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Cohen-Tannoudji on mutually exclusive (?) events

  1. May 24, 2015 #1
    I was looking at what Cohen-Tannudji has to say on compatibility of observables.
    Assumptions: ## A,B## are operators such that ##[A,B]=0 ## and we denote ## |a_i \,b_j\rangle## to be states for which ##A | a_i \, b_j \rangle= a_i | a_i \, b_j \rangle##, ##B | a_i \, b_j \rangle= b_j | a_i \, b_j \rangle##.
    We start with the state ##|\psi \rangle= \sum_{i,j} c_{i,j} | a_i \, b_j \rangle ##.
    The whole discussion starts with the following:
    "The probability for finding ## a_1 ## is ## P(a_1)= \sum_{j} |c_{1,j}|^2 ##." (Page 232 in a 1977 edition.)
    Have I forgotten something fundamental? I thought that the amplitudes ## \langle a_1 \, b_{j'} | \psi \rangle ## and ## \langle a_1 \, b_j | \psi \rangle ## are mutually exclusive for ##j' \neq j ##, so that according to quantum rules for probability
    $$P(a_1)= \big| \sum_j \langle a_1 \, b_{j} | \psi \rangle \big|^2= \sum_j \sum_{j'} \langle a_1 \, b_j | \psi \rangle \langle \psi | a_1 \, b_{j'} \rangle. $$
    I see no reason as to why ##j'= j ## should hold.
    My apologies for the slightly dull question, but I'm a bit lost.
     
  2. jcsd
  3. May 24, 2015 #2

    mathman

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    I'm completely ignorant of the physics here. However when dealing with probabilities, mutually exclusive means the probability of both happening is 0.
     
  4. May 24, 2015 #3
    Thanks for the reply, yeah.
    Let ## a,b## be exclusive events. For classical physical things ## P(a\mathrm{ \, or \, }b)= P(a) + P(b)##. In quantum physics, however, we have ## A(a\mathrm{ \, or \, }b)= A(a) + A(b)## where ##A \in \mathbb{C}## is a 'probability amplitude' so that ## P(x)= A(X) A^*(X)##, star denoting a complex conjugate, for some event ## X##. This definition will bring so called interference terms when compared with the classical case (they disappear in the classical limit).
    In my case, ## | a_i \, b_j \rangle ## are vectors.. some physical states, in fact, that have well-defined (=certain) values for some observables ##A,B## (so, a physical state always has some value or a distribution of values for both). We have different states for different ## i## and ## j##. A term ##\langle a_1 \, b_j | \psi \rangle## is actually just such an amplitude for the state ## | \psi \rangle## having the value ## a_1 ## for ## A## and the value ## b_j## for ## B##. I'm trying to determine the total probability to find that the value of ## A## for ## |\psi \rangle## is ## a_1##. As I see it, ##|\psi \rangle## can have that value while having ##b_1,b_2,b_3,... ## for ## B##.
     
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