I have a system of two spin 1/2 particles in a superposition of spin states in the z-direction given by:

$\psi = \frac{1}{2} |+ +\rangle + \frac{1}{2} |+ -\rangle + \frac{1}{\sqrt{2}} |- -\rangle$

where $+$ designates spin up, $-$ designates spin down and the first particle's state is the first term in each ket and the second particles' state is the second term in each ket. If I measure the spin on the first particle and get a value of $-\hbar / 2$ (corresponding to a spin down state) is the new state of the particles simply

$\psi = | - - \rangle$

meaning that the first particle is now "set" to being spin down? And if I determine the spin on the first particle to be spin up, would the subsequent state be

$\psi = \frac{1}{\sqrt{2}} |+ +\rangle + \frac{1}{\sqrt{2}} |+ - \rangle$ ?

Basically, my question is once I make a measurement of a spin of a particle, does the wavefunction stay collapsed on the spin determined? And does having a second particle affect this in any way?

  • $\begingroup$ Take a look at problem 3-24 [Sakurai] if you have a copy. $\endgroup$
    – Abolfazl
    Jan 20 '14 at 22:55
  • 1
    $\begingroup$ I don't have a copy of Sakurai. $\endgroup$ Jan 20 '14 at 22:59
  • $\begingroup$ Here $\endgroup$
    – Abolfazl
    Jan 20 '14 at 23:09
  • $\begingroup$ I don't think that's legal. $\endgroup$ Jan 20 '14 at 23:12

Yes, you've correctly understood what happens. Let's try to be more mathematically precise to be confident that this is the case.

I'll be using tensor product notation in this answer (the Hilbert space of the system is just the tensor product of the spin-$\frac{1}{2}$ Hilbert space with itself).

The projective measurement postulate says

For an observable $O$ with spectral decomposition $O = \sum_i\lambda_iP_i$, where $P_i$ is the projector onto the eigenspace of $O$ corresponding to eigenvalue $\lambda$, the possible outcomes of the measurement are the eigenvalues of the observable, and given that outcome $\lambda_i$ occured, the state of the system immediately after measurement is \begin{align} \frac{P_i|\psi\rangle}{\sqrt{\langle\psi|P_i|\psi\rangle}} \end{align}

For a two-spin-$\frac{1}{2}$ system, the $z$-component of the spin of the first particle is represented by the following observable: \begin{align} S_z\otimes I \end{align} where $I$ is the identity on the Hilbert space of the second particle. Now, recall that $S_z$ has the following spectral decomposition \begin{align} S_z = -\frac{\hbar}{2}P_-+\frac{\hbar}{2}P_+ \end{align} where $P_-$ and $P_+$ are projectors defined as \begin{align} P_-=|-\rangle\langle-|, \qquad P_+=|+\rangle\langle+| \end{align} and it follows that the spectral decomposition of $S_z\otimes I$ is \begin{align} S_z\otimes I = -\frac{\hbar}{2}P_- \otimes I+\frac{\hbar}{2}P_+ \otimes I+ \end{align} Upon measurement of $I\otimes S_z$ on the state $|\psi\rangle$ given in the question, obtaining the result $-\hbar/2$ indicates that the state got projected as follows upon the measurement: \begin{align} |\psi\rangle \to \frac{P_-\otimes I|\psi\rangle}{\sqrt{\langle\psi |P_-\otimes I|\psi\rangle}} = \frac{\frac{1}{\sqrt{2}}|--\rangle}{\sqrt{\frac{1}{2}}} = |--\rangle \end{align} If the value $+\hbar/2$ had been obtained, then the state would have been projected has follows: \begin{align} |\psi\rangle \to \frac{P_+\otimes I|\psi\rangle}{\sqrt{\langle\psi |P_+\otimes I|\psi\rangle}} = \frac{\frac{1}{2}|++\rangle + \frac{1}{2}|+-\rangle}{\sqrt{\frac{1}{4}+\frac{1}{4}}} = \frac{1}{\sqrt{2}}|++\rangle + \frac{1}{\sqrt{2}}|+-\rangle \end{align}

  • $\begingroup$ The postulate you quoted said that "the state of the system immediately after the measurement" is the projection you mentioned. What about at some time t after the measurement? Do the particles stay in the projected state indefinitely? $\endgroup$ Jan 20 '14 at 23:40
  • $\begingroup$ @Joshua It depends on the Hamiltonian of the system. If the state immediately after the measurement is an eigenstate of the Hamiltonian, then it will stay in the projected state. Otherwise, it will not. $\endgroup$ Jan 20 '14 at 23:50
  • $\begingroup$ I thought whenever we measured the state of a system it would collapse to an eigenstate of the Hamiltonian. Based on what you said, all measurements will force a system to indefinitely stay in one of the Hamiltonian's eigenstates. $\endgroup$ Jan 20 '14 at 23:57
  • $\begingroup$ @Joshua The first sentence is right; when you make a measurement on the system, it collapses to an eigenstate, but not just any old eigenstate. It collapses to an eigenstate of the observable being measured. If that observable happens to be something other than the Hamiltonian, then the state post-measurement may not be an eigenstate of the Hamiltonian. $\endgroup$ Jan 21 '14 at 0:04
  • $\begingroup$ So specifically whenever I measure energy (the eigenvalue value of the Hamiltonian) the subsequent state is an eigenstate of the Hamiltonian and all subsequent measures of energy will give the same value as the first measurement, even if the initial state before measurement was a superposition of many eigenstates? $\endgroup$ Jan 21 '14 at 0:08

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