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!

Complex probability in QM system?

  1. Sep 10, 2009 #1
    1. The problem statement, all variables and given/known data

    Consider a quantum system described by a basis [tex]\mid 1 \rangle[/tex] and [tex]\mid 2 \rangle[/tex].

    The system is initially in the state: [tex]\psi_i = \frac{i}{\sqrt3} \mid 1 \rangle + \sqrt{\frac{2}{3}} \mid 2 \rangle[/tex].

    (a) Find the probability that the initial system is measured to be in the state: [tex]\psi_f = \frac{1 + i}{\sqrt 3} \mid 1 \rangle + \frac{1}{\sqrt{3}} \mid 2 \rangle[/tex]

    2. Relevant equations

    The basis is assumed to be orthonormal, hence [tex]\langle 1 \mid 1 \rangle = \langle 2 \mid 2 \rangle = 1[/tex]

    Probability is calculated as [tex](\langle \psi_f \mid \psi_i \rangle)^2[/tex]

    3. The attempt at a solution

    Calculating this, I get a complex answer. I'm not sure but I think a probability (a real observable) should be a real number. Is that right?

    The answer I get is [tex]\frac{2 + 2\sqrt{2}}{9}(1+i)[/tex].
  2. jcsd
  3. Sep 10, 2009 #2
    I just tried to normalize the two states and found that they are both unit-normal already...

    This is confusing.
  4. Sep 10, 2009 #3


    User Avatar
    Homework Helper
    Gold Member

    Surely you mean:

    [tex]\frac{\langle \psi_i \vert \psi_f \rangle \langle \psi_f \vert \psi_i \rangle}{\langle \psi_i \vert \psi_i \rangle \langle \psi_f \vert \psi_f \rangle}=\vert\langle \psi_i \vert \psi_f \rangle \vert^2[/tex]

    (for normalized [itex]\psi_i[/itex] and [itex]\psi_f[/itex])....right?:wink:
  5. Sep 10, 2009 #4
    Thanks for the response ggh.

    Since my states are normalized, the denominator drops to 1. The numerator seems correct. I think you are right that I missed the absolute value signs.


    After a quick calculation, I see that's where I screwed up. I get a real answer if I take the absolute value before squaring.

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Complex probability in QM system?
  1. QM probability (Replies: 22)

  2. Composite Systems - QM (Replies: 3)