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Measurement and Unitarity in QM
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Here is a rather long and somewhat technical note of it. I apologize
for not giving a very detailed and down to earth explanation. Sorry about having to introduce density matrix. But I find it is the most closest analogy to the Liouville theorem, ----------------------------------------------------------------------------- ----------------------------------------------------------------------------- Roughly speaking, unitarity means conservation of probability. Suppose I start with a state |psi>. After a while, the state evolves to U|psi>. Then, conservation of probability means that <psi|U^+ U |psi>=<psi|psi>. It requires the evolution matrix to be unitary, i.e., U^+ U=1. More strictly speaking, unitarity is somewhat like the Liouville theorem. It is the conservation of some probability 'flow'... It is the one of the biggest mysteries that measurement irreversibly destroy this flow. This can be illustrated just by the double-slit experiment. ---------------------------------------------------- Before doing that, just to be systematic, let me introduce a little bit of language, the density matrix. For any single quantum mechanical state |psi>, define density matrix by W=|psi><psi|. First, it is a matrix. Given any base |1>, |2>....|n>....., matrix element W_mn = <m|psi><psi|n>. It is called density matrix since the expectation value of any operator O is <O>=Tr (W O) where the trace is carried out over any complete basis. This is again easy to see as <O> =<psi|O|psi>= Sum_n,m <psi|m><m|O|n><n|psi> = Sum <n| ( |psi><psi|) O |n> = Trace (W O). Therefore, if you like, density matrix is a statistical weight (which is analogous to the quantity in Liouville Theorem). We can also show that Tr(W)=Sum <n|psi><psi|n> = Sum <psi|n><n|psi>= <psi|psi>=1. Moreover, we can show that Tr(W W )=Tr(W W W....)=1 ---------------------------------------------------------------- Now the first important result, a sort of 'quantum Liouville' theorem: Any unitary evolution will 'conserve' density matrix of a single quantum mechanical state. Let's understand this more carefully. After evolution U, the density matrix will be W -> U|psi><psi|U^+ Now it is easy to show Tr(W)=1 is not affected by the evolution since Tr (U|psi><psi|U^+ ) = Tr (U^+ U|psi><psi| ) = Tr ( |psi><psi|)=1. Moreover, Tr(W W) should also not to be affected by any unitary evolution. ... This is what we mean by 'conserve'. ----------------------------------------------------------------- Now we come back to double-slit experiment. Say there are slit 1 and 2. The incoming particle is in a state 1/sqrt(2) *(|1>+|2>). Then if we put a screen after the slit, we see interference pattern... The density matrix for the incoming state is 1/2(|1>+|2>)(<1|+<2|), or, in a matrix form in the base of |1> and |2> | 1/2 1/2 | | | . | 1/2 1/2 | one can easily verify that this staisfies Tr(W)=1, Tr|W^2|=1,.... Notice that the off-diagonal entry in the density matrix is very important. It is the memory of a state which could be both 1 and 2, both here and there. It related to all quantum mechanical magic/mystery/misery/misunderstanding about coherence, pure state, etc. It is crucial to make Tr(W^2)=1. Now suppose we really want to find out what is going on. In particular, we put detectors at each slit to find out which slit the particle does pass through. Quantum mechanics rule (it is the RULE, no explanation ) says that too bad, now we have made the damn measurement. We destroyed the coherence. The wave-packet of every partile passing through collapses. The result would be EITHER |1> OR |2> (not BOTH 1 AND 2). We all know that it means no interference, and so on. Suppose we still want to construct a density matrix for the system after that measurement, it would have to be | 1/2 0 | | | . | 0 1/2 | since there would be no interference anymore. Immediately, we see that Tr(W^2)=1/2 not 1!! We have proved that any unitary evolution cannot change Tr|W^2|. Therefore, something which is not unitary must have happened during the measurement. >我记得，有人认为要把环境和测量仪器（包括观测者）考虑在内作为一个更大的 >系统来考察，而不是象传统那样只分析被观测者，只是不知道这种考虑管不管用。 This might be called decoherence propelled by Wheeler company. I don't know that much about it. Thinking naively, if all matter obey schrodinger equation (or whatever Hermitian generator), then the evolution will always be unitary.... including enviroment means that the dimension of the density matrix is much larger. In principle, the off-diagonal entries could move to other places. therefore, if we just focus on the 2 by 2 submatrix, it is possible that it is diagonal. However, the coherence is somewhere else. This does not seem solve the basic measurement problem especially if we are allowed to think of the universe as a big pure state... This would force us to think of some multiworld theory or something like that. maybe we are not allowed to say universe a pure state since gravity is very mysterious. but this is certainly just speculation. anyway, I hope this will tell us something new but I don't see even a vague hint of it. I am still too young to work on that... |