# Time Evolution of a Spin State

I am stuck at paragraph 2.1.5 of Modern Quantum Mechanics - Sakurai

We have a simple system - spin $$\frac{1}{2}$$ system subjected to an external magnetic field $$\mathbf{B}=(0,0,B)$$. The time-evolution operator is calculated to be $$\mathcal{U} (t,0)=\exp{\Big(\frac{-i\omega S_zt}{\hbar}\Big)}$$

If we apply this time-evolution operator to some arbitrary state $$|\alpha \rangle = c_{+}|+\rangle + c_{-}|-\rangle$$, we see that the state evolves to $$|\alpha;t\rangle = c_{+} \exp{\big(\frac{-i\omega t}{2}\big)} |+\rangle + c_{-} \exp{\big(\frac{i\omega t}{2}\big)} |-\rangle$$

Now, let the initial state of the system be $$S_{x}^{+} = |S_x;+\rangle = \frac{1}{\sqrt{2}} |+\rangle + \frac{1}{\sqrt{2}} |-\rangle$$ at time $$t=0$$. Using the above, we can see that after time $$t$$, this state will evolve to $$|S_x^{+};t\rangle = \frac{1}{\sqrt{2}}\exp{\big(\frac{-i \omega t}{2}\big)}|+\rangle + \frac{1}{\sqrt{2}}\exp{\big(\frac{i \omega t}{2}\big)}|-\rangle$$

After this, they've calculated the probability of the system which was initially in $$S_{x}^{+}$$ to be found in $$S_{x}^{+}$$ and also the probability of the same initial system to be found in $$S_{x}^{-}$$ as $$|\langle S_{x};+ | {S_x^{+};t}\rangle|^2$$ and $$|\langle S_{x};- | {S_x^{+};t}\rangle|^2$$ respectively.

How I understood this is $$|\langle S_{x};+ | {S_x^{+};t}\rangle|^2$$ is the probability that the state $$|S_x^{+};t\rangle$$ upon measurement collapses to $$S_{x}^{+}$$ and $$|\langle S_{x};- | {S_x^{+};t}\rangle|^2$$ is the probability that the state $$|S_x^{+};t\rangle$$ upon measurement collapses to $$S_{x}^{-}$$. Do I understand this correctly? or are they simply calculating the transition probability?

If yes, then isn't it true that a state can only collapse to one of its basis and not to any arbitrary state? If that is true, then how can $$|S_x^{+};t\rangle$$ collapse to $$S_{x}^{+}$$ and same with the other case?

• Isn't transition probability from $|S_{x};+\rangle$ to $|S_{x};-\rangle$, the same as the probability for that very state to collapse to $|S_{x};-\rangle$ upon measurement at time t ? Commented Oct 22, 2023 at 3:05
• The collapse of a wave function to a state depends on your measurement apparatus. For example, the act of measurement is equivalent to projecting the state your system onto a particular basis. Commented Oct 22, 2023 at 3:18

How I understood this is $$|⟨𝑆_𝑥;+|𝑆^+_𝑥;𝑡⟩|^2$$ is the probability that the state $$|𝑆^+_𝑥;𝑡⟩$$ upon measurement collapses to $$𝑆^+_𝑥$$ and $$|⟨𝑆_𝑥;−|𝑆^+_𝑥;𝑡⟩|^2$$ is the probability that the state $$|𝑆^+_𝑥;𝑡⟩$$ upon measurement collapses to $$𝑆^−_𝑥$$ . Do I understand this correctly? or are they simply calculating the transition probability?

If yes, then isn't it true that a state can only collapse to one of its eigenstates and not to any arbitrary state? If that is true, then how can $$|𝑆^+_𝑥;𝑡⟩$$ collapse to $$𝑆^+_𝑥$$ and same with the other case?
A state doesn't have eigenstates. An $$\textbf{operator}$$ has eigenstates. When you measure an observable, the state collapses to one of the eigenstates of the corresponding operator. For example, the operator which represents the $$x$$-component of the spin has eigenstates $$|S_x;+\rangle$$ and $$|S_x;-\rangle$$. So when you measure the $$x$$-component of the spin (on any arbitrary state), the state collapses to one of these eigenstates.
Notice the time-evolved state in the OP, $$|S_x^+;t\rangle$$, is generally not an eigenstate of the operator $$S_x$$. Measuring the spin in the $$x$$-direction causes the time-evolved state to collapse to one of these eigenstates with probabilities given in the OP.
• So, there were two places where I messed up. 1. I was thinking along the lines that a state can only collapse to one of its basis states (which I incorrectly wrote as eigenstates). This is not true. 2. I did talk about measurement but missed with what observable are we measuring it. We measure it with $S_x$ and because $S_x^{+}$ and $S_x^{-}$ are the two eigenstates of $S_x$, it collapses to either one of them with certain probability. It all makes sense now. Thank you so much! :)