Problem 8 - A Modern Approach to Quantum Mechanics

Question

A spin- \(\frac{1}{2}\) particle, initially in a state with \(S_{n}=\hbar / 2\) with \(\mathbf{n}=\sin \theta \mathbf{i}+\cos \theta \mathbf{k}\), is in a constant magnetic field \(B_{0}\) in the \(z\) direction. Determine the state of the particle at time \(t\) and determine how \(\left\langle S_{x}\right\rangle,\left\langle S_{y}\right\rangle\), and \(\left\langle S_{z}\right\rangle\) vary with time.

Step-by-Step Solution

Answer

The state evolves as \( |\psi(t)\rangle = e^{-i\omega_0 t/2} (\cos(\theta/2)|+z\rangle + \sin(\theta/2)e^{-i\omega_0 t}|-z\rangle) \). Expectation values are \( \langle S_x \rangle = \frac{\hbar}{2}\sin \theta\cos(\omega_0 t), \langle S_y \rangle = \frac{\hbar}{2}\sin \theta\sin(\omega_0 t), \langle S_z \rangle = \frac{\hbar}{2}\cos \theta \).

Step 1: Define Initial State

The initial state of the spin-\( \frac{1}{2} \) particle is given as \( S_n = \frac{\hbar}{2} \). The direction \( \mathbf{n} \) is a unit vector, which can be expressed in terms of angles as \( \mathbf{n}=\sin \theta \mathbf{i}+\cos \theta \mathbf{k} \). The initial state vector is \( | \psi(0) \rangle = \cos \frac{\theta}{2} | +z \rangle + \sin \frac{\theta}{2} | -z \rangle \).

Step 2: Hamiltonian and Time Evolution

The Hamiltonian of the system is \( H = -\gamma \mathbf{B} \cdot \mathbf{S} = -\gamma B_0 S_z \), where \( \gamma \) is the gyromagnetic ratio and \( \mathbf{B} = B_0 \mathbf{k} \). The time evolution of the state is given by \( | \psi(t) \rangle = e^{-iHt/\hbar} | \psi(0) \rangle \).

Step 3: Solve for Time Evolution Operator

The operator \( e^{-iHt/\hbar} \) simplifies to \( e^{i\omega_0 t/2} | +z \rangle \langle +z | + e^{-i\omega_0 t/2} | -z \rangle \langle -z | \), where \( \omega_0 = \gamma B_0 \). Appling this operator to \( | \psi(0) \rangle \), we find \( | \psi(t) \rangle = e^{-i\omega_0 t/2} \left( \cos \frac{\theta}{2} e^{i\omega_0 t/2} | +z \rangle + \sin \frac{\theta}{2} e^{-i\omega_0 t/2} | -z \rangle \right) \).

Step 4: Calculate Expectation Values

The expectation values of the spin components are given by \( \langle S_x(t) \rangle = \hbar \sin \theta \cos(\omega_0 t)/2 \), \( \langle S_y(t) \rangle = \hbar \sin \theta \sin(\omega_0 t)/2 \), and \( \langle S_z(t) \rangle = \hbar \cos \theta/2 \). These expressions show oscillations with frequency \( \omega_0 \).

Key Concepts

Spin-1/2 Particles

Spin-1/2 particles are fundamental components of quantum mechanics, often used to represent the simplest quantum systems.
These particles possess a 'spin', a form of intrinsic angular momentum, despite lacking any actual rotation in space.

A spin-1/2 particle exhibits two possible spin states: "spin-up" and "spin-down".

• "Spin-up" is aligned with an external magnetic field.
• "Spin-down" is opposed to an external magnetic field.

In quantum mechanics, these states are represented as basis vectors, |+z⟩ for "spin-up" and |-z⟩ for "spin-down."
Each state corresponds to eigenstates of the spin operator, with eigenvalues +ɧ/2 and -ɧ/2.

In practice, when we know a spin-1/2 particle’s specific orientation, as expressed by a unit vector \( \mathbf{n} = \sin\theta \mathbf{i} + \cos \theta \mathbf{k} \), we determine its initial state as a combination of the basis states,
i.e., \( | \psi(0) \rangle = \cos \frac{\theta}{2} | +z \rangle + \sin \frac{\theta}{2} | -z \rangle \).
This state exists as a quantum superposition, a key quantum concept illustrating how a particle can be in multiple states simultaneously.

Magnetic Field

A magnetic field in quantum mechanics significantly influences the behavior of spin-1/2 particles.
When such a particle interacts with a magnetic field, it experiences a torque, resulting in a precession of its spin.

The precession of spin is captured by the Hamiltonian, a function that describes the total energy of the system.

• For a magnetic field pointing in the z-direction, the Hamiltonian is \( H = -\gamma B_0 S_z \).
• Here, \( \gamma \) represents the gyromagnetic ratio, a fundamental property of the particle that denotes how strongly it interacts with the magnetic field.

In this setup, the interaction of the spin's magnetic moment \( \mathbf{S} \) with the magnetic field \( \mathbf{B} \) causes the particle's quantum state to evolve over time.

This interaction not only affects the state of the particle, but also influences its observable properties, such as its spin projections along different axes.

Time Evolution

Time evolution in quantum mechanics describes how the state of a system changes over time under the influence of its Hamiltonian.
For a spin-1/2 particle in a magnetic field, the time evolution follows a predictable pattern governed by Schrödinger's equation.

• The time evolution of the state is given by the formula \( | \psi(t) \rangle = e^{-iHt/\hbar} | \psi(0) \rangle \).
• The operator \( e^{-iHt/\hbar} \) is the so-called time-evolution operator, which modifies the initial state \( |\psi(0)\rangle \) with time \( t \).

This operator results in oscillations over time, captured by trigonometric components in the final expressions for the spin expectation values:

• \( \langle S_x(t) \rangle = \hbar \sin \theta \cos(\omega_0 t)/2 \)
• \( \langle S_y(t) \rangle = \hbar \sin \theta \sin(\omega_0 t)/2 \)
• \( \langle S_z(t) \rangle = \hbar \cos \theta/2 \)

These results show how the observables associated with the particle's spin are dynamic over time, reflecting the spin's precession around the magnetic field at frequency \( \omega_0 = \gamma B_0 \).