Homework Help is Here – Start Your Trial Now!
Science
Advanced Physics
Consider the following operators on a Hilbert space V³(C) : [ 0-i 0 -i 1. i LT √2 010 101 010 i 0 What are the corresponding eigenstates of L₂? , Liz 8] 100 000 00-1

Consider the following operators on a Hilbert space V³(C) : [ 0-i 0 -i 1. i LT √2 010 101 010 i 0 What are the corresponding eigenstates of L₂? , Liz 8] 100 000 00-1

icon
Related questions
Question
Consider the following operators on a Hilbert space \( \mathbb{V}^3(\mathbb{C}) \): \[ L_x = \frac{1}{\sqrt{2}} \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}, \quad L_y = \frac{1}{\sqrt{2}} \begin{bmatrix} 0 & -i & 0 \\ i & 0 & -i \\ 0 & i & 0 \end{bmatrix}, \quad L_z = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{bmatrix}. \] **Questions:** 1. What are the corresponding eigenstates of \( L_z \)? 2. What are the normalized eigenstates and eigenvalues of \( L_x \) in the \( L_z \) basis? **Explanation of Diagrams:** There are no diagrams or graphs present in the image. The image contains mathematical matrices which define operators \( L_x \), \( L_y \), and \( L_z \) used in quantum mechanics to represent observable quantities in a Hilbert space.
Transcribed Image Text:Consider the following operators on a Hilbert space \( \mathbb{V}^3(\mathbb{C}) \): \[ L_x = \frac{1}{\sqrt{2}} \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}, \quad L_y = \frac{1}{\sqrt{2}} \begin{bmatrix} 0 & -i & 0 \\ i & 0 & -i \\ 0 & i & 0 \end{bmatrix}, \quad L_z = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{bmatrix}. \] **Questions:** 1. What are the corresponding eigenstates of \( L_z \)? 2. What are the normalized eigenstates and eigenvalues of \( L_x \) in the \( L_z \) basis? **Explanation of Diagrams:** There are no diagrams or graphs present in the image. The image contains mathematical matrices which define operators \( L_x \), \( L_y \), and \( L_z \) used in quantum mechanics to represent observable quantities in a Hilbert space.
Expert Solution
Step 1: Required to find the eigenstates of L_z

Required to find the eigenstates of L_z

steps

Step by step

Solved in 4 steps with 4 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer