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Eigenfunctions and Operators: Excursions in Vector Space

Rahul Bhattacharya
13th April, 2014


[Note: This abridged article is excerpted from the lectures given by Rahul Bhattacharya as part of the FEM program in London and New York between 2009 and 2011.]

In Quantitative Finance we study eigenvectors and eigenvalues within the context of interest rate modeling, Principal Components Analysis (PCA) and many other important topics, including multi-asset Monte Carlo simulation. Eigenvectors and eigenvalues are extremely important mathematical concepts that have a wide variety of applications not only in quantitative finance but also in economics, electrical engineering, computer science and quantum mechanics. Eigenvalues and eigenvectors play a very important role in the mathematics behind Google's web search engine that we use almost every day. And yet, eigenvalues and eigenvectors are not easy topics to comprehend, especially for those undergraduate students taking a first course in Linear Algebra.

Eigenvectors and eigenvalues are first introduced to undergrad students within the framework of vector spaces, matrices and systems of linear equations in Linear Algebra courses and many a times the students, even though they understand the concepts behind systems of linear equations and matrices, they are unable to reconcile to the notions of eigenvalues and eigenvectors. Perhaps, it is easier to introduce the notions of eigenvectors and eigenvalues by relating them to the notion of functions. After all, a vector space , in which we study matrices and eigenvectors, is nothing but a space of all continuous functions on the closed interval . And sometimes, it is much simpler to understand functions as opposed to vectors and matrices.

Eigenfunctions are very similar to and related to eigenvectors through the concept of finite dimensional and infinite dimensional vector spaces. Let's keep things simple and try to understand the fundamental notion of an eigenfunction.

An eigenfunction is a special kind of a function, , where is a variable, such that if it is operated on by a mathematical operator, say, , then it yields a constant multiple of itself. In other words, if is a mathematical operator (a mathematical operator does things like summation, multiplication, transformation, differentiation, etc.) then if acts on the function we get some constant multiple of .

In the above, is a constant (a scalar). We have used the generalized notation for the operator because we want to denote it as a transformation.

Can we think of any such function and any such an operator? Yes, we can. Take, for example, the function where, is a constant. If we differential this function – find out the first mathematical derivative of with respect to – then we get the following:

Thus, we see that if is the mathematical operator then , where is a constant, is an eigenfunction of this operator because where . The operator, is called the differential operator and in general is represented as . An operator need not be just a differential operator. Say, we define an operator, , which performs the operation of multiplying any function, by the variable . Now, let's take the case of a Dirac Delta function, which is defined around a constant point, , and is given by: . This delta function is zero for all values of x except for where it is infinite. This function is formally defined as:

What happens when we multiply the operator , with the delta function ? We get a constant multiple of the delta function, i.e. where, is a constant.

Therefore, is the eigenfunction of the operator .

In Quantum Mechanics, eigenfunctions play a very important role. In fact, both the differential operator, and the functions that we mentioned above have very special meaning within the context of quantum mechanics. In fact, the operator is known as a position operator in quantum mechanics and the differential operator, , is altered slightly to generate what is known as a momentum operator.

We are not going to go into detail here and in this session we'll restrict ourselves to some very fundamental math concepts only (elsewhere, in other lectures in FEM program as well as in some Computational Physics lectures, I have talked about eigenfuctions and their relevance in quantum mechanics in detail). Suffice it to say, that the differential operator, , is actually a quantum momentum operator (something that represents the momentum of a sub-atomic particle) and the eigenfunction , which is actually written as , represents the "state", or the "momentum state" of the particle.

Let's very briefly talk about the momentum operator and its eigenfunction. Take a complex function, , which is given by:

Where, is the imaginary number, is a constant (and related to Planck's constant) and is another constant. Now, let us define an operator as:

Then, we can immediately see that where, is a constant given by . This follows from simple rules of differentiation:

Hence, we see that is an eigenfunction of the operator because when we operate on the function it yields a constant multiple of itself.

Let's take another very similar complex function, , such that it is expressed as:

Here, is an imaginary number, is a constant associated with classical momentum of a particle and is a constant that is related to Planck's constant. is known as a wave function. Again, define a new momentum operator (quantum mechanical momentum), , that is given by

Then, we can see that when the operator p acts on the function the result is a constant multiple of .

Therefore, is an eigenfunction of the momentum operator .

Reference:

  1. Bedaque, Paulo, Quantum Physics, University of Maryland, College Park (Lecture Notes
  2. Fitzpatrick, Richard, Quantum Mechanics, University of Texas at Austin (Lecture Notes)
  3. Fedaka, Willian A. and Prentis, Jeffrey J., The 1925 Born and Jordan paper “On quantum mechanics”, American Association of Physics Teachers, October 2008.
  4. Gray, Alfred, Mexxino, Michael and Pinsky, Mark A., Introduction to Ordinary Differential Equations with Mathematica, Springer-Verlag, New York, 1997.
  5. Strang, Gilbert, Linear Algebra and its Applications, 4th edition, Thomson Brooks/Cole Cengage learning, 2006
  6. Penrose, Roger, The Road to Reality, Vintage Books, 2004

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