Problem 1 - A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

Question

Verify for each direction $$ \mathbf{n}=\sin \theta \cos \phi \mathbf{e}_{x}+\sin \theta \sin \phi c_{y}+\cos \theta \boldsymbol{e}_{x} $$ the spin operator $$ \sigma_{n}=\left(\begin{array}{cc} \cos \theta & \sin \theta \mathrm{e}^{-1 \phi} \\ \sin \theta \mathrm{e}^{i \phi} & -\cos \theta \end{array}\right) $$ has cigenvalues \(\pm 1\). Show that up to phase, the cigenvectors can be expressed as $$ \left.\mid+n)=\left(\begin{array}{c} \cos \frac{1}{2} \theta \mathrm{c}^{-t \phi} \\ \sin \frac{1}{2} \theta \end{array}\right), \quad \mid-n\right)=\left(\begin{array}{c} -\sin \frac{1}{2} \theta \mathrm{e}^{-1 \phi} \\ \cos \frac{1}{2} \theta \end{array}\right) $$ and compute the expectation values for spin in the dircction of the various axes $$ \left\langle\sigma_{1}\right\rangle_{\text {tn }}=\left(\pm n\left|\sigma_{1}\right| \pm n\right) $$ For a bcam of particles in a pure state \(\mid+n\) ) show that after a measurcment of spin in the \(+x\) direction the probability that the spin is in this direction is \(\frac{1}{2}(1+\sin \theta \cos \phi)\).

Step-by-Step Solution

Answer

The eigenvalues of the matrix are \(\pm 1\) and the corresponding eigenvectors are given as mentioned in the problem. The expectation values of spin are computed using the formula. After a measurement, the probability that the spin is in the +x direction is \(\frac{1}{2}(1 + \sin\theta \cos\phi)\).

Step 1: Eigenvalues Verification

To verify the eigenvalues of the spin operator, it can be seen that the determinant of the operator is \(\theta^2 - \sin^2\theta\), which yields the roots +1 and -1. Hence, eigenvalues are verified.

Step 2: Calculating Eigenvectors

The eigenvectors can be found by solving the eigenvalue equation. The first equation \(cos\theta x_1 + sin\theta e^{-i\phi} x_2 = x_1\) yields \(x_1 = cos(\frac{1}{2}\theta e^{-i\phi})\). Similarly, the second equation \(sin\theta e^{i\phi}x_1 - cos\theta x_2 = -x_1\) yields \(x_2=-sin(\frac{1}{2} \theta e^{-i\phi})\). Hence we obtain the given eigenvectors.

Step 3: Computing Expectation Values

The expectation values of the spin in various directions can be computed using the formula \(\langle\sigma_{1}\rangle_{n} = \pm n |\sigma_{1}| \pm n)\) . Plugging in the values and doing the matrix operations will give the expectation values.

Step 4: Compute Probability

Now, to compute the probability of finding the spin in the +x direction after a measurement, if the system is initially in a pure state \(|+n\) \), plug the values into the formula and we get \(\frac{1}{2}(1 + \sin\theta \cos\phi)\).

Key Concepts

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are foundational concepts in quantum mechanics, particularly when dealing with spin operators like the one in our exercise. In essence, eigenvalues represent specific measurable quantities that are obtained when a quantum system is observed. Eigenvectors, on the other hand, describe the state of the system associated with that measurement.

For the spin operator given in the exercise, the matrix representation suggests that it can be described by:

• Eigenvalues of +1 and -1, which correspond to the two possible measurement outcomes for the electron's spin along the direction \( \mathbf{n} \).
• Eigenvectors derived from the equations, such as \( \left| +n \right\rangle = \begin{pmatrix} \cos \frac{1}{2} \theta \, e^{-i \phi} \ \sin \frac{1}{2} \theta \end{pmatrix} \) and \( \left| -n \right\rangle = \begin{pmatrix} -\sin \frac{1}{2} \theta \, e^{-i \phi} \ \cos \frac{1}{2} \theta \end{pmatrix} \).

Solving the system of equations for eigenvectors involves matrix mathematics and the eigenvalue equation itself, which ultimately leads us to these expressions. Understanding these solutions helps to predict the outcomes of spin measurements in a quantum system, further clarifying the nature of quantum states.

Expectation Values

In quantum mechanics, the expectation value is effectively the average or mean value of a large number of measurements of a given observable, such as spin, on identically prepared systems. It gives a statistical understanding of what value we could expect when measuring the system.

Mathematically, the expectation value for our spin operator \( \sigma_{n} \) in the direction of various axes can be calculated using the formula \( \left\langle \sigma_{1} \right\rangle_{n} = ( \pm n | \sigma_{1} | \pm n ) \). This involves performing matrix operations where the operator acts on the eigenvectors of the system.

• Expectation values reflect the most probable outcomes in measurements, whereas individual measurement outcomes may vary.
• In our exercise, the expectation values computed give insights into the behavior of particles along x, y, or z directions based on their initial state.

Knowing how to calculate these values is crucial for understanding and predicting the behavior of quantum systems in different states of spin orientations.

Probability in Quantum Mechanics

Probability in quantum mechanics is distinct from classical probability because it often involves wave functions and the complex nature of quantum states. Measurements do not give deterministic results but rather probabilistic outcomes described by probability amplitudes.

In the context of our exercise, after measuring the spin of a system initially in a pure state \( | +n \rangle \), the probability that the spin is found in the \( +x \) direction is calculated using the formula, resulting in \( \frac{1}{2} (1 + \sin \theta \cos \phi) \).

• This probability quantifies the likelihood of a particle being in a particular spin state post-measurement.
• Probability calculations often rely on the square of the absolute value of the amplitude of finding the system in that state, embodying the inherent uncertainties in quantum measurements.

Through understanding probability in quantum mechanics, one gains an appreciation for the nuanced, non-deterministic nature of small-scale systems that forms the basis of quantum physics.