What are the eigenvalues of the Pauli spin matrix Sigma sub Y when measuring spin along the y-axis?
The eigenvalues of the Pauli spin matrix Sigma sub Y, when measuring spin along the y-axis, can be determined by solving the eigenvalue equation associated with this matrix. Before delving into the specifics, let's first establish some foundational knowledge.
In the field of quantum information, spin is a fundamental property of elementary particles. It is a quantum mechanical angular momentum that can be measured along different axes, such as the x, y, or z-axis. The Pauli spin matrices, named after Wolfgang Pauli, are a set of three 2×2 matrices that represent the spin operators for spin-1/2 particles.
The Pauli spin matrix Sigma sub Y, denoted as σ_y, is one of these matrices. It is defined as:
σ_y = [[0, -i],
[i, 0]]
To find the eigenvalues of σ_y, we need to solve the eigenvalue equation:
σ_y |ψ⟩ = λ |ψ⟩
where |ψ⟩ is the eigenvector and λ is the corresponding eigenvalue.
Let's begin by writing out the eigenvalue equation explicitly:
[[0, -i],[i, 0]] |ψ⟩ = λ |ψ⟩
Expanding this equation, we have:
[0*|ψ⟩ – i*|ψ⟩] = λ*|ψ⟩[i*|ψ⟩ + 0*|ψ⟩] = λ*|ψ⟩
Simplifying further, we obtain two equations:
-i|ψ⟩ = λ|ψ⟩
i|ψ⟩ = λ|ψ⟩
Now, let's solve these equations for the eigenvalues.
For the first equation, we have:
-i|ψ⟩ = λ|ψ⟩
Rearranging, we get:
(λ + i)|ψ⟩ = 0
Since |ψ⟩ cannot be the zero vector (as it represents a physical state), we must have (λ + i) = 0. Solving for λ, we find:
λ = -i
For the second equation, we have:
i|ψ⟩ = λ|ψ⟩
Rearranging, we get:
(λ – i)|ψ⟩ = 0
Again, |ψ⟩ cannot be the zero vector, so we have (λ – i) = 0. Solving for λ, we find:
λ = i
Therefore, the eigenvalues of the Pauli spin matrix σ_y, when measuring spin along the y-axis, are λ = -i and λ = i.
To illustrate this concept, consider the following example: Suppose we have a spin-1/2 particle in the state |ψ⟩ = (1/sqrt(2))(|+⟩ + |-⟩), where |+⟩ and |-⟩ represent the eigenstates of σ_z. If we measure the spin along the y-axis, we will obtain either the eigenvalue -i or i with corresponding probabilities determined by the squared magnitudes of the inner products between |ψ⟩ and the eigenstates of σ_y.
The eigenvalues of the Pauli spin matrix σ_y, when measuring spin along the y-axis, are λ = -i and λ = i. These eigenvalues play a important role in quantum information and provide insights into the behavior of spin-1/2 particles.
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