Consider a set of electrons with an initial spin state
$$|\psi\rangle=\left[\begin{array}{l}
0 \
1
\end{array}\right]$$
If we send these electrons into an SGx device, what is the probability a given electron will be determined to have its spin anti aligned with the $+x$ direction?
A) 0
B) $1 / 4$
C) $1 / 2$
D) $\sqrt{1 / 2}$
E) 1

Question

Quizplus · Accepted Answer

The given initial spin state $|\psi\rangle = \left[\begin{array}{l} 0 \ 1 \end{array}\right]$ represents a spin-down state in the z-direction. When measuring the spin in the x-direction, the probabilities are determined by projecting this state onto the eigenstates of the $S_x$ operator. The eigenstates of $S_x$ for spin $+\frac{1}{2}$ and $-\frac{1}{2}$ (aligned and anti-aligned with the x-direction, respectively) are equal superpositions of the z-basis states, but with different phase factors for the spin-up and spin-down components. The probability of finding the electron in either state (aligned or anti-aligned) is the square of the amplitude of the projection of $|\psi\rangle$ onto the respective eigenstate of $S_x$. Since the initial state is a pure z-direction spin-down state, it is an equal superposition of the x-direction spin states, leading to a $1/2$ probability of finding the electron spin anti-aligned with the $+x$ direction.

Consider a Set of Electrons with an Initial Spin State ∣ψ⟩=[01]|\psi\rangle=\left[\begin{array}{l}0 \\1\end{array}\right]∣ψ⟩=[01​]

Consider a Set of Electrons with an Initial Spin State $|\psi\rangle=\left[\begin{array}{l}0 \\1\end{array}\right]$