Properties of Hermitian Operators | Quantum Science Philippines
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Properties of Hermitian Operators

by BEBELYN A. ROSALES


Linear operators in quantum mechanics may be represented by matrices. A type of linear operator of importance is the so called Hermitian operator.  An operator is Hermitian if each element is equal to its adjoint. Most quantum operators, for example the Hamiltonian of a system, belong to this type.

Now linear operators are represented by its matrix elements. We can therefore easily look at the properties of a Hermitian operator by looking at its matrix representation. A particular Hermitian matrix we are considering is that of below. We can calculate the determinant and trace of this matrix .

The determinant and trace of a Hermitian matrix

A. The determinant and trace of the matrix  are shown below as:

where , so that

and,

B. Next we then calculate the eigenvalue of . Their sum and product of its eigenvalues are shown to be consistent with its determinant and trace.

To get its eigenvalues, we solve the eigenvalue equation:






Hence, we can easily see that


These results are therefore consistent with the answers in part A.

Eigenvalues and eigenvectors of a Hermitian operator

C. Knowing its eigenvalues, we can solve for the eigenvectors of . Within the degenerate sector, we construct two linearly independent eigenvectors. We do this by making the eigenvectors orthogonal to each other. Then we finally normalize all three eigenvectors so that their magnitudes are unity.

Beginning with the

We solve first the eigenvector for =0;



Solving equations (1) and (2) simultaneously leads to

and get

Now, solving equations (2) and (3) yields

and get

Substituting to equation (1),


and we therefore get .

Since is abitrary, we can choose . With this choice we now have



Therefore the eigenvector corresponding to the eigenvalue 0 is

.

Now, solving the eigenvector for , we have



Also since  and are arbitrary,

We can choose
and
and get,

or we can also choose
and ;
and get,

Note that we have two eigenvalues which are equal to 3. To solve the corresponding eigenvector, we need to use the Gram Schmidt procedure which is outlined below.

Let

Normalizing,

The corresponding normalized eigenvectors for , , and are then

The Unitary Transformation

D. We now construct the unitary matrix that diagonalizes the matrix .
We can also show explicitly that the similarity transformation reduces to the appropriate diagonal form where its eigenvalues can be read directly from its diagonal elements.

Given the eigenvectors

we can construct the unitary matrix by having these eigenvectors as elements, thus:

the adjoint of this matrix is then given by

.

We can apply a similarity transformation of the form

Hence the matrix is transformed into its diagonal form:

About the Author:

BEBELYN A. ROSALES is studying for her masters degree in physics at the Mindanao State University-Iligan Institute of Technology (MSU-IIT) in Iligan City, Philippines. She hopes to continue with her doctoral studies in computational and experimental physics in a university abroad.

13 Responses to “Properties of Hermitian Operators”

  1. jude Says:

    Congratulations bebelyn. It was fun reading your article. I just want to make minute comment.
    * I believe the general form of Gram-Schmidt and similarity transformation should be shown beforehand.

  2. Ancelie Says:

    hi bebs,

    you did a very nice article. the solutions to the problems are presented in detailed manner. it makes the article long though but it is good ad comprehensible. keep up the good work.i like it.

  3. Cariel Says:

    Hi Bebelyn. You have done a nice job about the Properties of Hermitian Operators. It is very detailed. I am just confused of the notation you are using on the adjoint of the unitary matrix. Can it be changed?

  4. pearl francisco Says:

    Hi bebelyn, I must say that it is indeed a nice article. The presentation of the properties of hermitian operators are clearly stated.

  5. jessica Says:

    hello bebelyn,

    The article you made is very nice and very comprehensible. Just want to make comment on the alignments of your equations on the latter part for a dandier view.

  6. bien Says:

    There is a missing equation that is very fundamental in your presentation, I guess that was the equation that Simon meant.. Its’very important bebz.

  7. Taylor Series Expansion of Hermitian and Unitary Operators Says:

    […] the matrix representation of an operator, the procedure in extracting the eigenvalues and corresponding eigenvectors of this operator was […]

  8. Gibson T. Maglasang Says:

    Congratulations! For a job well done. I really appreciate it. I fully understand now the concept of hermitian operators and its properties are deeply inculcated in my mind. Two thumbs up to all of you guys.

  9. Sandra L. Manulat Says:

    What are the requirements in using the Gram-Schmidt procedure???Is it necessary for all cases of finding the eigenvector???Thank You.

  10. Majvell Kay Odarve Says:

    Hello Ms. Bebelyn,

    This article is really a great help in my understanding of Hermitian operators. I just have a query on the part where you calculated the eigenvector for the degenerate states. What is the basis for selecting the values of the arbitrary variables x_2 and x_3? Is there any way of directly knowing that the values to be used leads to orthoganal vectors or is it really necessary to perform the Gram Schmidt procedure? Thank you so much and God bless you.

  11. Hananish Joy Says:

    Thank you for the effort of showing the properties of the hermitian operators in a way that we could understand. I just want to clarify, if we have degenerate states, we need to use Gram Schmidt, right?

    Also, the given matrix can not be seen. I think there’s something wrong in the code there.

    thanks for making Hermitian matrices simpler to understand.

  12. catherine therese quiñones Says:

    Hi bebelyn! The way you presented your article is really student friendly. Nice job and keep it up! However, there are things you missed (just minor ones) like putting “det” before the matrix on the first equation of part A and another “det” before the matrix on the second equation of part B. I also have some questions if you don’t mind. I am confused about the degenerate eigenvalues (ie w=3). For w1 you chose x2=0 and x3=1 and for w2 you chose x2=1 and x2=0. What are your basis of choosing the latter values of x2 and x3? Will the result of the Gram-Schmidt be affected if we use other values of x2 and x3? Another thing, In obtaining the trace of the Hermitian matrix, you solved it in two ways right? First is by summing up the diagonal elements and the other is by adding up the eigenvalues. My question is, are these procedures also valid for non-Hermitian matrices? But anyway, thanks for that procedures. It’s amazing that we can also obtain the trace not just by doing diagonalization which is quite long. I hope those techniques are also valid for non-Hermitian matrices.

  13. Thu Vien So Says:

    What are the requirements in using the Gram-Schmidt procedure???Is it necessary for all cases of finding the eigenvector???Thank You.

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