17. Formalism of Quantum Mechanics: Eigenvectors and Eigenvalues

Consider a complex vector space; every linear transformation $\hat{T}$ has vectors |α>, which are transformed into simple multiples of themselves:

$\hat{T}|\alpha >=\lambda |\alpha >$                                     (17.1)

The vectors |α> are the eigenvectors of the transformation and the complex number λ is the eigenvalue. The null vectors, are ‘trivial solutions’ of Equation (17.1) and doesn’t count. Therefore, any nonzero multiple of eigenvector is still an eigenvector with the same eigenvalue.

The eigenvector equation, with respect to a particular basis, in matrix form is given by:

Ta = λa                                                                    (17.2)

Or:

(T – λ1)a = 0                                                               (17.3)

Here, 0 is the zero matrix. By the assumption that a is nonzero, the matrix (T – λ1) must in fact be singular, which means that its determinant is equal to zero:

det (T – λ1) = 0                                                            (17.4)

An algebraic equation for λ can be obtained by expansion of the determinant:

${{C}_{n}}{{\lambda }^{n}}+{{C}_{n-1}}{{\lambda }^{n-1}}+$ …$+{{C}_{1}}\lambda +{{C}_{0}}=0$                                          (17.5)

This is called the characteristic equation for the matrix, where the coefficients C_i depend on the elements of T and its solutions determine the eigenvalues. Equation (17.5) is an n^th-order equation that has n (complex) roots. The corresponding eigenvectors can be constructed by plugging each λ back into Equation (17.2) and solve for the components of a. We are going to show you how it goes by mean of an example.