Find the eigenvalues and eigenvectors of σy=( 0 -i i 0 ). Suppose an electron

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Find the eigenvalues and eigenvectors of σy=( 0 -i ...

Find the eigenvalues and eigenvectors of $\sigma_{y}=\left(\begin{array}{cc}0 & -i \\ i & 0\end{array}\right)$. Suppose an electron is in the spin state $\left(\begin{array}{c}\alpha \\ \beta\end{array}\right)$. If $s_{y}$ is measured, what is the probability of the result $\hbar / 2 ?$

Key Concepts

Quantum Spin

Quantum spin is an intrinsic form of angular momentum carried by particles such as electrons. Unlike classical angular momentum, spin does not have any classical analog and is quantized, meaning its measurements yield discrete values. The representation of a spin state in a two-dimensional complex vector space allows for the application of operators like the Pauli matrices to determine outcomes of spin measurements.

Quantum Measurement and the Born Rule

Quantum measurement involves the collapse of a quantum state onto one of the eigenstates of the observable being measured. The Born rule states that the probability of a particular measurement outcome is given by the square of the amplitude (modulus squared) of the projection of the state vector onto the eigenstate associated with that outcome. This rule is essential for calculating the likelihood of obtaining a specific eigenvalue when a quantum observable is measured.

Eigenvalues and Eigenvectors

In linear algebra, an eigenvalue of a matrix is a scalar such that there exists a nonzero vector (the eigenvector) that, when the matrix is applied to it, yields the scalar multiplied by the vector. This concept is critical in quantum mechanics because observables are represented by operators whose eigenvalues correspond to the measurable quantities, and eigenvectors represent the possible states of the system after measurement.

Pauli Matrices

The Pauli matrices are a set of three 2x2 complex matrices that are used to describe the spin operators for spin-1/2 particles. They are fundamental in quantum mechanics for representing spin observables. In this context, sigma_y is one of the matrices, and its eigenvalues and eigenvectors define the possible outcomes and corresponding states for measurements of the y-component of spin.

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