Show that the operators \(\hat{S}_{i}=\frac{1}{2} \sigma_{i}\), where \(\sigma_{j}\) are the three Pauli spin-matrices, $$ \hat{S}_{x}=\frac{1}{2}\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right), \quad \hat{S}_{y}=\frac{1}{2}\left(\begin{array}{rr} 0 & -i \\ 1 & 0 \end{array}\right) \quad \text { and } \quad \hat{S}_{2}=\frac{1}{2}\left(\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right), $$ satisfy the same algebra as the angular momentum operators, namely $$ \left[\hat{S}_{x}, \hat{S}_{y}\right]=i \hat{S}_{2}, \quad\left[\hat{S}_{y}, \hat{S}_{z}\right]=\hat{S}_{x} \quad \text { and } \quad\left[\hat{S}_{z}, \hat{S}_{x}\right]=i \hat{S}_{y} . $$ Find the eigenvalue(s) of the operator \(\hat{S}^{2}=\frac{1}{4}\left(\hat{S}_{x}^{2}+\hat{S}_{y}^{2}+\hat{S}_{2}^{2}\right)\), and deduce that the eigenstates of \(\hat{S}_{2}\) are a suitable representation of a spin-half particle.

Step by step solution

01

Identify the Commutation Relations

We need to check if the given operators \($$\hat{S}_x, \hat{S}_y, \hat{S}_z$$\) satisfy the following commutation relations:$$\left[ \hat{S}_x, \hat{S}_y \right] = i \hat{S}_z$$,$$\left[ \hat{S}_y, \hat{S}_z \right] = i \hat{S}_x$$,$$\left[ \hat{S}_z, \hat{S}_x \right] = i \hat{S}_y$$.

02

Title - Verify Commutator [Sx, Sy]

Calculate the commutator [Sx, Sy]:$$\left[ \hat{S}_x, \hat{S}_y \right] = \hat{S}_x \hat{S}_y - \hat{S}_y \hat{S}_x$$.Using matrix multiplication,$$\hat{S}_x \hat{S}_y = \begin{pmatrix} 0 & \frac{1}{2} \ \frac{1}{2} & 0 \end{pmatrix} \begin{pmatrix} 0 & -\frac{i}{2} \ \frac{i}{2} & 0 \end{pmatrix} = \begin{pmatrix} \frac{i}{4} & 0 \ 0 & -\frac{i}{4} \end{pmatrix}$$.Similarly,$$ \hat{S}_y \hat{S}_x = \begin{pmatrix} 0 & -\frac{i}{2} \ \frac{i}{2} & 0 \end{pmatrix} \begin{pmatrix} 0 & \frac{1}{2} \ \frac{1}{2} & 0 \end{pmatrix} = \begin{pmatrix} -\frac{i}{4} & 0 \ 0 & \frac{i}{4} \end{pmatrix}$$.Then compute the commutator as:$$\left[ \hat{S}_x, \hat{S}_y \right] = \begin{pmatrix} \frac{i}{4} & 0 \ 0 & -\frac{i}{4} \end{pmatrix} - \begin{pmatrix} -\frac{i}{4} & 0 \ 0 & \frac{i}{4} \end{pmatrix} = \begin{pmatrix} \frac{i}{2} & 0 \ 0 & -\frac{i}{2} \end{pmatrix} = i \hat{S}_z$$.

03

Title - Verify Commutator [Sy, Sz]

