Product of Hermitian Operators

We find the operator that is Hermitian conjugate of the product of the operators F and L . So, we consider

Letting,

then

Letting,

then

So,

or

Hence, taking the adjoint of the product of matrix operators corresponds to reverse the order of the matrices, complex conjugate and transpose each matrix, then matrix multiply them. So, the resulting expression will be to complex conjugate each of the two matrices, matrix multiply them, and then transpose:

Summarily, taking the adjoint of the product of operators reverses the order of all the factors in a product and takes the adjoint of each factor independently. The adjoint of a sum is simply the sum of the adjoints.

Continuous Spectrum

For the case of a continuous spectrum then 2.1 can be rewritten:

In this case, F is an integral operator and

the kernel of the operator that should be understood as the generalized matrix operator for the case of a continuous spectrum. Letting in 2.19,

then

We find the action of the operator F on the amplitude {£ a):

Suppose we swap the integral with the operator F then We can then examine this as an integral with the kernel:

So, further we understand the product of the operators 2.25 in the sense of 2.24.

Schturm-Liouville Problem: Eigenstates and Eigenvalues

Letting F„, be an observable that in basis states |/r„) then we construct an operator F , corresponding to the given observable:

This equation is called the spectral representation of the operator F .

We consider the operator F defined in 2.26 and acting on the basis states|/<„):

From 1.23 of chapter 1, then 2.27 becomes the Schturm-Liouville problem: or

If an operator of a physical quantity acts on an eigenstate of that quantity (i.e., on the eigenstate of that state for which the physical quantity has a defined value say Fm) then the resultant is the product of the eigenvalue and the eigenstate of that quantity.

The Schturm-Liouville problem in equation 2.29 may simply be rewritten:

The parameters /r are called eigenvalues or the proper values of the operator F . They may form a countable set of discrete spectra of eigenvalues. We may also have a continuous spectrum of eigenvalues. Considering 2.29 or 2.30, the eigenvector direction in the inner product space is left invariant by the action of the operator F . It is instructive to note that an operator can have multiple eigenvectors, and the eigenvectors need not all be different.

CONCLUSION: The observable value of a physical quantity is the eigenvalue of the operator of that quantity. The corresponding eigenfunction of the operator of the physical quantity is the wave function of that state for which that quantity has a defined value.

We write relation 2.30 in different representations:

1. Choosing the representation, say {|2„)} and then project the vector equation 2.30: onto the various orthonormal basis vectors |/r) for a discrete spectrum:

We insert the closure relation between F and |2„): or

We observe from here that {|^>} is the set of eigenstates of the Hermitian operator F with eigenvalues /r. When the particle is in the arbitrary state |/„) then measurement of the variable corresponding to the operator F will yield only the eigenvalues {//} of F . This measurement will yield the particular value // for that variable with relative probability |(/„|/()|2- The system will then change from state |/„) to state |//) as a result of the measurement taken. 'I hat is to say, the eigenvalues of F are the only measurable quantities with the measurement outcome being fundamentally probabilistic. The relative probability of the particular allowed outcome /r is obtained by finding the projection of |A„) onto the corresponding eigenstate |^>. This implies that, if |i„) is an eigenstate of F , the measurement will always yield the corresponding eigenvalue. The measurement process itself changes the state of the particle to the eigenstate |/r) corresponding to the measurement outcome ц.

Suppose

So, 2.104 becomes or

This is a set of linear homogeneous equations (characteristic equation) with the unknown coefficients Ck being the components of the eigenvector in the chosen representation. The quantity F„* is the matrix element of the matrix F . Relation 2.36 is a system of N equations,

with к unknown coefficients Ck,

As 2.36 is a linear and homogenous equation, then it has non-trivial solutions when the determinant of the coefficient matrix vanishes yielding the following characteristic or secular equation:

Here, F is the N x N matrix constituting the matrix elements F„A. and I, the unit matrix. As we are working in the /с-dimensional space, then 2.39 is an k-th order polynomial equation in // and so would possess к solutions for corresponding to all the eigenvalues of the operator F . Tire equation 2.39 has к roots that may be real or imaginary, distinct or identical. Since the characteristic equation is independent of the given representation then the eigenvalues of the operator are obviously roots of its characteristic equation. From the eigenvalues it is then easy to find the eigenstates from equation 2.36 that for Hermitian operators F will be a linear combination of к independent eigenstates.

2. The representation that relates a continuous spectrum:

As the kernel ^|F|4’^ has the form in 2.25 then

We consider 2.33 for the case of a discrete spectrum where the matrix element is diagonalized:

Similarly, for the case of a continuous spectrum then

Suppose in 2.97 and 2.29 we consider the eigenvalue F„ to have the value 2„ then we have the following operator for a discrete spectrum:

and for the continuous spectrum and for a mixed spectrum

Example:

We consider an example of an eigenvalue problem where the operator:

We solve the Schturm-Liouvilie equation: where

A

We represent the operator in the form of a series:

If we consider this and the Schturm-Liouvilie equation, then we may find the solution in the form

From here considering that the wave function should be univalent then we let

Thus,

and

 
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