by BIENVENIDO M. BUTANAS JR.
Some properties and examples of Hermitian and unitary operators were discussed in previous posts given by Mary Jane Madulara and Bebelyn Rosales. Hermitian operators are such that its matrix elements are equal to the elements of its corresponding adjoint matrix. Also that a matrix is classified as a unitary if its adjoint is equal to its inverse, among other properties.
Given the matrix representation of an operator, the procedure in extracting the eigenvalues and corresponding eigenvectors of this operator was shown.
An example of a unitary transformation which allows a Hermitian matrix to become diagonal by constructing a unitary matrix from its eigenvectors is also shown in the procedure involving the simultaneous diagonalization of two Hermitian matrices.
Here we will show another commonly used mathematical expression in the form of Taylor Series. Taylor series are used to define functions and operators in diverse areas of mathematics. In particular, this is true in areas where the classical definitions of functions break down. Using Taylor series, one may define analytical functions of matrices and operators such as matrix exponential or matrix algorithm (See for example http://en.wikipedia.org/wiki/Taylor_expansion#Taylor_series_as_definitions).
Here we show a few calculations involving the Taylor series expansions of two matrices.
To do these calculations, we use the fact that functions of matrices are defined by their Taylor expansions, i.e.:
a. We can then find the expression exp(M) if given that
by substituting into the Taylor series expansion and doing the necessary matrix multiplications, we get
In the last expression above, it is easily seen that the fourth term yields zero. Therefore, the subsequent terms which are multiples of this are also equal to zero. Hence, we arrive at the result,
For the next example, we take the matrix M to be
Again we apply Taylor series expansion to get
re-grouping these terms we obtain
We can now see that each expression in the braces can be represented in compact form as
Next we show that
We first take the first given matrix Mand note that
we get the determinant also and substituting these calculated values we arrive at the following results
For the next sample matrix, we have
so we have
evaluating the right hand side, this becomes
Functions of Hermitian and Unitary Operators
Lastly, we show that if H is hermitian, show that is unitary.
The proof of the above statement, starts with the definition of hermitian operators, that is;
and that of unitary operators which is: . We let then we can show that .
Simplifying the last equation, we have
but and simplifying the left hand side of the equation, we get
so and . Therefore, is unitary.
ABOUT THE AUTHOR:
BIENVENIDO BUTANAS JR. is a graduate student in physics at the MSU-Iligan Institute of Technology. He is fascinated with quantum mechanics in particular and would like to do finish more advanced studies in the near future.