- #1
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 ?
A.σ where A is an arbitrary vector ?
Finding the Eigen Value of A.σ Vector Product
- Thread starter rupesh57272
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In summary, the conversation discussed the eigenvalues of the product of a vector with Pauli matrices. The product results in a 3x3 matrix that can have eigenspinors in the direction of ±A with eigenvalue |A|. The conversation also clarified that the eigenspinors are the same for Sz and S-z, with positive eigenvectors for spin-up and negative eigenvectors for spin-down.
- #2
tiny-tim
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welcome to pf!
hi rupesh57272! welcome to pf!
i don't follow you
…
hi rupesh57272! welcome to pf!
i don't follow you
…A.σ is a vector, so how does it have eigenvalues?
- #3
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He means a sort of 'scalar' product, which would be (after performing the sum) a 3x3 matrix which can have eigenvalues.
[tex] \vec{A}\cdot\vec{\sigma} = A_{x}\sigma_x + A_{y}\sigma_y + A_{z}\sigma_z [/tex].
[tex] \vec{A}\cdot\vec{\sigma} = A_{x}\sigma_x + A_{y}\sigma_y + A_{z}\sigma_z [/tex].
- #4
tiny-tim
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ohh!
then won't they be eigenspinors rather than eigenvectors, in the directions of ±A, and with eigenvalue |A| ?
then won't they be eigenspinors rather than eigenvectors, in the directions of ±A, and with eigenvalue |A| ?
- #5
rupesh57272
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Sorry I forgot to mention that it is scalar product of a Vector and Pauli Spin matrices. What is the Eigen Value of it ?
- #6
rupesh57272
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I think it should be ±|A|
- #7
tiny-tim
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sorry, yes, ±|A|
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)
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)
1. What is an Eigen Value?
An Eigen Value is a special number that represents the scale factor of a vector when it is transformed by a matrix. It is often denoted by the Greek letter lambda (λ).
2. What is an A.σ Vector Product?
The A.σ Vector Product is the product of a matrix A and a vector σ, where the vector is multiplied on the right side of the matrix. This results in a new vector that is a linear combination of the columns of the matrix, with the coefficients being the elements of the vector.
3. Why is it important to find the Eigen Value of A.σ Vector Product?
Finding the Eigen Value of A.σ Vector Product allows us to understand the behavior of the matrix and vector system. It helps us identify important properties such as the direction and magnitude of the vector transformation, and can be used in various applications such as image processing and data compression.
4. How do you find the Eigen Value of A.σ Vector Product?
The Eigen Value of A.σ Vector Product can be found by solving the characteristic equation det(A-λI) = 0, where A is the matrix and λ is the Eigen Value. This equation will result in one or more values for λ, which are the Eigen Values of the matrix A.
5. Can the Eigen Value of A.σ Vector Product be complex?
Yes, the Eigen Value of A.σ Vector Product can be complex. This occurs when the matrix has complex eigenvalues, which can happen when the matrix is not symmetric or when it has repeated eigenvalues. In these cases, the Eigen Values will have a real and imaginary component.
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