![]() ![]() Position- momentum uncertainty relation, which can be stated as follows: It is one of the primary features that distinguishes quantum and classical mechanics and gives rise to " quantum weirdness". ![]() The Heisenberg Uncertainty Principle is a very important and central part of quantum mechanics. This is part of what makes them so useful in representing different kinds of data.Ĭuriously, it turns out that the bell curve that describes the normal distribution is the best compromise between localization in space and frequency! This function as a wave function gives the equality in the uncertainty principle inequality. Of 'compromises' in this respect, being moderately localized in both time and frequency. In fact, one way to look at wavelets is that they are kind It is impossible to have a function that has a precisely defined frequency, but is only non-zero at a single point. So the uncertainty principle follows from the fact that the sine function and Dirac delta function are not the same function. On the other hand, a particle with a precisely defined momentum would have a wave function that looks like a sine wave (The Fourier transform must be a Dirac delta function). This kind of a function is called the "Dirac delta function" in math jargon. Looks like a 'spike', which is zero everywhere except where the particle is. So, a particle with a precisely defined position would have a wave function that Similarly, the probability density of finding the particle with a momentum around p is given by conj(psif(p))*psif(p), where psif(p) is the Fourier transform of psi(x). conj(psi(x))*psi(x), the square of the absolute value of psi, describes the probability density of finding the particle around the place x. In quantum mechanics _everything_ about the position and velocity of a particle is described by a complex-valued function of its position, wavefunction psi(x). Our 'interference through measurement' that the romantics like so much is never even an issue. If there is streaking, then the speed measurement will be much much less accurate than had the picture been taken at a slower speed, which gives a better sample. If we take a picture with a high-speed device, we'll get a clear picture of the bullet's position, but no indication of how fast it's going. We can't easily measure its position, but by noting the time of exposure and measuring the streak, we can tell how fast it's going. We can take a picture with a normal camera, and see a streak on the film where it passes. Whenever we measure a particle, we always (by definition) use some sort of filter that gives us the position and velocity of the particle to some degree, and these degrees of error are inherently dependent on each other.Īnother way to see it: Think of a photograph of a bullet in flight. And where a step function tells you something about magnitude, it is homogeneous over time, and cannot be pinned down anywhere. The problem is that while a pure impulse is accurate in position, its area is indeterminate. When you take the velocity, it looks like a step function. When you take the position of your particle, it looks like an impulse. And since people use oscilloscopes to measure this stuff anyway, it's probably pretty close to the truth. I found that an explanation based on control systems helped me the most. The HUP also explains why a transporter will never work (though those tricky people at Star Trek mention something about a Heisenberg Compensator. ![]() Now, instead of classical determinism, we can do no better than quantum determinism - meaning if it's at all possible to write such equations then they will have to come in the form of the probabilities where particles could be. But Uncertainty broke down the movement of particles to probability functions (known as wave functions). ![]() In the Newtonian world, it was still possible that the grand equation for all particles could be written out starting at the beginning of time and and tell exactly what things would be like in the future : determinism. At least not in my house ( young man!).)Īs for the Romantics, though Heisenberg didn't begin the Downfall of Scientific Method, he did, however, end classical determinism. The effect is symmetrical so attempting to measure velocity we'll naturally affect direction (roughly equivalent to the seed bouncing off your knife or something - though I don't think tomato seeds ever reach such high velocities. In other words, the measurement of position directly affects velocity (being speed and direction). Well, what happens, as soon as you locate their position ( you poke 'em) they start to move. Let's say that tomato seeds are our metaphorical particles and in order to determine their position you must touch them with your knife. The best metaphor I've heard for the HUP is: tomato seeds. ![]()
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