Tag Archives: quantum mechanics

Mecânica Quântica

Mecânica Quântica (QM) is the physics of small things. How do they behave and how do they interact with each other? Conspicuously absent from this framework of QM is why. Why small things do what they do is a question QM leaves alone. E, if you are to make any headway into this subject, your best bet is to curb your urge to ask why. Nature is what she is. Our job is to understand the rules by which she plays the game of reality, and do our best to make use of those rules to our advantage in experiments and technologies. Ours is not to reason why. Realmente.

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Uncertainly Principle

The uncertainty principle is the second thing in physics that has captured the public imagination. (The first one is E=mc^2.) It says something seemingly straightforward — you can measure two complimentary properties of a system only to a certain precision. Por exemplo, if you try to figure out where an electron is (measure its position, que é) more and more precisely, its speed becomes progressively more uncertain (ou, the momentum measurement becomes imprecise).

Where does this principle come from? Before we can ask that question, we have to examine what the principle really says. Here are a few possible interpretations:

  1. Position and momentum of a particle are intrinsically interconnected. As we measure the momentum more accurately, the particle kind of “spreads out,” as George Gamow’s character, Senhor. Tompkins, puts it. Em outras palavras, it is just one of those things; the way the world works.
  2. When we measure the position, we disturb the momentum. Our measurement probes are “too fat,” por assim dizer. As we increase the position accuracy (by shining light of shorter wavelengths, por exemplo), we disturb the momentum more and more (because shorter wavelength light has higher energy/momentum).
  3. Closely related to this interpretation is a view that the uncertainty principle is a perceptual limit.
  4. We can also think of the uncertainly principle as a cognitive limit if we consider that a future theory might surpass such limits.

Tudo certo, the last two interpretations are my own, so we won’t discuss them in detail here.

The first view is currently popular and is related to the so-called Copenhagen interpretation of quantum mechanics. It is kind of like the closed statements of Hinduism — “Such is the nature of the Absolute,” por exemplo. Accurate, pode ser. But of little practical use. Let’s ignore it for it is not too open to discussions.

The second interpretation is generally understood as an experimental difficulty. But if the notion of the experimental setup is expanded to include the inevitable human observer, we arrive at the third view of perceptual limitation. Neste ponto de vista, it is actually possible to “derive” the uncertainty principle.

Let’s assume that we are using a beam of light of wavelength \lambda to observe the particle. The precision in the position we can hope to achieve is of the order of \lambda. Em outras palavras, \Delta x \approx \lambda. In quantum mechanics, the momentum of each photon in the light beam is inversely proportional to the wavelength. At least one photon is reflected by the particle so that we can see it. Assim, by the classical conservation law, the momentum of the particle has to change by at least \Delta p \approx constant\lambda from what it was before the measurement. Assim, through perceptual arguments, we get something similar to the Heisenberg uncertainty principle \Delta x \Delta p = constant.

We can make this argument more rigorous, and get an estimate of the value of the constant. The resolution of a microscope is given by the empirical formula 0.61\lambda/NA, where NA is the numerical aperture, which has a maximum value of one. Assim, the best spatial resolution is 0.61\lambda. Each photon in the light beam has a momentum 2\pi\hbar/\lambda, which is the uncertainty in the particle momentum. So we get \Delta x \Delta p = (0.61\lambda)(2\pi\hbar) \approx 4\hbar, approximately an order of magnitude bigger than the quantum mechanical limit. Through more rigorous statistical arguments, related to the spatial resolution and the expected momentum transferred, it may possible to derive the Heisenberg uncertainty principle through this line of reasoning.

If we consider the philosophical view that our reality is a cognitive model of our perceptual stimuli (which is the only view that makes sense to me), my fourth interpretation of the uncertainty principle being a cognitive limitation also holds a bit of water.

Reference

The latter part of this post is an excerpt from my book, O Unreal Universo.

Sex and Physics — De acordo com Feynman

Física passa por uma época de complacência de vez em quando. Complacência origina de um sentimento de completude, um sentimento que descobrimos tudo o que há para saber, o caminho é claro e os métodos bem-entendido.

Historicamente, essas crises de complacência são seguidas por uma rápida evolução que revolucionam a forma física é feito, mostrando-nos o quão errado temos sido. Esta lição humilhante da história é, provavelmente, o que levou Feynman dizer:

Tal era da complacência existia na virada do século 19. Personas famosas como Kelvin observou que tudo o que restava a fazer era fazer medições mais precisas. Michelson, que desempenhou um papel crucial na revolução a seguir, foi aconselhado a não entrar em uma “mortos” área como física.

Quem teria pensado que em menos de uma década no século 20, que iria completar mudar a forma como pensamos de espaço e tempo? Quem no seu perfeito juízo iria dizer agora que vamos voltar a mudar as nossas noções de espaço e tempo? Faço. Então, novamente, ninguém nunca me acusou de um juízo!

Outra revolução teve lugar no decorrer do século passado — Mecânica Quântica, que acabou com a nossa noção de determinismo e desferiu um duro golpe para o paradigma do sistema-observador da física. Revoluções similares vai acontecer de novo. Não vamos segurar nossos conceitos como imutável; eles não são. Não vamos pensar em nossos velhos mestres como infalível, pois eles não são. Como o próprio Feynman recorda, física, sozinha, detém mais exemplos da falibilidade de seus antigos mestres. E eu sinto que uma completa revolução no pensamento está atrasado agora.

Você pode estar se perguntando o que tudo isso tem a ver com sexo. Bem, Eu apenas pensei que o sexo iria vender melhor. Eu estava certo, não fui eu? Quero dizer, você ainda está aqui!

Feynman também disse,

Foto por "Caveman Chuck" Coker cc

Einstein on God and Dice

Although Einstein is best known for his theories of relativity, he was also the main driving force behind the advent of quantum mechanics (QM). His early work in photo-voltaic effect paved way for future developments in QM. And he won the Nobel prize, not for the theories of relativity, but for this early work.

It then should come as a surprise to us that Einstein didn’t quite believe in QM. He spent the latter part of his career trying to device thought experiments that would prove that QM is inconsistent with what he believed to be the laws of nature. Why is it that Einstein could not accept QM? We will never know for sure, and my guess is probably as good as anybody else’s.

Einstein’s trouble with QM is summarized in this famous quote.

It is indeed difficult to reconcile the notions (or at least some interpretations) of QM with a word view in which a God has control over everything. em QM, observations are probabilistic in nature. Isto quer dizer, if we somehow manage to send two electrons (in the same state) down the same beam and observe them after a while, we may get two different observed properties.

We can interpret this imperfection in observation as our inability to set up identical initial states, or the lack of precision in our measurements. This interpretation gives rise to the so-called hidden variable theories — considered invalid for a variety of reasons. The interpretation currently popular is that uncertainty is an inherent property of nature — the so-called Copenhagen interpretation.

In the Copenhagen picture, particles have positions only when observed. At other times, they should be thought of as kind of spread out in space. In a double-slit interference experiment using electrons, por exemplo, we should not ask whether a particular electron takes on slit or the other. As long as there is interference, it kind of takes both.

The troubling thing for Einstein in this interpretation would be that even God would not be able to make the electron take one slit or the other (without disturbing the interference pattern, que é). And if God cannot place one tiny electron where He wants, how is he going to control the whole universe?