Finding the Eigen Value of A.σ Vector Product

Click For Summary
SUMMARY

The eigenvalue of the scalar product of an arbitrary vector A with the Pauli matrices, represented as A.σ, is definitively ±|A|. This scalar product results in a 3x3 matrix formed by the sum of the components of A multiplied by the corresponding Pauli matrices. The eigenvalues correspond to eigenspinors aligned with the direction of vector A, confirming that the eigenvalues are indeed ±|A|.

PREREQUISITES
  • Understanding of Pauli matrices and their properties
  • Knowledge of vector algebra in quantum mechanics
  • Familiarity with eigenvalues and eigenvectors
  • Basic concepts of spinors in quantum physics
NEXT STEPS
  • Study the properties of Pauli matrices in quantum mechanics
  • Learn about the mathematical formulation of eigenvalues and eigenvectors
  • Explore the concept of spinors and their applications in quantum theory
  • Investigate the implications of scalar products in quantum mechanics
USEFUL FOR

Quantum physicists, students of quantum mechanics, and anyone interested in the mathematical foundations of spin and eigenvalues in quantum systems.

rupesh57272
Messages
6
Reaction score
0
Can anyone tell me what is eigen value of product of a vector with pauli matrices i.e
A.σ where A is an arbitrary vector ?
 
Physics news on Phys.org
welcome to pf!

hi rupesh57272! welcome to pf! :smile:

i don't follow you :redface: …

A.σ is a vector, so how does it have eigenvalues? :confused:​
 
He means a sort of 'scalar' product, which would be (after performing the sum) a 3x3 matrix which can have eigenvalues.

\vec{A}\cdot\vec{\sigma} = A_{x}\sigma_x + A_{y}\sigma_y + A_{z}\sigma_z.
 
ohh!

then won't they be eigenspinors rather than eigenvectors, in the directions of ±A, and with eigenvalue |A| ?
 


Sorry I forgot to mention that it is scalar product of a Vector and Pauli Spin matrices. What is the Eigen Value of it ?
 
I think it should be ±|A|
 
sorry, yes, ±|A| :smile:

eg for Sz, or for S-z, the two eigenspinors are the same …

spinor in the z direction (which we call spin-up, with positive eigenvector, for Sz and spin-down, with negative eigenvector, for S-z)

spinor in the minus-z direction (which we call spin-down, with negative eigenvector, for Sz and spin-up, with positive eigenvector, for S-z) :wink:​
 

Similar threads

Undergrad Are Dot Products and Dirac Brackets Interchangeable in Vector Mathematics?
  • · Replies 10 ·
Replies
10
Views
1K
Undergrad Understanding tensor product and direct sum
  • · Replies 11 ·
Replies
11
Views
3K
Undergrad Are density matrices part of a real vector space?
  • · Replies 9 ·
Replies
9
Views
3K
Undergrad Inner and Outer Product of the Wavefunctions
  • · Replies 8 ·
Replies
8
Views
4K
Undergrad Dot product of two vector operators in unusual coordinates
  • · Replies 14 ·
Replies
14
Views
3K
Undergrad Notation for vectors in tensor product space
  • · Replies 17 ·
Replies
17
Views
3K
Undergrad Degenerated perturbation theory
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Density Operators of Pure States
  • · Replies 21 ·
Replies
21
Views
3K
Undergrad Cyclic rotation of the cross product involving derivation
  • · Replies 12 ·
Replies
12
Views
2K
Graduate Vector and Axial vector currents in QFT
  • · Replies 1 ·
Replies
1
Views
2K