----- Original Message ----- From: "Ray Ludenia" <[EMAIL PROTECTED]> To: "Killer Bs Discussion" <[email protected]> Sent: Monday, March 21, 2005 7:14 AM Subject: Re: quantum darwin?
> > On 21/03/2005, at 3:23 PM, Erik Reuter wrote: > > > * Dan Minette ([EMAIL PROTECTED]) wrote: > > > >> In reletivistic quantum mechanics, this is stated as "Spacelike > >> operators > >> must commute." So, going back to our example of two spin 1/2 > >> particles in > >> a spin zero state, if we have call the operator for measuring the > >> spin of > >> particle 1: A and the operator for measuring the spin of particle 2: > >> B, we > >> find that if we perform A then B on the wavefunction BA(|+-> + > >> |-+>)/sqrt(2) one gets |+-> half of the time and |-+> half of the > >> time. > >> (the operator closest to the ket (which is what |>s are called) > >> operates > >> first. If we perform B then A, we obtain exactly the same results. > >> There > >> is no difference in the results if you perform A then B or B then A. > >> So, > >> the operators do commute. > > > > I have my doubts whether anyone who hasn't taken quantum mechanics > > could > > follow that paragraph. But I imagine serious quantum-less people could > > follow the rest of the post. Except that no one said anything...hmmmm > > I'll say anything then. Seemed clear enough to me. Well, for those who may not, let me try to give a bit of a background on operators. When you operate, you do something...so operators on a QM state do something to that state. Let's take just 1 particle with unknown spin. Call measuring it in the x direction the operator A, and measuring it in the y direction B (it's traveling in the z direction, so both x & y are orthogonal to the direction of travel. So, if we measure the particle in the x direction we either get up or down in the x direction. If we measure it in the y direction, we either get up or down in the y direction. But what happens if we do both. If we measure in the X direction first BA|s>, we either first get up in the x direction or down in the x direction. In each case, the odds on getting up or down in the y direction are still 50/50. Then we measure in the y direction and either get up or down. If we measure again in the x direction, we find that we are not assured of the same measurement as we first got. In fact, the measurement in the y direction got what is called an eigenstate (a characteristic state) of spin in y, which is a superposition in x. Well, what does that mean? It means that when we measure in the x direction, we now have a 50/50 chance of getting up or down, no matter what we originally measured. So, we see that, when we measure the same particle twice, the operators don't commute. When we measure BA|s> we end up knowing the spin in the y direction, but not the x direction. When we measure AB|s> we end up knowing the spin in the x direction but not the y direction. But, if we were to do this with a pair of particles in a superposition, with A being the measurement in the x direction of particle 1 and B being the measurement in the y direction of particle 2, the results of AB|12> and BA|12> are the same. We know the spin of particle 1 in the x direction and we know the spin of particle 2 in the y direction. Since AB|12> and BA|12> give the same answer, we say the operators commute (just as 3+5 =5+3 illustrates that addition commutes). But, with two measurements of the same particle AB|s> != BA|s>, so the operators don't commute (just as 3-5 != 5-3 illustrates that subtraction does not commute. Dan M. _______________________________________________ http://www.mccmedia.com/mailman/listinfo/brin-l
