Question

In class we we briefly discussed how quantum mechanical systems can be represented in matrix (vector) form. Here you will find the eigenvalues and eigenvectors of a system composed of two electrons (both spin 1/2) by constructing the appropriate matrices. You have already worked with the 2x2 Pauli matrices that describe spin 1/2 in the basis α and β. Now it is time to construct the 4x4 matrices which describe the spin of two electrons in the basis formed by αα, βα, αβ , ββ (often called the tensor product basis).

a. Construct a 4x4 matrix describing S² = ˆS · ˆS
b. Find the eigenvalues of S² = ˆS · ˆS and the common eigenvectors of S² = ˆS · ˆS and ˆSz = ˆsz1 + ˆsz2. Which states are a spin-singlets and spin-triplets and why?

Answer 1

To construct the 4x4 matrix describing S² = ˆS · ˆS, take the tensor product of the spin 1/2 operators σ_x, σ_y, and σ_z. The resulting matrix for S² is [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]. The eigenvalues of S² are 1, and the common eigenvectors with ˆSz are αα and ββ (spin-singlets) and βα and αβ (spin-triplets).

Step-by-step explanation:

To construct the 4x4 matrix describing S² = ˆS · ˆS, we need to consider the tensor product basis formed by αα, βα, αβ, and ββ. In this basis, we can represent the spin 1/2 operators as 2x2 matrices:

σ_x = [[0, 1], [1, 0]], σ_y = [[0, -i], [i, 0]], σ_z = [[1, 0], [0, -1]].

To form the 4x4 matrix for S², we take the tensor product of these matrices, resulting in the matrix:

S² = [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]].

The eigenvalues of this matrix are all equal to 1, and the common eigenvectors of S² and ˆSz are the basis vectors αα and ββ. These states are spin-singlets because their total spin is 0, and they are not affected by spin interactions. The spin-triplet states are βα and αβ, which have total spin 1 and are affected by spin interactions.