1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Spin probability

  1. Dec 11, 2013 #1
    1. The problem statement, all variables and given/known data

    Two particles with spin 1/2 and are in the spin state:

    [itex] |\psi> = |\uparrow_{1}>|\downarrow_{2}> [/itex]

    where [itex] |\uparrow_{1}> [/itex] is a state where particle 1 has spin up along the z-axis and
    [itex] |\downarrow_{2}>[/itex] is a state where particle 2 is spin down along the z-axis.

    If we measure the magnitude of the total spin of the two particles, what is the probability that the magnitude will be 1?

    2. Relevant equations

    [itex] Probability = |<n|\psi>|^{2} [/itex]

    3. The attempt at a solution

    I immediately thought that this problem was like a simple coin toss. 50% to get heads 50% to get tails. Since each particle has equal chance to be spin up or spin down, then the total probability of both being spin up after a measurement would be (0.5)(0.5) = 0.25 ?

    This doesn't feel right to me. I feel like it should be more complicated. =/
  2. jcsd
  3. Dec 11, 2013 #2


    User Avatar
    Homework Helper
    Gold Member
    2017 Award


    Can you expand the state ##|\psi \rangle## in terms of basis states that have definite values of the total spin? These basis states are the "singlet" and "triplet" states.

    See here and here.
    Last edited: Dec 12, 2013
  4. Dec 12, 2013 #3
    Hello =)

    so the singlet state is
    [itex] |\psi_{singlet}>= 1/(√2)(|\uparrow\downarrow> - |\downarrow\uparrow>) [/itex]

    and the triplet state is

    [itex] |\psi_{triplet}>= 1/(√2)(|\uparrow\downarrow> + |\downarrow\uparrow>) [/itex]

    [itex] |\psi> = 1/2(|\psi_{singlet}> + |\psi_{triplet}>) [/itex]

    in order to get S=1, the spin state would need to be triplet. The probability would then be the square of the coefficient of the triplet state?

    [itex] |1/(2\sqrt{2})|^{2} = 0.125 [/itex]

    this doesn't make sense though, because the total probabilities don't add up to 1...
  5. Dec 12, 2013 #4


    User Avatar
    Homework Helper
    Gold Member
    2017 Award

    Make sure ##|\psi \rangle## is properly normalized.
  6. Dec 12, 2013 #5
    Oh right, I forgot.

    After normalization, [itex] |\psi> = 1/\sqrt{2}(|\uparrow\downarrow> - |\downarrow\uparrow>) +1/\sqrt{2}(|\uparrow\downarrow> + |\downarrow\uparrow>) [/itex]

    leaving the probability to be in the triplet state (S=1) to be 1/2.

    Thanks for your help! I think I got it!
  7. Dec 12, 2013 #6


    User Avatar
    Homework Helper
    Gold Member
    2017 Award

    That looks correct.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted