In quantum mechanics, a particle with both position and spin can be described in the state space $L^2(\mathbb{R}) \otimes \mathbb{C}^2$. An orthonormal basis for such a state space is given by the set $\{|x\rangle \otimes |\uparrow\rangle, |x\rangle \otimes |\downarrow\rangle\}, x \in \mathbb{R}$, where $|x\rangle$ represents the position state and $|\uparrow\rangle$, $|\downarrow\rangle$ represent the spin-up and spin-down states, respectively. The general state of such a particle is described by the vector
$$ |\psi\rangle = \int \psi_{\uparrow}(x)|x\rangle \otimes |\uparrow\rangle + \psi_{\downarrow}(x)|x\rangle \otimes |\downarrow\rangle \, dx, $$where $\psi_{\uparrow}(x)$ and $\psi_{\downarrow}(x)$ are complex-valued functions representing the probability amplitudes for finding the particle at position $x$ with spin-up and spin-down, respectively.
When a measurement is performed on this system, the state $|\psi\rangle$ collapses according to the nature of the measurement:
1. Spin Measurement: If the spin is measured, the particle's state collapses onto one of the spin eigenstates, either spin-up or spin-down, with probabilities determined by $|\psi_{\uparrow}(x)|^2$ and $|\psi_{\downarrow}(x)|^2$, respectively. For example, if spin-up is measured, the post-measurement state becomes $\int \psi_{\uparrow}(x)|x\rangle \otimes |\uparrow\rangle \, dx$.
2. Position Measurement: If the position is measured and found to be at a specific point $x_0$, the particle's state collapses to a state localized at $x_0$, maintaining its spin superposition: $|x_0\rangle \otimes (\psi_{\uparrow}(x_0)|\uparrow\rangle + \psi_{\downarrow}(x_0)|\downarrow\rangle)$.
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Author of the notes: Antonio J. Pan-Collantes
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