Calculate the commutator [Sy, Sz]:$$\left[ \hat{S}_y, \hat{S}_z \right] = \hat{S}_y \hat{S}_z - \hat{S}_z \hat{S}_y$$.Using matrix multiplication,$$\hat{S}_y \hat{S}_z = \begin{pmatrix} 0 & -\frac{i}{2} \ \frac{i}{2} & 0 \end{pmatrix} \begin{pmatrix} \frac{1}{2} & 0 \ 0 & -\frac{1}{2} \end{pmatrix} = \begin{pmatrix} 0 & \frac{i}{4} \ \frac{i}{4} & 0 \end{pmatrix}$$.Similarly,$$ \hat{S}_z \hat{S}_y = \begin{pmatrix} \frac{1}{2} & 0 \ 0 & -\frac{1}{2} \end{pmatrix} \begin{pmatrix} 0 & -\frac{i}{2} \ \frac{i}{2} & 0 \end{pmatrix} = \begin{pmatrix} 0 & -\frac{i}{4} \ -\frac{i}{4} & 0 \end{pmatrix}$$.Then compute the commutator as:$$\left[ \hat{S}_y, \hat{S}_z \right] = \begin{pmatrix} 0 & \frac{i}{4} \ \frac{i}{4} & 0 \end{pmatrix} - \begin{pmatrix} 0 & -\frac{i}{4} \ -\frac{i}{4} & 0 \end{pmatrix} = \begin{pmatrix} 0 & \frac{i}{2} \ \frac{i}{2} & 0 \end{pmatrix} = i \hat{S}_x$$.

04

Title - Verify Commutator [Sz, Sx]

Calculate the commutator [Sz, Sx]:$$\left[ \hat{S}_z, \hat{S}_x \right] = \hat{S}_z \hat{S}_x - \hat{S}_x \hat{S}_z$$.Using matrix multiplication,$$\hat{S}_z \hat{S}_x = \begin{pmatrix} \frac{1}{2} & 0 \ 0 & -\frac{1}{2} \end{pmatrix} \begin{pmatrix} 0 & \frac{1}{2} \ \frac{1}{2} & 0 \end{pmatrix} = \begin{pmatrix} 0 & \frac{1}{4} \ -\frac{1}{4} & 0 \end{pmatrix}$$.Similarly,$$ \hat{S}_x \hat{S}_z = \begin{pmatrix} 0 & \frac{1}{2} \ \frac{1}{2} & 0 \end{pmatrix} \begin{pmatrix} \frac{1}{2} & 0 \ 0 & -\frac{1}{2} \end{pmatrix} = \begin{pmatrix} 0 & -\frac{1}{4} \ \frac{1}{4} & 0 \end{pmatrix}$$.Then compute the commutator as:$$\left[ \hat{S}_z, \hat{S}_x \right] = \begin{pmatrix} 0 & \frac{1}{4} \ -\frac{1}{4} & 0 \end{pmatrix} - \begin{pmatrix} 0 & -\frac{1}{4} \ \frac{1}{4} & 0 \end{pmatrix} = \begin{pmatrix} 0 & \frac{1}{2} \ -\frac{1}{2} & 0 \end{pmatrix} = i \hat{S}_y$$.

05

Title - Compute Eigenvalue of S^2

Find the eigenvalue of \(\hat{S}^2\) where $$\hat{S}^2 = \frac{1}{4}( \hat{S}_x^2 + \hat{S}_y^2 + \hat{S}_z^2 )$$.First, calculate the squares of the operators,$$\hat{S}_x^2 = \frac{1}{4}\begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}\begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} = \frac{1}{4}\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} = \frac{1}{4}I$$,$$\hat{S}_y^2 = \frac{1}{4}\begin{pmatrix} 0 & -i \ i & 0 \end{pmatrix}\begin{pmatrix} 0 & -i \ i & 0 \end{pmatrix} = \frac{1}{4}\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} = \frac{1}{4}I$$,$$\hat{S}_z^2 = \frac{1}{4}\begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix}\begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix} = \frac{1}{4}\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} = \frac{1}{4}I$$.Then compute the sum,$$\hat{S}^2 = \frac{1}{4}(\frac{1}{4}I + \frac{1}{4}I + \frac{1}{4}I) = \frac{1}{4}(\frac{3}{4}I) = \frac{3}{16}I$$.The eigenvalue of \(\hat{S}^2\) is thus \(\frac{3}{16}\).

