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Homework Help: Expectation value of an observable

  1. May 9, 2007 #1
    1. The problem statement, all variables and given/known data

    We have an observable A, that has eigen vectors l a1 > and l a2 > , with eigenvalues a1 and a2 respectively. A second observable B has eigenvectors l b1 > and l b2 > with eigenvalues b1 and b2 respectively. The eigenstates of B can be written in terms of the eigenstates of A as:

    l b1 > = 3/4 l a1 > + sqrt(7)/4 l a2 >
    l b2 > = -sqrt(7)/4 l a1 > -3/4 l a2 >

    If the particle is in state l b2 > , what are the expectation values <A> and <B> in terms of a1, a2, b1 and b2

    2. Relevant equations

    3. The attempt at a solution

    expectiation value of, say A, is given by:

    <A> = <b2 l A l b2 > since the particle is in the state b2

    now how do I proceed to find <A> ? Is A a 2x2 matrix?
  2. jcsd
  3. May 9, 2007 #2


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    Science Advisor
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    Gold Member

    Just apply A to |b2> using the fact that

    A |a1> = a1 |a1>

    A |a2> = a2 |a2>

    and then using orthonormality of |a1> and |a2>
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