Write down the 3 ×3 matrix that represents Sx for a spin-one...

Write down the $3 \times 3$ matrix that represents $S_{x}$ for a spin-one system in the basis in which $S_{z}$ is diagonal (i.e., the basis states are $|0\rangle$ and $|\pm\rangle$ with $S_{z}|+\rangle=|+\rangle$, etc. $)$ A beam of spin-one particles emerges from an oven and enters a Stern-Gerlach filter that passes only particles with $J_{z}=\hbar$. On exiting this filter, the beam enters a second filter that passes only particles with $J_{x}=\hbar$, and then finally it encounters a filter that passes only particles with $J_{z}=-\hbar$. What fraction of the particles stagger right through?

Write down the $3 \times 3$ matrix that represents $S_{x}$ for a spin-one system in the basis in which $S_{z}$ is diagonal (i.e., the basis states are $|0\rangle$ and $|\pm\rangle$ with $S_{z}|+\rangle=|+\rangle$, etc. $)$
A beam of spin-one particles emerges from an oven and enters a Stern-Gerlach filter that passes only particles with $J_{z}=\hbar$. On exiting this filter, the beam enters a second filter that passes only particles with $J_{x}=\hbar$, and then finally it encounters a filter that passes only particles with $J_{z}=-\hbar$. What fraction of the particles stagger right through?

Key Concepts

Projection Postulate and Quantum Probability Amplitudes

The projection postulate states that measurement of a quantum system projects its state onto one of the eigenstates of the measured observable. In the context of the sequential Stern–Gerlach filters, the probabilities of observing a particular spin component in a subsequent filter are calculated using the square modulus of the corresponding probability amplitudes. These amplitudes arise from the inner product between the state vector before the measurement and the eigenstates of the next measurement operator, providing the link between the abstract state transformations and experimentally observable outcomes.

Stern–Gerlach Experiment and Sequential Spin Measurements

The Stern–Gerlach experiment demonstrates the quantization of angular momentum and is used to spatially separate particles according to their spin projections along a specific axis. When particles pass through successive Stern–Gerlach filters aligned along different axes (for example, first along z, then x, and then z again), the measurement process forces the quantum state to collapse into the eigenstates of each measured operator. This sequential measurement and the associated probabilities of state transitions are at the heart of understanding how quantum states evolve under measurement.

Spin-1 Operator Matrices

In quantum mechanics, particles with spin one are described by 3-dimensional Hilbert spaces and their spin operators are represented by 3×3 matrices. These matrices (Sx, Sy, Sz) satisfy the angular momentum commutation relations. For example, the operator Sx in the basis where Sz is diagonal is expressed as a specific symmetric matrix that couples the eigenstates of Sz, illustrating the intrinsic connection between the different components of spin angular momentum.

Choice of Basis and Diagonalization

The representation of quantum operators depends on the chosen basis for the Hilbert space. In problems involving spin, it is common to choose the basis that diagonalizes one of the operators—typically Sz—so that the eigenvalues associated with measurements along a fixed direction are immediately known. This basis is used to express other operators, like Sx, whose matrices then have off?diagonal elements that encapsulate the mixing between the eigenstates of Sz.

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