ECE 487 Lecture 12 : Tools to Understand Nanotechnology III
Class Outline:
•Inverse Operator •Unitary Operators •Hermitian Operators Things you should know when you leave… Key Questions • What is an inverse operator? • What is the definition of a unitary operator? • What is a Hermetian operator and what do we use them for?
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
Let’s begin with a problem…
Prove that the sum of the modulus squared of the matrix elements of a linear operator  is independent of the complete orthonormal basis used to represent the operator.
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
Last time we considered operators and how to represent them…
•Most of the time we spent trying to understand the most simple of the operator classes we consider…the identity operator.
•But he was only one of four operator classes to consider.
1. The identity operator…important for operator algebra.
2. Inverse operators…finding these often solves a physical problem mathematically and the are also important in operator algebra.
3. Unitary operators…these are very useful for changing the basis for representing the vectors and describing the evolution of quantum mechanical systems.
4. Hermitian operators…these are used to represent measurable quantities in quantum mechanics and they have some very powerful mathematical properties.
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
So, let’s move on to start discussing the inverse operator…
Consider an operator  operating on some arbitrary function |f>. If the inverse operator exists, then it is just Â-1.
This means we can do Which is just another some simple things… way of identifying the identity operator. •The operator  in general will take an input vector and do something to it.
•The inverse operator does exactly the opposite and restores the original vector.
•Since the operator can be represented as a matrix, finding the inverse of the operator reduces to finding the inverse of a matrix.
•However, this is only true if the inverse exists.
•There is not any guarantee that the inverse does exist.
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
We can see this through a simple example using the projection operator…
Let’s define the projection operator:
•This guy does not have any inverse because he is “projecting” all input vectors onto only one axis in the space. Which one? The one corresponding to the vector |f>.
•So this is a “many to one” mapping in vector space.
•There is no way of knowing anything about the input vector except its component along this axis.
•We do not have enough information to reconstruct the original input vector as it could be one of many.
•This is not a particularly useful operator in quantum mechanics and this is probably the last we will speak of him.
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
While inverse operators are a bit lame, unitary operators are much more useful…
How do we define a Basically, its inverse is its Hermetian transpose unitary operator: or adjoint.
Mathematically, we can form this by…
So what is a unitary operator doing?
•It is not changing the length of the vector on which it operates.
•More generally, if we operate on two vectors with the same unitary operator, it does not change the inner product.
•The conservation of length property follows as a special case which we will cover later.
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
It’s later now, so let’s start thinking about the conservation of lengths and inner product under unitary operation…
Start with some unitary operator, Ũ, and a couple of vectors |fold> and |gold>. Now let’s form some new vectors by operating with Ũ. We get…
In conventional matrix or vector-matrix multiplication with real matrix elements, we know Where the superscript denotes the transpose.
With complex elements, things are a bit different when we consider matrix or operator multiplication. We get… Or with matrix-vector multiplication…
But so what, it doesn’t look like we have gotten any closer to being able to prove any of these conjectures about conservation of lengths and inner products!
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
But by using our definitions of the unitary transform, we can do a bit of math…
Here’s our definitions: and
Let’s use these and take the inner product:
And now for some simple math…
So, now we get what we want…
•The inner product doesn’t change if both vectors are transformed this way.
•And the length of the vector is not changed by a unitary operator.
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
Before, we mentioned that these unitary operators were useful. Now we should start thinking about how they are useful…
Most of the time when you see these guys, it’s because we are changing the basis sets or representations or coordinate axes.
•To see this, let’s take some old vector |fold> and represent him as an expansion on the functions |ψn> as a column vector:
Each of these c’s are the projections of |fold> on the orthogonal coordinate axes in the vector space labeled with |ψ>’s
Fine, now suppose that we want to represent this vector on a new set of orthogonal axes… •Changing the axes is equivalent to changing the basis set of functions. •It doesn’t change the vector that we are representing. •It does change the column of numbers used to represent the vector. M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
For example, suppose that the original vector was actually the first basis vector in the old basis: Then in the new representation, the elements of the Let’s write him as: column of numbers would be the projections of this vector on the various new coordinate axes.
Mathematically, this Or for our specific example looks like: under this change of basis:
But we don’t have to stop here, we can do this for all basis vectors |ψn> and we can get the correct transformation if we define a matrix, like this one…
Where each of these elements in an inner product or the projection of the basis vector on our new axes.
Now we can define our new column of numbers:
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
But be sure to remember that |fold> and |fnew> are still the same vector in the vector space…
•The only thing that we’ve done is change the representation (coordinate axes) and the column of numbers.
