Can the eigenvalue of a state vector in quantum mechanics be thought of as a a phase if the state vector is a wave?
Eigenvalues of state vectors represent values of physical quantities, position, linear momentum, energy, spin projection, etc. Now, think for yourself, the value of the linear momentum of a particle can be regarded as a phase? By the way, phase is a number without dimensions, while a physical quantity has dimensions, cm, g, sec, etc.
@sofia thank you very much.How about phase difference as an analogy for eigenvalues?
The Eigenvalue of an operatore can be complex; thus it has a magnitute and a phase. An example is the Eigevalue of a mass matrix : the real part gives the mass or energy and the imaginary part as the decay width. The Eigenvalue can not always be a simple phase because then the magnitute will be one.
The relative phases among wavefunctions are important and have observable effects in interference experiments. There are many examples of this phenomenon. For example, the interference of two coherent spin states will have an interference term which depends on the difference of the phases.
The eigenvalues of unitary operators are phases-the eigenvalues of hermitian operators are real numbers. These are standard results of linear algebra. Momentum of a particle in quantum mechanics is represented by a hermitian operator, therefore its eigenvalues are real numbers-assuming appropriate boundary conditions, however. So the answer to the question is found, as usual, by specifying the boundary conditions and checking the properties of the operator. Eigenvalues can't be ``thought of'' as anything. Nor does the term ``eigenvalue of a state vector'' mean anything: eigenvalues and eigenvectors are properties of operators.
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