By Sol Wieder (Auth.)
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It must be stressed that the eigenvalues and the eigen vectors are characteristic to the operator and are independent of the basis used to calculate them. In certain cases where a matrix display is not feasible, it is still possible to deduce the eigenvalues and eigenvectors without using a specific basis. This is especially important when the dimensionality of the space Ν is large. IV Eigenvalue-Eigenvector Algebra for H e r m i t i a n O p e r a t o r s The theorems to be discussed below are quite independent of any basis that might be used in a representation.
As we shall see in Chapter 3, quantum mechanics requires that the generalized coordinates play an equal role with their corresponding canonically conjugate momenta. The required classical formalism must formally involve equations of motion which are symmetric in qj and Pj. Lagrange's equations (and Newton's law) treat the coordinates differently from their momenta in that only time derivatives of momenta appear in (2-32). Hamilton's formalism, to be dis cussed below, provides the symmetry in pj and qj that we need.
The Hermitian nature of operator observables assures the reality of these eigenvalues (Theorem I). Postulate 4 The state of any physical system is characterized by a state vector of unit length in a complex space. > = l). (3-50) Postulate 4 is quite remarkable in that it restricts the precision with which we can describe the state of a mechanical system. Equation (3-50) implies that the probabilistic picture of the system is determined by the relative orientation of the state vector with respect to the various eigenvectors of operator observables.