Hamiltonian Matrix Representation With Magnetic Field

In this problem, we're dealing with a situation where the magnetic field is at some angle relative to the z-axis, but we've written the Hamiltonian in the spin up spin down basis. Part A: Obtain the matrix representation of $H$ in the spin up spin down basis. Part B: Find the eigenvalues and normalized eigenvectors of $H$. Part C: Verify that the normalized eigenvectors of $H$, denoted as $E_1$ and $E_2$, satisfy the Dirac completeness relation.

This problem explores the fascinating relationship between magnetic fields and quantum mechanical operators, specifically within the context of spin systems. When dealing with the Hamiltonian of a system, understanding how to represent it in a matrix form using the spin up and spin down basis is crucial. This basis, which refers to the quantum states associated with a particle's spin orientation, provides a framework for analyzing how quantum states evolve in a magnetic field. The components of the matrix representation will reflect how the magnetic field influences the energy levels based on its orientation to the z-axis. A deep understanding of how to switch between different bases and interpret the resulting matrices is fundamental in quantum mechanics. Finding the eigenvalues and eigenvectors is a core part of solving quantum problems as they represent measurable quantities, such as energy levels, and the states of the system corresponding to these measurements.

Solving for eigenvalues provides critical insights into the quantized nature of a quantum system. In solving this problem, students will practice calculating these using linear algebra principles, reinforcing important problem-solving skills like matrix diagonalization and understanding the physical implications of these mathematical results. Furthermore, verifying that the eigenvectors satisfy the Dirac completeness relation not only confirms the mathematical coherence of the solution but also illustrates how quantum states form a complete basis set that span the space in which quantum mechanics operates. This allows any state of the system to be expressed as a combination of its eigenstates, a fundamental concept for quantum state analysis and quantum information theory.