Question

Problem 4.32 (a) Find the eigenvalues and eigenspinors of Sy. (b) If you measured Sy on a particle in the general state x (Equation 4.139), what values might you get, and what is the probability of each? Check that the probabilities add up to 1. Note: a and b need not be real! (C) If you measured S, what values might you get, and with what probabilities? Х y?

Answer 1

The student's question concerns the quantum state, expectation values, and properties of spin in quantum mechanics, with a particular focus on the eigenvalues and eigenspinors of the y-component of spin angular momentum (Sy) and other related measurements in a quantum system.

Step-by-step explanation:

The quantum state and the related concepts of eigenvalues and eigenvectors pertain to the field of quantum mechanics, which is a subdiscipline of physics. If we specifically address the student's question about the Sy operator, which refers to the y-component of spin angular momentum, the answer deals with quantum mechanics and the properties of spin in particle physics. For a spin-1/2 particle like an electron, the Sy operator will have two eigenvalues corresponding to the spin 'up' and 'down' states along the y-axis, typically given by ℏ/2 and -ℏ/2. The associated eigenvectors or eigenspinors can be found using the representation of the Sy operator in terms of Pauli matrices.

Regarding the expectation value of measurements in quantum mechanics, if a measurement of Sy is conducted on a general quantum state described by some combination of eigenspinors, the possible outcomes are the eigenvalues of Sy, each with a probability determined by the square of the coefficients (a and b in the general state) in the state vector's expansion. Similarly, if Sx is measured, the observed values and their probabilities will also be based on the projection of the general state onto the eigenvectors of Sx.

Lastly, for questions involving a free particle described by a wave function involving trigonometric functions, the energy of the particle can be determined by applying the time-independent Schrödinger equation. To find the expectation values of different observables, such as kinetic energy or position squared, the respective operators are applied to the state, and the results integrate across all space.

Answer 2

The eigenvalues and eigenspinors of
`\(S_y\)` represent the potential values and corresponding states when measuring the spin in the y-direction, while measuring
`\(S_y\)` on the state
`\(|x\rangle\)` provides probabilities for obtaining specific spin values, and measuring
`\(S_y^2\)` yields probabilities for the square of the spin in the y-direction.

The operator
`\(S_y\)` represents the spin operator in the y-direction for a quantum system. The eigenvalues and eigenspinors of
`\(S_y\)` can be found by solving the eigenvalue equation:

`\[ S_y | \psi \rangle = \lambda | \psi \rangle \]`

(a) Eigenvalues and Eigenspinors of
`\(S_y\)`:

The eigenvalues of
`\(S_y\)` are the possible values you could measure when measuring the spin component in the y-direction. The eigenspinors are the corresponding eigenstates associated with these eigenvalues.

(b) Measurement of
`\(S_y\)` on a Particle in the State
`\(| x \rangle\):`

Given a general state
`\(| x \rangle\)` (Equation 4.139), measuring
`\(S_y\)` on this state might yield specific values corresponding to the eigenvalues of
`\(S_y\)`. The probabilities of obtaining each eigenvalue are determined by the coefficients of
`\(| x \rangle\)` in the basis of eigenvectors of
`\(S_y\)`. To find these probabilities, you'd need to decompose
`\(| x \rangle\)` into the basis of eigenvectors of
`\(S_y\)`.

(c) Measurement of
`\(S_y^2\)`:

The measurement of
`\(S_y^2\)` is the square of the
`\(S_y\)` operator, indicating the possible values you might obtain when measuring the square of the spin in the y-direction. Similar to the previous case, the probabilities of obtaining each possible value of
`\(S_y^2\)` are determined by the coefficients of
`\(| x \rangle\)` in the basis of eigenvectors of
`\(S_y^2\)`.

Solving for the eigenvalues and eigenspinors of
`\(S_y\)` involves the application of the specific spin operators used in quantum mechanics. If you have a specific quantum system or spin state, I could provide a numerical solution or steps for a particular scenario to find the eigenvalues, eigenspinors, and associated probabilities.

Question: