ece 487 lecture 10 : tools to understand nanotechnology i

ECE 487Lecture 10 : Tools to

Understand Nanotechnology I

Class Outline:

•Dirac Notation•Vector Spaces

• How can we represent integrations and summations shorthand?

• What is a vector space?• How does a vector space differ

from my normal geometrical space?

M. J. Gilbert ECE 487 – Lecture 1 0 02/1 7/1 1

Things you should know when you leave…

Key Questions


Before we leave time-dependence, let’s do a quick problem to make sure that we understand some of the concepts from the last lecture…

Tools to Understand Nanotechnology- I

In many of the interesting situations we wish to analyze, the mass is not constant in space. This occurs specifically in analysis of semiconductor heterostructureswhere the effective mass is different in different materials. For the case where mass m(z) varies with z, we can postulate the following Hamiltonian:

( ) ( )zVzzmz

H +

∂∂

∂∂

−=1

2ˆ

2

i. Show that this Hamiltonian leads to the conservation of particle density.

ii. What are the boundary conditions used for analyzing a step potential using this new Hamiltonian?


Alright, enough time dependence. Time to think more generally about what tools we’ll need to analyze current and future nanodevices…


•Thus far, we have introduced quantum mechanics through the Schrödinger equation and the spatial and temporal wavefunctions that are solutions to it.

•But quantum mechanics is much broader than this as we already know of situations where the Schrödinger equation does not apply, like to photons.

•Therefore, we must seek a more general mathematical formalism to go much farther.

•This new formalism that we will develop is mostly based on linear algebra as in

•Matrix algebra•Fourier transforms•Solutions to differential equations•Integral equations•Analysis of linear systems in general

•Here we assume at least the matrix version of linear algebra.


But it is even more general than that…


•The formalism of quantum mechanics is based on linear algebra because quantum mechanics is apparently absolutely linear in certain specific ways.

•As in quantum mechanical “amplitude”

•To generalize linear algebra for quantum mechanics we introduce some shorthand notations, especially Dirac’s “bra-ket” notation, but the underlying concepts still remain the same.

•The mathematical approach here is deliberately informal and the emphasis is placed on grasping the core concepts and way of visualizing the mathematical operations.

•The major goals of this mathematical approach are to visualize quantum mechanics and to develop an intuitive understanding of quantum mechanics that extends to a broad range of problems.


Let’s begin to formulate this new approach to quantum mechanics by considering a function f(x):


•This function essentially maps one set of numbers to another set of numbers.

•In other words, the argument, x, of the function is mapped to another number, the result of the function, f(x).

•The fundamentals of this concept are not changed for functions of multiple variables or for functions with complex number or vector results.

Therefore, we can imagine…

•The set of possible values of the argument is a list of numbers, and the corresponding set of values of the function is another list.

•For example, we could take the set of real numbers and, presuming we know a list of possible arguments, we can write these functions out as a column vector.


For example, we could specify the function at a discrete set of points spaced evenly by some small amount, δx…


or

•If we consider breaking this into small enough changes over a sufficient enough range, then it could be useful for some calculation.

•An easy example would be that of in integral.


So the question then becomes, how do we visualize these vectors?


•Suppose that the function, f(x), is approximated by its values at three points: x1, x2, and x3.

•Let’s represent it as a vector:

•Then it is easy to visualize the function in normal geometrical space:


It’s not quite this easy most of the time because many different functions in quantum mechanics are complex and not real.


It’s also compounded by:

•The fact that there are many different elements in a vector, perhaps even an infinite number.

•The space may even need to be large, possibly of infinite number of dimensions.

•However, we can still visualize the quantum mechanical state as a vector in space.


Let us now begin to introduce some new notation to help us in our analysis of nanotechnology…


Let’s introduce the so-called Dirac “bra-ket” notation:

Is referred to as a “ket” and it refers to an appropriate form of our column vector.

For the case of our function, f(x), we could define the “ket” as:

•More strictly, this is the case for when δx goes to zero.

•We have incorporated the δx to ensure that the vector is properly normalized.

• The concept is still that the function is a vector list of numbers.


In just the same way, we can also define a “bra” to refer to a row vector…


•Note what is different about our row vector…we have taken the complex conjugate of the different entries.

•The vector has many names associated with it…

•Hermetian adjoint•Hermetian transpose•Hermetian conjugate•Adjoint

Of the vector: A common notation used to indicate the Hermitianadjoint is to use the character as a superscript.


So, how do we form the Hermitian adjoint?


We can do this by reflecting about a 45o line then taking the complex conjugate of all of the elements…

So, basically the “bra” is the Hermitian adjoint of the “ket”

•So now we have a new shorthand definition to reduce or time.

•But is it useful for anything beyond that?


Consider again our definition of f(x) as a vector…


With our previous definitions, we can write:

Which we can rewrite as a summation:

•Writing this as a vector multiplication eliminates the need to write a summation or integral.

•These operations are now implicit in the vector multiplication.

•The shorthand for the vector product of the “bra” and the “ket” is


We can also use the same type of shorthand when we are dealing with two different types of functions…


•This kind of product is normally referred to as an “inner product”.

•Inner because it takes two vectors and turns them into a number or smaller entity.

•More likely, the inner product refers to the fact that the “bra” and “ket” combinations give the look of a closed set and nothing more.


Another important concept that we have dealt with so far is the expansion of a wavefunction on a basis set. How does our new notation deal with this concept?


In other words, suppose that the function is not represented directly as a set of values for each point in space, but as an expansion in a complete orthonormal basis…

We could write this function down as a “ket” but this “ket” has an infinite number of values:

And the “bra” becomes:

So, let’s think about what we’ve done…

•When we write the function in a different form we say that we have changed representations.•Everything about the vector is the same as before.•We have just changed the axes we use to represent the function.


Just as before we can expand out our functions to evaluate the probability density…



How do we evaluate the expansion coefficients in these equations when we use the “bra - ket” notation?


•It’s easy to do because we choose the set of functions ψn(x) to be orthonormal.

•Since ψn(x) is just another function, it is easy to write it as a ket.

•To evaluate the coefficients cm, we need to premultiply by the bra.

Using bra-ket notation, we can write f(x) as:

Using bra-ketnotation, we can drop the x.

•Cn is just a number so we can move it around.•Multiplication by a vector and a number is commutative, but vectors and matrices, generally, is not.


Just so that we’re clear, why use bra-ket notation?


•It is an easy and general way of writing the underlying linear algebra operations we need to perform.

•It can be used whether we are thinking about representing functions as continuous functions in some space or summations over basis sets.

•It will also continue to be useful as we consider other quantum mechanical attributes that are not represented in normal geometric space.

Thus far in this lecture, and in general, we have been interested in representing the state of a quantum mechanical system.

•The set of numbers represented by the bra and the ket vector represents the stat of the system.

•Typically, the ket vector is referred to as the state vector of the system and the bra is the adjoint of the state vector.


But when we are thinking about these concepts, one thing is clear…we need a space for our vectors to live in.


For a vector with three components, we can imagine a three dimensional Cartesian space.

It’s easy to think about this as a line in space with projected lengths along the axes.

What if we have an infinite number of orthogonal axes?

Just as we can label the axes in conventional space, we can change the labels on the axes to reflect the kets


Ok, so now we have some space in which the basis vectors live. What now? What are the properties of such spaces?


•The geometrical space has a vector dot product that defines both the orthogonality of the axes:

And the components of a vector along those axes:

•The vector space has an inner product that defines the orthogonality of the basis functions and their components:


There are more properties…


The addition of vectors is both commutative and associative…

The vectors in the vector space are linear…

The inner product is linear…

The inner product is also linear in superposition of vectors…


And more properties…


•There are well defined lengths to the vectors…

•The vectors are both compact and complete…

•Any vector in the space can be represented to an arbitrary degree of accuracy as a linear combination of the basis vectors (completeness or compactness)

•The inner product for the geometrical space is different than for the vector space.


All of the elementary mathematical properties above, other than the inner product, are sufficient to define these two spaces (Vector and Hilbert) as “linear vector spaces”…


•With the properties of the inner product these are what is referred to as “Hilbert spaces”

•The Hilbert space is the space in which the vector representation of the function exists (Cartesian space is where geometrical vectors exist).

•The main differences between our vector space and geometrical space are:•Our components can be complex and not only real.•We can have more dimensions.

•Our vector space can also be called a function space as a vector in this space is a representation of a function.

•The set of basis vectors (basis functions) that can be used to represent vectors in this space is said in linear algebra to “span” the space.

ece 487 lecture 10 : tools to understand nanotechnology i

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