge about the computer game, cultural context etc. The explanation of the dialogue has to refer to the relevant context.

So why do people refer to brains when they talk about behaviour? Part of the explanation may be confusion, but that can’t be the whole explanation. Anyone capable of writing an article can read. Anyone who can read can learn about the theory of computation and the theory of how people create knowledge. So a writer could think about such ideas and refrain from writing false stuff about the brain. People actually write about the behaviour of teenagers in terms of brain because of a reckless disregard for the truth, and a desire to denigrate teenagers.

Consider the first paragraph of the article:

Teenagers may know full well how important final exams are – but that won’t stop some putting in minimal effort. This may be because their brains aren’t developed enough to properly assess how high the stakes are, and adapt their behaviour accordingly.

Some teenagers don’t put a lot of effort into exams. Instead of asking those teenagers about their priorities the writer attributes this behaviour to their brains. So then the writer doesn’t have to ask questions like the following. Do exams actually matter? Why might a person value some other activity over passing an exam?

The article continues:

Catherine Insel, at Harvard University, and her team asked adolescents between the ages of 13 and 20 to play a game while lying in an fMRI brain scanner. In some rounds of the game, participants could earn 20 cents for a correct response, while an incorrect one would cost them 10 cents. But in rounds with higher stakes, correct responses were worth a dollar, and wrong answers lost the participants 50 cents.

The team found that while the older volunteers performed better in the high stakes rounds, the younger ones didn’t – their performance didn’t change in line with whether the stakes were low or high. And the older the volunteers were, the more improved their performance was. “Interestingly, the ability to adjust performance according to the stakes at play emerged gradually across adolescence,” says Insel.

When the team looked at the brain activity of the volunteers, they found that their ability to improve their performance was linked to how developed their brains were. A region called the corticostriatal network seemed to be particularly important. This is known to connect areas involved in reward to those that control behaviour, and continues to develop until we are at least 25 years old.

A person’s priorities and ideas will in general change how he uses his brain and so would change patterns of brain activation. So studies like this don’t imply that a teenager’s brain explains his behaviour and ideas.

There are other explanations to consider. For example, parents will often take property and money away from children and teenagers. So teenagers and children may value money less since they think they won’t be allowed to use it to fulfil their own preferences.

This sort of “study” is careless junk science.

UBI and principles

Sam Altman of Y Combinator wants universal basic income (UBI). UBI is a policy where the government hands out money to people unconditionally.

Altman explains what problem he is trying to solve in an interview:

Spectacle: So the idea is that basic income is an answer. But what’s the question that it’s trying to answer?
Sam Altman: The question I’m interested in: How do we unlock maximum human potential? I think resources are distributed incredibly unequally around world. Obviously, there are a lot of people who could do great things that would benefit all of us. Create art, start companies and yet they can’t. I think 50 years from now it will look pretty ridiculous that we motivated people by fear of not being able to eat, not having a place to sleep, to work in jobs they hate for bosses they hate, or not do what they really wanted.
I think YC was an example that I saw firsthand of something like a basic income that allowed people to pursue what they wanted to pursue. Airbnb and Reddit would not exist had they not gotten basic income from YC.
One thing I didn’t realize at the time, but now I do looking back, is that Y Combinator itself is very much like a basic income. When I went through YC, $12,000 got invested in my company. We used that to live on. Buy some servers, do some projects, but mostly to live on. And everybody else got that as well. Some people totally fail. But some people create companies that they never otherwise would’ve been able to start.

He then talks about UBI having a branding problem of being associated with socialism and handouts:

Spectacle: There’s a huge branding problem—especially in the U.S.—of people who will basically say, “These are handouts.”

Sam: “This is socialism…”

Spectacle: Yeah, how do you change their minds?

Sam: What I would propose is a model like a company where you get a share in U.S. Inc. And then, instead of getting a fixed fee, you get a percentage of the GDP every year. As the whole country does better, you do better. I think that is how you align everybody. That message you can get a lot of people behind, even people who traditionally hate welfare, hate socialism.

Spectacle: Profit sharing in the United States?

Sam: Some version of that.

People often have trouble dealing with government policy issues, like UBI. Some people may get more money from UBI and have more stuff at least in the short term. Some of the gainers may be poor and UBI might improve what they can buy a lot. You might expect government deficits to go up or something like that. But who is to say whether all these effects balance out or tend to favour one side or the other?

