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Fri Apr 20 2001 at 02:21:40 (16.6 years ago )
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Mon Nov 13 2017 at 15:32:16 (5.9 days ago )
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mission drive within everything
Add to understanding, not just information.
specialties
Physics, Math
school/company
University of Maryland
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Every passing minute is another chance to turn it all around
most recent writeup
July 7, 2008
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I chose not to choose life, I chose something else.
















Feel free to comment on my nodes using the E2 annotation tool and send me a /msg about it.

I'm a graduate student at the University of Maryland, working on my PhD in Physics. In addition to physics, I'm interested in science in general, math, philosophy, and really most other academic subjects from history to anthropology to economics, though I won't claim to know much outside of the hard sciences. When I'm not slaving away in the salt mines, I like to spend the rest of my time watching movies, playing sports, and going to punk and ska shows...oh, and reading E2.

Thoughout most of my contact with E2 I've mostly been a lurker. I lurked for a while before actually becomming a user. This is, I think, the best advice to those who are new to E2 (or really any online forum): Lurk first, spend time to not just read the FAQs, but look around at what's there. Being back in grad school has made it tough to find any time to node. I also don't generally bother to node things unless I think I can do a fairly complete and coherent job (though I still fail). I generally stick to "node what you know", because I don't really have time to node what I don't. I think noding arguments (e.g. on politics) is a good thing, as long as the argument is well put an sticks to points that are decidable and supported, so that the debate has a finite length. Obviously inadquitely supported arguments just lead to everbody contributing their two cents. ...oh, and if you haven't noticed, I can't spell worth a damn. On my nodes I usually try to spell check but mistakes still happen.

Contact Information:

ICQ Number:   13367880
AOL IM Screen Name:   DeusXMakna


Below is the original version of my very first node, the Heisenberg Uncertainty Principle. You can see the new revamped version there. If you do look at both, please let me know if you think it's an improvement.

Heisenberg Uncertainty Principle

Here are an informal explanation of the Heisenberg Uncertainty Principle, the formal explanation, and the proof. First, however, to describe this, we need to be clear on exactly what we mean by uncertainty. Quantum mechanics describes the behavior of the wave function for a system, which tells you the probability of getting a particular result when you measure a particular observable quantity. For example, it tells you the probability of finding the electron in a hydrogen atom within a given radius of the center of the atom (position is the observable quantity). For such a probability distribution you can define an expectation value, denoted <O> for an observable O, which is the value you'd get if you averaged the results of measurements made on many systems all in that quantum state. It's basically the average value. You can then define an uncertainty for that average value, denoted dO here, which would be the standard deviation (roughly speaking, the spread) of the results from many such measurements around the expectation value. Also, below <= means "less then or equal to", and >= means "greater then or equal to".


Informal Explanation

The Heisenberg Uncertainty Principle says that in quantum mechanics, if you have two observable quantities of a system, then there will in general be some lower bound to the certainty with which both values can be known for both observables, mathematically, for two observables A and B, dA*dB >= L. In general, this lower bound is not zero, meaning that the less the uncertainty in one the greater the uncertainty in the other. The lower limit is defined by the commutator of the two observables, denoted [A,B], which roughly measures how similarly each acts in the specific system, so the L I refer to is dependent on both of the observables being discussed and the state of the system. This is defined for any pair of observables, which can include things like position, momentum, energy, angular momentum, and spin.

The form of the Heisenberg Uncertainty Principle that is normally discussed is the statement of it for position and momentum. For position x and momentum p the lower bound on certainty turns out to be independent of the state, and results in the uncertainty relation

dx*dp >= hbar/2

So, the more accurately you know something's position, the less accurately you know its momentum, and vice versa.

There are a few misconseptions around about the uncertainty principle. First of all, there is a energy-time uncertainty relation, but it is not the same as the one discussed here (the one used for position and momentum), although it does have a similar mathematical form. It's different because time is not an observable, meaning that you don't measure a system to find its time like you would to find, say, its momentum. Rather, you think of time as existing independent of the system, as something that the observer keeps track of. Also, the meaning of uncertainty in that relation is different.

Another common misconception is that the Heisenberg Uncertainty Principle is equivalent to the statement, "You can't measure a system without changing it." In fact, it applies to unmeasured states and does not really take account of the effect of measurement. The effects of measurement are a part of other areas of quantum theory, like quantum measurement theory, the interpretation of quantum mechanics, and quantum decoherence. Many people seem to think that the uncertainty principle is saying that if you measure one quantity for a system this will disturb the system, thus messing up your second measurement, but, in fact, it is saying that if you make the measurement of the first quantity on one system and the measurement of the second on a completely separate system, then as long as they both started out in the same state before measurement you will have this uncertainty relation. The Heisenberg Uncertainty Principle is actually just a statement about how quantities relate in quantum mechanics. Take the case of position-momentum uncertainty: In classical mechanics, it makes sense to imagine a particle with any combination of position and momentum. In quantum mechanics, however, a particle having a certain wave function in terms of position implies a specific wave function in terms of momentum (the Fourier transform of the position wave function). The uncertainty principle tells us something about how these position and momentum distributions relate; it tells us that the more narrowly peaked around one average value you make the position state, the more spread out away from any particular value the momentum state will be. And the reverse, of course, is also true.


