What are the eigenvalues of the Pauli spin matrix Sigma sub Z when measuring spin along the z-axis?
The eigenvalues of the Pauli spin matrix Sigma sub Z, when measuring spin along the z-axis, can be determined by solving the eigenvalue equation for this matrix. The Pauli spin matrices are a set of three 2×2 matrices commonly used in quantum mechanics to describe the spin of particles. The Sigma sub Z matrix represents the spin operator along the z-axis.
To find the eigenvalues of Sigma sub Z, we start by writing down the eigenvalue equation:
Sigma sub Z |v⟩ = λ |v⟩
where |v⟩ is the eigenvector and λ is the corresponding eigenvalue. We can express the Sigma sub Z matrix explicitly as:
Sigma sub Z = |1 0|
|0 -1|
where the entries represent the matrix elements. Substituting this expression into the eigenvalue equation, we get:
|1 0| |v1⟩ = λ |v1⟩
|0 -1| |v2⟩ |v2⟩
This leads to the following system of equations:
1*v1 = λ*v1
0*v1 – 1*v2 = λ*v2
Simplifying these equations, we obtain:
v1 = λ*v1
-v2 = λ*v2
From the first equation, we see that v1 must be nonzero for a nontrivial solution. Therefore, we can set v1 = 1 without loss of generality. Substituting this into the second equation, we have:
-v2 = λ*v2
Rearranging the equation, we find:
(λ+1)*v2 = 0
For this equation to hold, either λ+1 = 0 or v2 = 0. If v2 = 0, then the eigenvector |v⟩ is proportional to (1, 0). If λ+1 = 0, then λ = -1, and the eigenvector |v⟩ is proportional to (0, 1).
Therefore, the eigenvalues of the Pauli spin matrix Sigma sub Z, when measuring spin along the z-axis, are λ = 1 and λ = -1. These eigenvalues correspond to the two possible outcomes of a spin measurement along the z-axis, which are spin up and spin down, respectively.
It is worth noting that the eigenvalues of the Pauli spin matrix Sigma sub Z are always ±1, regardless of the direction of the spin measurement axis. This property is a consequence of the fact that the Pauli spin matrices are Hermitian and have real eigenvalues.
The eigenvalues of the Pauli spin matrix Sigma sub Z, when measuring spin along the z-axis, are λ = 1 (spin up) and λ = -1 (spin down). These eigenvalues represent the possible outcomes of a spin measurement along the z-axis.
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