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B Notation meaning

  1. Aug 4, 2017 #1
    (posted in General Math in case this would be transferred elsewhere)

    In the thread, https://www.physicsforums.com/threads/mwi-and-path-of-single-electron.900851/page-4 someone posted:

    "This should be obvious; just do the same thing for subsequent measurements as we did for the initial measurements.

    For example, suppose we measure the spins of two electrons (call them electrons 1 and 2) in succession, both in the up/down direction. The total evolution looks like this (hopefully the notation is clear):

    ## \Psi_0 = \left( a_1 \vert u_1 \rangle + b_1 \vert d_1 \rangle \right) \left( a_2 \vert u_2 \rangle + b_2 \vert d_2 \rangle \right) \vert R_1, R_2 \rangle \vert O_{R1}, O_{R2} \rangle##

    ##\rightarrow \Psi_1 = \left( a_2 \vert u_2 \rangle + b_2 \vert d_2 \rangle \right) \left( a_1 \vert u_1 \rangle \vert U_1, R_2 \rangle \vert O_{U1}, O_{R2} \rangle + b_1 \vert d_1 \rangle \vert D_1, R_2 \rangle \vert O_{D1}, O_{R2} \rangle \right)##

    ##
    \rightarrow \Psi_2 = a_1 a_2 \vert u_1 \rangle \vert u_2 \rangle \vert U_1, U_2 \rangle \vert O_{U1}, O_{U2} \rangle + a_1 b_2 \vert u_1 \rangle \vert d_2 \rangle \vert U_1, D_2 \rangle \vert O_{U1}, O_{D2} \rangle \\ + b_1 a_2 \vert d_1 \rangle \vert u_2 \rangle \vert D_1, U_2 \rangle \vert O_{D1}, O_{U2} \rangle + b_1 b_2 \vert d_1 \rangle \vert d_2 \rangle \vert D_1, D_2 \rangle \vert O_{D1}, O_{D2} \rangle##

    In other words, each time a measurement happens, it creates another entanglement. So after two measurements, we have an entangled state containing four terms, one corresponding to each possible combination of the results of the two measurements."

    What does the notation or equations means? suppose initially we measure the spins of two electrons (call them electrons 1 and 2) in succession, both in the up/down direction.. what does the final evolution means.. what are the contents of the 4 terms?

    Thank you.
     
  2. jcsd
  3. Aug 4, 2017 #2
    btw.. a1, b1, a2, b2 are the probabilities
    u1, u2 d1, d2 are the actual spin conditions
    R is Ready
    O is Observer

    I guess I'm getting it after reading it over and over again.. what kind of math is this? differential equations?
     
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