Worse, lots of unpredictable stuff will happen. People will invent all sorts of clever ways to get as much money out of this system as possible. People will also adapt to whatever the means the government uses to finance UBI.

Faced with all this complexity, people tend to say that these issues should be sorted out by experiments, but this is a bad idea. If you can’t sort out the merits of some complicated set of possible effects of a policy, an experiment will have no clear results. And if there are a load of unanticipated results how are those supposed to be judged good or bad except by whim after the experiment? When somebody sez we need to decide a policy by experiment you should translate that to the following. “I have no argument that could explain to you why you should adopt this policy. But if you try it and people come to depend on it, then I may be able to blackmail you into keeping it, so I want to run an experiment.”

The best way to sort out issues like this is to understand them in terms of principles, not by doing “experiments”. A principle is a guideline about what you shouldn’t do and why you shouldn’t do it. A principle can be clearly understood and criticised and replaced if it is no good. A principle can also help you work out effects of a policy you wouldn’t expect without the principle.

The relevant principle in this case is private property rights. Property should only be transferred between people as a result of a voluntary agreement. If a transfer takes place without such an agreement you have the right to take property from the thief without his consent. If you do transfers without voluntary agreements, then you’re ignoring objections to the transfer, which is irrational.

How will UBI be provided? There are two ways it could be provided. (1) The government could take money from some people and give it to other people. (2) The government could print money or issue credit.

Both (1) and (2) involve taking resources from people by threatening them with force.

In case (1) if you refuse to give the government the money then the government will try to take more money from you (fines), or try to put you in prison and if you resist by force then the government may kill you. If you try to avoid prosecution in the courts, that will cost you a lot of money and your defence may fail even if you haven’t broken the law. So if the government comes after you, the best case scenario is you lose a load of money and property.

In case (2) the value of money will decline relative to other goods, so you may want to use some other medium of exchange that retains its value. You can’t refuse to take government money in settlement of a debt because of legal tender laws.

Whether governments would prosecute people who make their own money is unclear. The US government prosecuted Bernard vonNothaus for minting his own money saying that it could be mistaken for US dollars. Governments haven’t decided to prosecute bitcoin users so far. Bitcoin changes value a lot and I suspect lots of bitcoin owners are speculating that it will rise in value. Most people will not put much of their assets in bitcoin partly because they don’t know much about it and don’t understand it. I suspect governments don’t see bitcoin as a serious threat so they aren’t pursuing those who trade it in any serious way. Bitcoin transactions can often be tracked to the people who made them, so if governments decide bitcoin is a threat they could go after many users.

Western governments provide some useful services with the money they take, like police, courts, roads and national defence. They also hold elections in which the current government can be thrown out of power. So governments aren’t completely useless and have some means of correcting errors. But governments still take money from people without their consent. We shouldn’t want reform in the direction of government doing more stuff.

So UBI involves taking money from people by force and giving it to other people. UBI is a violation of property rights and should be opposed.

Let’s be a bit more specific about what’s bad about UBI. The people who are net donors to UBI are more productive than those who are net receivers. Taking stuff from the donors may prevent them from undertaking productive activities. It will also make the donor’s life worse cuz he will have fewer options. So UBI punishes competence and prevents some improvements in how stuff is produced.

The recipients are also harmed by UBI. A person can’t have stuff that isn’t produced, so the production-preventing aspects of UBI mean there is less stuff the recipient could have. Also, a person who is short of money in a free market is bad at making decisions. Hiding his incompetence from him by giving him money taken from other people by force will stop him from realising he should change his ways. If he had to rely on people privately giving him charity he might get some advice and help on how to be more productive. Since people in government get money through threats of violence, they are generally not going to be able to give good advice on how to be more productive. Another problem: since UBI is not given out by any rational standards, there is no reason why the government should continue to give the money. So recipients are making their lives dependent on totally arbitrary whims of government officials.

There is another side to principles. You can ask what principles a policy endorses and then see if they are any good. UBI involves taking money from some people by force and giving it to others. So UBI endorses the principle that hurting people who have done nothing wrong to help others is good and necessary. This is the same as the principle behind other government redistribution schemes. So if you want to understand the principle properly, it’s useful to look at the societies in which it has been taken most seriously. It helps explain the state of Soviet Russia, North Korea, Venezuela etc. This principle pits people against one another in a bitter war for survival. Not a metaphorical war, but an actual war involving guns and bombs and destruction of property and murder. UBI is not just an isolated vague fuzzy warm idea. UBI entails an evil and destructive view of the world and should be rejected as evil garbage.