Finding the Eigen Value of A.σ Vector Product

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The discussion revolves around finding the eigenvalues of the scalar product A.σ, where A is an arbitrary vector and σ represents Pauli matrices. Participants clarify that A.σ results in a 3x3 matrix, which can indeed have eigenvalues. The eigenvalues are identified as ±|A|, corresponding to eigenspinors aligned with the vector A. The conversation highlights the relationship between the direction of the vector and the resulting eigenstates in quantum mechanics. Understanding these eigenvalues is crucial for applications in quantum physics.
rupesh57272
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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 ?
 
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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:​
 

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