Eigen Value of an Operator

The eigenvalues of an operator refer to the values that satisfy the equation:

A|ψ⟩ = λ|ψ⟩

where A is the operator, |ψ⟩ is the eigenvector, and λ is the eigenvalue. 

The eigenvalue determines how a particular eigenvector is transformed by the operator.

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Eigen value or the constant value?

If the value of physical property obtained when Wave function is operated by corresponding operator is a constant, we call the value as eigen value. 

And the wave function corresponding to that eigen value can is canned eigen function.

When Wave function is multiplied by the operator, a value is obtained which can be constant or variable. If the value is a constant, it is eigen value. If the value is variable, we have got to take expectation value of the operator. Expectation value is covered here.

Finding eigen values of the Operator:

To find the eigenvalues of an operator, one can solve the eigenvalue equation by determining the values of λ that satisfy the equation. This can be done by rearranging the equation:

(A - λI)|ψ⟩ = 0

where I is the identity matrix. By finding the values of λ that make the determinant of (A - λI) equal to zero, you can determine the eigenvalues of the operator.