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Measuring sum of two components of spin angular momentum

  1. Nov 21, 2013 #1
    1. The problem statement, all variables and given/known data

    What are the possible results of the measurement of the sum of the x and z components of the spin angular momentum of a spin-1/2 particle?

    Sx = Spin angular momentum operator x
    Sz = Spin angular momentum operator x


    2. Relevant equations


    3. The attempt at a solution

    I started by applying the spin-up eigenket to the sum of the spin angular momentum operators.

    (Sx + Sz) | up > = Sx | up > + Sz | up > = -1/2h_bar + 1/2h_bar = 0

    The text I'm using proves that Sx | up > = 1/2h_bar | down >
    and like wise Sx | down > = 1/2h_bar | up >

    Does this mean if you were to probe the particle for spin up in the x direction you would actually see spin down?

    then I applied the spin down eigen-ket to pull the possible eigenvalues from the operators.

    (Sx + Sz) | down > = Sx | down > + Sz | down > = +1/2h_bar + (-1/2h_bar) = 0

    I feel that there is something wrong with how I've gone about calculating this.. I know that Sx and Sz don't commute therefore you cannot measure their eigenvalues simultaneously. Since I get zero did I show this correctly?

    Also, the eigenkets for spin up and spin down are considered the eigenvectors, correct?

    I am still trying to get the hang of the linear algebra and what everything means so any help would be amazing!

    Thanks
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 21, 2013 #2

    vela

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    One problem with your work is that saying, for example, that ##\hat{S}_x \lvert \text{up} \rangle = \frac{\hbar}{2}## doesn't make sense. This is akin to writing
    $$\frac{\hbar}{2}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix} = \frac{\hbar}{2}.$$ The lefthand side is a vector, but the righthand side is a scalar.

    No. Rid yourself of the notion that taking a measurement means applying the associated operator to a state.

    The result of a measurement is one of the eigenvalues of the associated observable. Given the state of a particle, you can calculate the probability amplitude for a particular result by finding the projection of this state in the direction of the corresponding eigenstate. For example, if you were measuring the spin in the z-direction, you would need to know the eigenvalues of ##\hat{S}_z##, which are ##\pm \hbar/2##, and the corresponding eigenstates, ##\lvert + \rangle## and ##\lvert - \rangle##. If the particle is in state ##\psi##, the probability amplitude for finding the particle in the spin-up state is ##\langle + \vert \psi \rangle##, which is generally a complex number. The probability is given by square of the modulus of the amplitude.
     
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