Calculating Eigenkets from Matrix w/ Orthonormal Basis

[mkbh_10]

How am i supposed to write eigenkets of an operator whose matrix is given to me given that the two ket vectors form an orthonormal basis .

 

[CompuChip]

Eigenkets is just the "quantum mechanical" name for eigenvectors. So you find the eigenvectors, and then make them orthonormal (e.g. apply the Gram-Schmidt orthonormalisation process)

 

[Entropia]

To calculate the eigenkets of an operator given its matrix, you can use the following steps:

1. Determine the eigenvalues of the matrix by finding the roots of the characteristic equation det(A-λI)=0, where A is the matrix and λ is the eigenvalue.

2. For each eigenvalue, solve the equation (A-λI)|ψ>=0 to find the corresponding eigenvector |ψ>.

3. Normalize the eigenvectors by dividing each component by the square root of the sum of their squares, to ensure that they form an orthonormal basis.

4. The resulting normalized eigenvectors will be the eigenkets of the operator.

In summary, you can use the given orthonormal basis to find the eigenkets by first finding the eigenvalues and then solving for the corresponding eigenvectors. The normalization step ensures that the eigenvectors form an orthonormal basis, which is essential for performing further calculations and analyses.