Quantum Science Philippines
Quantum Science Philippines

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.

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Sticky: Basics of Linear Vector Spaces

by CARIEL O. MONTALBAN


In quantum mechanics, I have learned that the wavefunctions, , reside in Hilbert’s space.  What is Hilbert’s space? I guess to answer this question requires exploring the basic  properties of Hilbert’s space.

Hilbert’s space is a linear vector space whose elements, entities or components obey certain rules or axioms.  This means firstly than you can add these elements and the resulting sum is also as a member or entity of  that  space. Secondly, you can multiply the elements with any arbitrary scalar and the product yields something which is also a component of that same space.  Additionally, the operations of addition and multiplication obey definite rules. These rules are called axioms for addition and multiplication.

By means of simple problems discussed below, I illustrate these axioms which are obeyed by a linear vector space and to which the wavefunctions, , of quantum mechanics  belongs.

As a simple example, let us consider the set of all entities of the form where are real numbers. Do these form a linear vector space? First, we have know how these elements are added and how they multiply with scalars. If their addition and multiplication are defined respectively as follows:

;

and

,

we can then verify that the axioms required for a linear vector space are satisfied in this case.

From the addition operation, we can write the null vector of the set as:

.

Also from the multiplication operation, we can then write down the inverse of simply as  .

We can now verify that all four axioms for addition of elements of the set are satisfied.

First Axiom: Commutativity Property

The operation of addition in a linear vector space is commutative; which means that we don’t care about the order in which the elements are added because we always get the same result.  This axiom is written as:

(i)

Our proof is as follows. Let and .

Then,

.

Thus in a linear vector, the addition of vectors is commutative.

Second Axiom: Associative Property

The operation of addition in a linear vector space is associative which means that we don’t care about the order in which two elements are added to the third one because we always get the same result. This axiom is expressed as:

(ii) .

To prove this in the case of the set , we let

Then,

.

Therefore the addition of vectors in a linear vector space is associative.

Third Axiom: Existence of an identity element

The third requirement for a set to be a linear vector space is that the identity element exists. The identity element is defined as

(iii) .

The identity element of the set is therefore none other than the null vector

To show this property, we just apply the definition of addition hence

.

Fourth Axiom: Existence of an inverse

The inverse of a vector should exist in a linear vector space. The inverse is defined by the statement

(iv) .

For the set we can then verify the existence of an inverse as follows:

Examples of non-vector spaces

From the four axioms of addition of linear vector space, we can further make the following observations.

(1) If are required to be positive numbers, we can’t construct a vector space because Axiom (iv) will not be satisfied.

(2) The vectors of the form do not form a linear vector space. To show this, we let

where are all real numbers.

Then by Axiom (i),

.

Thus, does not form a linear vector space. The closure property is clearly violated since

.
About the Author:
CARIEL O. MONTALBAN finished his B.S. in Physics from Mindanao State University-Iligan Institute of Technology (MSU-IIT), Iligan City, Philippines in March 2008 and is now a graduate student of the same university. He hopes to become an active researcher in the field of experimental physics in the future.

Posted in Linear Vector Space, Quantum Physics, Quantum Science Philippines | 24 Comments »


The Birth of Quantum Mechanics


” The Heisenberg-Bohr concepts leave us all breathless, and have made a deep impression on all theoretically oriented people.”           -Albert Einstein, 1926


The period 1905 to 1925 was a great time for the world’s leading physicists in the race to understand the quantum nature of matter. To explain so many curious and undigestible phenomena about radiation, atoms, molecules and solid materials, some groups worked together and sometimes compete with another. The climax happened around 1925 when the structure of quantum mechanics was finally laid down.

Louis de Broglie,1887-1961

Louis de Broglie

This started with Louis de Broglie‘s conjecture in 1924 that particle-like objects such as electrons should display wave properties. Indeed if light which is initially thought to be a wave can behave as a particle or quantum, why not those objects which we normally conceive of as particles display wave-like properties? Why not indeed?

Erwin Schrodinger,1887-1961

Erwin Schrodinger

Shortly after de Broglie introduced his concept of matter waves, Erwin Schrodinger proposed an answer to the question of what happens to the matter waves when a force acts on it. He came up with a wave equation now known as Schrodinger’s Equation that lies at the heart of quantum mechanics.

Given a particle and the force that acts on it, Schrodinger’s equation  gives the possible waves associated with this particle at a given position and time. And this is designated by the hardest working symbol in modern physics: the wave function .

Max Born,1882-1970

Max Born

That Schrodinger would be mistaken in the physical interepretation of the wave function is only one of  the many curious twists in this very interesting and engaging conversation.  What took Max Born to interpret the absolute square of as probability density for finding electrons and not  matter density as  Schrodinger intimated?

Werner Heisenberg,1901-1976

Werner Heisenberg

Pascual Jordan,1902-1980

Pascual Jordan

Months before Schrodinger was to write down his famous equation, Max Born, with his young students Werner Heisenberg and Pascual Jordan, already created  an entirely different approach from that of Schrodinger.  The matrix formulation of quantum mechanics developed by Born’s group in Gottingen, Germany described matter  and radiation as discrete particles.

