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.
Other recent questions and answers regarding EITC/QI/QIF Quantum Information Fundamentals:
- Are amplitudes of quantum states always real numbers?
- How the quantum negation gate (quantum NOT or Pauli-X gate) operates?
- Why is the Hadamard gate self-reversible?
- If measure the 1st qubit of the Bell state in a certain basis and then measure the 2nd qubit in a basis rotated by a certain angle theta, the probability that you will obtain projection to the corresponding vector is equal to the square of sine of theta?
- How many bits of classical information would be required to describe the state of an arbitrary qubit superposition?
- How many dimensions has a space of 3 qubits?
- Will the measurement of a qubit destroy its quantum superposition?
- Can quantum gates have more inputs than outputs similarily as classical gates?
- Does the universal family of quantum gates include the CNOT gate and the Hadamard gate?
- What is a double-slit experiment?
View more questions and answers in EITC/QI/QIF Quantum Information Fundamentals