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I'm a beginner in quantum mechanics, and I'm a bit confused about states and the probability to measure certain values. I would like to understand at least the following simplified situation:

Suppose the operator $S_z$ is measuring the spin in the $z$ direction of a free particle. Let $$\{e_s\},\quad s = -L, -L+1, \dots, L-1, L$$ be a basis of eigenstates for $S_z$, with $S_z(e_s) = s e_s$.

If at time $t<0$ the system is constantly in the pure state $P_{\psi}$, where $\psi$ is one basis vector, how can I compute the probability that the measuring the spin in the $z$ direction has result $-L/2$?

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    $\begingroup$ " S_z is measuring the spin in the x direction" - did you mean the z direction? Is the particle completely free or is there some sort of magnetic field its in? $\endgroup$
    – DJBunk
    Apr 27, 2012 at 14:27
  • $\begingroup$ Yes, it was a mistake. It is z. By the way the particle is free $\endgroup$
    – Abramodj
    Apr 27, 2012 at 14:31
  • $\begingroup$ Why $L/2$ specifically? $\endgroup$
    – David Z
    Apr 28, 2012 at 0:51

2 Answers 2

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The propability $P$ that a system in state $|\Psi\rangle$ will eventually transit into state $|\Theta\rangle$ is given by: $$ P = | \langle\Theta | \Psi \rangle |^2 $$

So for example if you have a System in Spin-X-State "up" denoted by $ |\Psi\rangle \hat{=} \frac{1}{\sqrt{2}} (1, 1)^T$, the propability of getting Spin-Z-State "up" (e.g. by measuring along the z-axis) denoted by $ |\Theta\rangle \hat{=} (1, 0)^T$ is: $$ P = | \frac{1}{\sqrt{2}} (1, 0) \cdot (1, 1)^T |^2 = \frac{1}{2} $$

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  • $\begingroup$ Note that this answer is about pure states. For arbitrary mixed states, you'd need the trace (as in the other answer). $\endgroup$
    – Martin
    Mar 16, 2015 at 13:05
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Ok, I've got it!

Using the bra-ket notation we have

enter image description here

Therefore the required probability is given by the square norm of the inner product between $\psi$ and $e_{-L/2}$.

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    $\begingroup$ Comment to the answer(v1): There seems to be something wrong with the spin normalization. If say, $L=1/2$, then $e_{s=-L/2=-1/4}$. But $s$ is supposed to be half-integral, cf. the question (v3). $\endgroup$
    – Qmechanic ♦
    Apr 27, 2012 at 19:51
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    $\begingroup$ Comment to the answer: it is not worthwhile to show "manipulations" that are not manpulations, but formal exercizes. You could have said the answer without any of the intermediate steps, as there are no intermediate steps. The definition of the inner product of two states is the amplitude for one state to be the same as the other, and the absolute square of this is by definition the probability.. $\endgroup$
    – Ron Maimon
    Jun 26, 2012 at 21:46

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