Particle with position and spin

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 $\{|x⟩\otimes |\uparrow ⟩,|x⟩\otimes |\downarrow ⟩\},x\in \mathbb{R}$, where $|x⟩$ represents the position state and $|\uparrow ⟩$, $|\downarrow ⟩$ represent the spin-up and spin-down states, respectively. The general state of such a particle is described by the vector

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 $|\psi ⟩$ 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 $\int {\psi }_{\uparrow }(x)|x⟩\otimes |\uparrow ⟩dx$.

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: $|x_0⟩\otimes ({\psi }_{\uparrow }(x_0)|\uparrow ⟩+{\psi }_{\downarrow }(x_0)|\downarrow ⟩)$.