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Chapter 10: Problem 16

Calculate the polarization vector explicitly for an arbitrary pure state of a spin 1 particle. Show that the length of this vector is less than or equal to unity. Find the condition on the state such that the length is equal to unity.

Short Answer

Expert verified
Pure state: |ψ⟩ = a|1⟩ + b|0⟩ + c|-1⟩. The polarization vector length is; \(≤1\). Condition for unity length: State is maximally polarized (b real or imaginary).

Step by step solution

01

- Write the Pure State of a Spin 1 Particle

Represent the pure state of a spin 1 particle using the state vector: \[ | \text{ψ} \rangle = a|1\rangle + b|0\rangle + c|-1\rangle \]. Here, \(a, b,\) and \(c\) are complex numbers satisfying the normalization condition: \[ |a|^2 + |b|^2 + |c|^2 = 1 \].
02

- Define the Polarization Vector

The polarization vector \(\mathbf{P}\) for a spin 1 particle is: \[ \mathbf{P} = \begin{pmatrix} \text{Re}(a^*b) + \text{Im}(a^*b) \hat{i} \ \text{Re}(a^*c) + \text{Im}(a^*c) \hat{j} \ \text{Re}(b^*c) + \text{Im}(b^*c) \hat{k} \end{pmatrix}. \]
03

- Calculate the Length of the Polarization Vector

To find the length of the polarization vector, compute: \[ ||\mathbf{P}|| = \frac{1}{2} \sqrt{3(\mathbf{S}^2 + \mathbf{S_3}^2 + \mathbf{S_2}^2)} \], where \(\mathbf{S}\) are the spin matrices.
04

- Show the Length is Less than or Equal to Unity

Calculate the upper bound: \[ ||\mathbf{P}||^2 = \text{Tr}(\rho Sn) \leq 1/2. \] Prove the length of the polarization vector \( \mathbf{P} \) is always less than or equal to 1.
05

- Condition for Unity Length

For the length to be equal to unity: The state must be a maximally polarized state which means \(a\) or \(c\) must be zero, and \(b\) must be real or imaginary.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pure State
In quantum mechanics, a pure state represents the most complete knowledge we have about a particle. For a spin 1 particle, the state is given by the vector sum: |ψ⟩=a|1⟩+b|0⟩+c|−1⟩. Here, the coefficients a, b, and c are complex numbers indicating the probability amplitudes of the particle being in the states |1⟩, |0⟩, and |−1⟩ respectively. A pure state means there is no mixture of other states.
Polarization Vector
The polarization vector ( is an important concept when studying the spin properties of particles. For a spin 1 particle, the polarization vector represents the expectation values of the spin components along x, y, and z directions. The formula for the polarization vector P is given by:
Normalization Condition
The normalization condition ensures that the total probability of finding the particle in any of its possible states is 1. For the pure state of a spin 1 particle, this condition is mathematically represented as |a|^2 + |b|^2 + |c|^2 = 1. This equation guarantees that the sum of the probabilities for all possible states (1, 0, and -1) equals one, maintaining the principle of conservation of probability in quantum mechanics.
Spin Matrices
Spin matrices are mathematical operators used to describe the spin components of a quantum particle. For a spin 1 particle, these matrices are larger in dimension compared to spin 1/2 particles. They are typically represented by 3x3 matrices. Commonly denoted as ⃗ S=(S_x,S_y,S_z), these matrices help determine the polarization vector and spin measurements.
Maximally Polarized State
A maximally polarized state for a spin 1 particle occurs when the polarization vector reaches its maximum possible length of unity. This happens when either a or c is zero and b is a real or imaginary number. In this state, the particle's spin is fully aligned along a certain direction, providing the maximum amount of information about the particle's spin.

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