06

Title - Deduce Spin-Half Eigenstates

The eigenstates of \(\hat{S}_z\) are found by solving the eigenvalue equation \(\hat{S}_z |\psi\rangle = \lambda |\psi\rangle\). Since \(\hat{S}_z = \frac{1}{2}\begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix}\), the eigenstates and eigenvalues are:\(\frac{1}{2}|1\rangle = |\uparrow\rangle\) with eigenvalue \(\frac{1}{2}\) and eigenstate \(\begin{pmatrix} 1 \ 0 \end{pmatrix}\),\(\frac{1}{2}|-1\rangle = |\downarrow\rangle\) with eigenvalue \(-\frac{1}{2}\) and eigenstate \(\begin{pmatrix} 0 \ 1 \end{pmatrix}\).These satisfy the spin-half particle representation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Spin Operators

In quantum mechanics, spin is a fundamental property allowing particles to have intrinsic angular momentum. Spin operators describe this property and can be mathematically represented using Pauli spin matrices. For a spin-half particle, the spin operators \(\hat{S}_x, \hat{S}_y, \hat{S}_z\) are defined as \(\hat{S}_i=\frac{1}{2} \sigma_i\), where \(\sigma_i\) are the Pauli matrices. These spin operators meet the angular momentum commutation relations similar to classical angular momentum. The Pauli matrices are given by: \[ \hat{S}_x = \frac{1}{2}\begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}, \ \hat{S}_y = \frac{1}{2}\begin{pmatrix} 0 & -i \ i & 0 \end{pmatrix}, \ \hat{S}_z = \frac{1}{2}\begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix} \].
These matrices help describe the behavior of spin in quantum systems, providing a basis for understanding complex particle spins.

Angular Momentum Commutation Relations

In quantum mechanics, operators for angular momentum components must satisfy specific commutation relations. These relations ensure the physical consistency of measurements of different components. For the spin operators \(\hat{S}_x, \hat{S}_y, \hat{S}_z\), the commutation relations are:

• \[\left[ \hat{S}_x, \hat{S}_y \right] = i \hat{S}_z\]
• \[\left[ \hat{S}_y, \hat{S}_z \right] = i \hat{S}_x\]
• \[\left[ \hat{S}_z, \hat{S}_x \right] = i \hat{S}_y\]

These commutation relations are key to showing that the spin operators obey the same mathematics as classical angular momentum operators, leading to the prediction and confirmation of quantum phenomena observed in experiments.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are critical in understanding how quantum systems behave when measured. For spin operators, solving the eigenvalue equation \(\hat{S}_i |\psi\rangle = \lambda |\psi\rangle\) identifies the possible states (eigenvectors) and the measured values (eigenvalues).
For the \(\hat{S}_z\) operator, the eigenvalue equation is: \[\hat{S}_z |\psi\rangle = \frac{1}{2}\begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix} |\psi\rangle\].
Solving it yields two eigenstates: \[|\uparrow\rangle = \begin{pmatrix} 1 \ 0 \end{pmatrix}, \ |\downarrow\rangle = \begin{pmatrix} 0 \ 1 \end{pmatrix}\], with corresponding eigenvalues \(\frac{1}{2}\) and \(-\frac{1}{2}\), respectively. These states represent the spin-up and spin-down orientations of a spin-half particle along the z-axis.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with particles at very small scales, like electrons and photons. It fundamentally differs from classical physics, with principles such as quantization, superposition, and entanglement. In this framework, properties like spin are described using mathematical objects called operators.
The behavior of quantum systems is dictated by the Schrödinger equation: \[i\hbar \frac{\partial}{\partial t} \Psi = \hat{H} \Psi\], where \(\Psi\) is the wave function and \(\hat{H}\) is the Hamiltonian operator. For spin, the Pauli matrices encapsulate the rotational symmetries and intrinsic angular momentum inherent in quantum particles. Understanding these concepts allows us to explore phenomena like atomic structure, quantum computing, and particle interactions at a fundamental level.