The two formulations of quantum mechanics were thought to be different but they were quickly proved to be equivalent by Schrodinger himself.  Soon thereafter, Paul Dirac incorporated the special  theory of  relativity with quantum mechanics and the ‘quantum field theory’ was born.

Paul Dirac,1902-1985

Paul Dirac

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Sticky: The Equation That Changed The World

“Common sense is that layer of prejudices we acquire before we are 16.”
– Albert Einstein


The physical world that was opening up in  1900 was revealed and seen in a radical light. Max Planck’s bold hypothesis that light was emitted in  bundles or  quanta of energy where each quantum’s energy is determined by the frequency was completely  without  precedence. Planck tried for a number of years to fit his quantum hypothesis into the fabric of  classical  physics but failed.

In 1905, Albert Einstein published a paper proposing that light was not only emitted in integral units or bundles of energy but it was also absorbed in such bundles – bundles that came to be known as photons. Again the energy of absorption was equal to the mysterious, h, Planck’s constant multiplied by the light’s frequency, f. Indeed this is the same equation of Max Planck where the quantum of action, h, was introduced for the first time in 1095: E= hf. This is the single equation that changed the world of physics.

Albert Einstein, 1879-1955

Albert Einstein

Nothing could have been more contrary to the prevailing ideas at that time concerning the transfer of energy of a wave. Since light is in the form of waves, it has to got transfer its energy and be absorbed through its intensity and not by units of its frequency. Einstein in this 1905 paper also provided an explanation for a well-known phenomenon known as photoelectric effect which is associated with the absorption and emission of electromagnetic radiation by matter. It was this work that earned Einstein in 1921  the Nobel Prize in physics  not his special theory of relativity  which was also published that same year.

The Rutherford-type model of the atom proposed by Niels Bohr in 1913 was an extraordinary success in accounting for the spectrum, stability and other aspects of the hydrogen atom. Its success hinged on the Einstein-Planck quantum relation. However Bohr’s theory failed when applied to helium and other atoms. Plus the fact that the theory contained inconsistencies that could not been resolved.

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Our Physical World Beckons To Be Understood

” … To free myself from the chains of the ‘merely personal’, from an existence which is dominated by wishes, hopes and primitive feelings. Out yonder there was this huge world, which exists independently of us human beings and which stands before us like a great, eternal riddle, at least partially accessible to our inspection and thinking.  The contemplation of this world beckoned like a liberation…”

– Albert Einstein



Lucretius, 94-55B.C.

Lucretius

It’s not yet too late to participate in an engaging conversation that is as as old as our civilization. Almost a hundred years before Christ, Lucretius wrote: “Nature resolves everything into its component atoms.” Yet already several hundred years before him Democritus (b. 460 BC) proposed that each atom cannot be further divided and each atom was not capable of change.
Democritus

Democritus

Fast forward two thousand years later. Lord Kelvin and J.J. Thomson gave the world a simple picture of the atom that can tested by experiment. These gentlemen proposed that the atom’s structure consists of electrons embedded like raisins in a dough or cake of equally positive charge.

William Thomson (Lord Kelvin)

William Thomson (Lord Kelvin)

J. J. Thomson

J. J. Thomson

It didn’t take long for Ernest Rutherford in 1911 to modify the cake model of the atom due to his discovery of the atomic nucleus. From his alpha scattering experiments, it was revealed that the atom has a positive charge concentrated minutely at its center with the electrons spread out over a large region beyond it. Much like the solar system with the sun at the center and the planets revolving around the sun. It this model of the atom that persists in the minds of many people today.

Ernest Rutherford

Ernest Rutherford

Yet this planetary model of the atom has a very serious flaw. According to the classical laws of physics the electrons going around the atom will emit radiation thereby losing its energy until it will spiral off and collapse to the center of the atom in a short time. Therefore this model cannot hold matter. It is not as stable as the planetary system we are in.

James Clerk Maxwell (1831-1879)

James Clerk Maxwell


Niels Bohr

Niels Bohr


Niels Bohr then proposed his famous theory of the hydrogen atom in 1913. His model restricts the electron in stable, allowed, circular orbits without emitting any radiation; thus making a robust atom. Now this idea is completely alien and so contrary to what everyone believed at that time. Nothing in James Clerk Maxwell‘s electrodynamics and Isaac Newton‘s mechanics would support Bohr’s contentions. When does the electron emit energy? It is only when the electron makes a transition from one allowed orbit to another that it radiates energy. And this energy is related to an equation postulated 13 years earlier.

Isaac Newton

Isaac Newton

Bohr’s radical proposal had its roots in the work of Max Planck who in 1900 introduced the earliest concept of the quantum of action to explain the age-old problem of radiation produced by a heated body or black body radiation.

Max Planck

Max Planck


Planck’s hypothesis is completely startling and without precedence. Light was emitted in bundles of energy where each bundle was related to the frequency of light, f, by E = hf. In this equation, h , Planck’s constant, the quantum of action was introduced for the first time into the body of physics. Planck’s hypothesis was very startling, completely unprecedented and completely in contradiction with old physics.

The quantum of action hypothesis of Planck marked the beginning of modern physics. We can say that the world was never the same after Planck.

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