Formal Explanation

For an operator O, let <O> be the expectation value of the operator and let dO be the standard deviation of O, so dO^2 = <O^2> - <O>^2. For a complex number z, let conj(z) be the complex conjugate of z and |z|^2 be the complex norm z*conj(z). Let adjoint(O) be the adjoint of O, also called the hermitian conjugate, and remember that all operators in this proof represent observables, so they are all hermitian (self-adjoint). Let the commutator of A and B be [A,B]=AB-BA. Finally, hbar is Planck's constant divided by twice pi. Then given two observable quantities represented by operators A and B:

(dA)^2*(dB)^2 >= |<[A,B]>|^2/4

This is the general statement of the Heisenberg Uncertainty Principle for any two quantities you can observe. The value of the lower limit of the uncertainty depends on what you're measuring, and it can be zero in some cases. This is not the same as the energy-time uncertainty relation because this is stated for observables of the system as represented by operators on the Hilbert space of the system. Time is not an observable, and it represented in quantum mechanics as a scalar parameter of the theory, and independent variable, not an operator. Also, the sense in which uncertainty is defined is different. The uncertainty principle is not a statement concerning measurement directly because it simply tells you how the standard deviation of the probability distribution changes as you change basis between the eigenstates of A and the eigenstates of B. It does not rely on any notions regarding measurement such as wave function collapse (or your preferred equivalent), the discontinuous evolution during measurement referred to by Von Neumann as "process 1".


Proof

For this proof, I will use Dirac notation, where a state of the system labeled n, a vector in the Hilbert space of the system, is denoted by the ket |n> and its dual is denoted by the bra <n|. The scalar product on a bra m with a ket n is then denoted <m|n>. Also, the expectation value of an operator O for a state n is <O> = <n|O|n>, where O is a hermitian operator on the Hilbert space.

Let the system be in a state |n>, and consider two observables represented by the operators A and B.



  • To prove the theorem as stated above, we will need to restate both sides of the inequality.
  • First, consider an operator
    A' = A-<A>. <A'^2> = <(A-<A>)^2> = <A^2-2*<A>*A+<A>^2> = <A^2>-(2*<A>)*<A>+<A>^2
  • Thus, <A'^2> = <A^2>-<A> = (dA)^2 And the same goes for B and B'.
  • Also, [A',B'] = [A-<A>,B-<B>] = [A,B]-[<A>,B]-[A,<B>]+[<A>,<B>].
  • <A> and <B> are just numbers so they commute with any other object; thus,
  • [A',B'] = [A,B]
  • Finally, both A' and B' are hermitian, since the Adjoint(A-<A>) = Adjoint(A)-Adjoint(<A>) = A-<A> given that A is hermitian and <A> must be a real number.
  • <[A',B']> = <A'B'-B'A'> = <A'B'>-<B'A'>
  • <B'A'> = conj(<adjoint(A')*adjoint(B')>) = conj(<A'B'>), since A' and B' are hermitian.
  • Thus, <[A',B']> = <A'B'>-Conj(<A'B'>) = 2*i*Im(<A'B'>), where Im(z) is the real number that is the coefficient of the imaginary part of z, meaning Im(z) = (z-conj(z))/(2*i)
  • |<[A',B']>|^2 = 4*|Im(<A'B'>)|^2 <= 4*|<A'B'>|^2 Since the norm of the imaginary part can't be more than the total.
  • |<A'B'>|^2 = |<n|A'B'|n>|^2
  • By the Cauchy-Schwartz inequality, for any two vectors |m> and |n>, |<m|n>|^2 <= |<m|m>|*|<n|n>|, so
  • |(<n|A')(B'|n>)|^2 <= |<n|A'A'|n>|*|<n|B'B'|n>| = |<A'^2>|*|<B'^2>| = <A'^2>*<B'^2> since <A'^2> and <B'^2> have to be positive because A' and B' are hermitian.

Now to put all this crazy crap together:


(dA)^2(dB)^2 = <A'^2>*<B'^2> >= |<A'B'>|^2 >= |<[A',B']>|^2/4 = |<[A,B]>|^2/4

So there it is. Not too enlightening, so that's why I saved it until last, but now you know. Of course, this is all based on formalism developed after Heisenberg stated the uncertainty principle, so he would have proved it a different way, but this is the basic scheme of the modern proof.


Other Stuff

So these are the few odds and end. Firstly, I'm new at this, so please /msg me if there's stuff you think should be fixed, in either content or style. And otherwise, if you're so inclined, just let me know what you think.

The last note is on Socialist Wolf's comment above. First of all, sending signals back in time is specifically forbidden in most "orthodox", i.e. mainstream, physical theories. Secondly, even if you could do that, it still wouldn't help, because as I explained in my write-up, it's not measuring the thing that causes the uncertainty relation to be true. It's an intrinsic property of the unmeasured state in quantum mechanics, and it applies even if you make measurements of two totally separate systems, so long as they're in the same state. In that case, timing of the measurments clearly isn't an issue.