•We could pose a simple analogy: say we have some sculpture of an arrow sitting at some angle with the floor.
•Then we can write something down for the length and direction.
•Move it somewhere else and the arrow stays the same but the way we describe it changes.
This is just so brilliant. Let it sink in for a second…ok. So how do you prove that an operator is unitary?
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
We can prove that some generic operator is unitary by writing the matrix in its sum form. Or…
•So we proved that the generic operator is unitary since its Hermetian transpose is its inverse. Multiply the two and we get the identity matrix back.
•Therefore, we can make any change in basis by defining a unitary operator.
•Or another way, any change in representation is a unitary transform.
•For completeness, notice how you can play some meaningless games with unitary transforms and the identity matrix…
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
We can make some additional general statements about unitary transforms… •Since they don’t change the inner product, once we change to any different basis then the inner product won’t change either.
•It’s just like a dot product of two regular vectors…the lengths and directions depend on the vectors but not on the coordinate axes.
So, what happens to the matrix of an operator if we change the basis?
•To start, just consider a simple expression
Where the f and g vectors are arbitrary. Also the “old” and “new” refer to the representations and not the vectors or operators.
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
Remember that the actual vectors and operators are not changed by the change we made in the representation…
•We only changed the sets of numbers which represent them.
•So then the result would not be changed by changing the representation.
•Thus far, we believe the following…
From this equation, we can deduce the following:
Or put a slightly different way:
So far, this provides a nice symmetry. The operators that change the quantum mechanical state are also unitary.
•And we already know that unitary operators are not changing the basis set. •They are really changing the quantum state of the system and the vector’s orientation in space.
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
Why do such operators pop up in our analysis of quantum mechanics in the first place? Consider a simple example where we are dealing with a system which contains just a single particle…
•We know that if we sum all of the occupation probabilities then we should obtain unity.
•If we expand the quantum mechanical state |ψ> on the basis |ψn> then we have: Assuming particle number is conserved, then:
•This sum is retained even as the system evolves in time.
•But the sum is just the length of the vector |ψ>.
•Therefore, a unitary operator is appropriate for describing changes that conserve the particle (like time evolution)
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
Now let’s talk about Hermitian operators…
•Hermitian operators are their own adjoint or self-adjoint. We can put this down mathematically as:
Or in terms of matrices…
And that
So the Hermiticity condition at the top of the slide implies that:
•For all i and j, from which we can conclude that the diagonal elements of this matrix are real.
•But so what if the operator is self-adjoint?
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
To understand the implications of a Hermitian operator being self-adjoint in terms of its action on functions in general, let’s start by analyzing a simple result:
Now take the Hermitian adjoint of this result:
•To proceed further, we need to remember the rules about adjoints of the products of matrices and, to a lesser extent, vectors as a special case of a matrix.
•In particular we need to use the following relation:
•As it applies to the specific case of the result we began with, the resulting matrix is a 1x1 matrix. In other words, it’s just a number which allows us to state:
Or in terms of general functions f and g:
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
Now if we use our rule on Hermicity, that we defined a few slides ago.
Namely: From which we have obtained:
•But this is just a general way of stating the Hermicity of the operator M.
•This statement is also true if the two functions are not orthogonal.
•So the case where the diagonal elements are all real is really just a special case of the more general one presented here.
For completeness, remember that we can represent both the functions not only in Dirac notation but also in integral form.
•Our general statement of Hermicity can be written as:
But we’re not quite finished defining Hermicity in integral form yet…
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
We can rewrite the right hand side of the integral equation by using the property of complex conjugates. Namely:
This allows us to rewrite our integral equation as:
Rearrange it a bit to obtain:
From here we can get the rest of the relations for Hermicity that we already have:
•For functions of a continuous variable, like in the above integral equation, can all be regarded as equivalent statements of Hermicity of the operator M.
•But notice how much nicer Dirac notation is as compared to the integral representation.
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
But the whole time, we’ve been studying Hermitian operators because they are related to actual measurements on systems. So they should have some special properties.
The special properties of Hermitian matrices are:
1. Reality of eigenvalues
2. Orthogonality of eigenfunctions with different eigenvalues
3. Completeness of the set of eigenvalues
Might as well start at the top and consider the reality of the eigenvalues…
•Start by supposeing |ψn> is a normalized eigenvector of the Hermitian operator M with eigenvalue μn.
Then by definition we have:
Multiply through by the bra and we get: M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
So now let’s apply what we already know about Hermitian operators to the problem at hand.
In particular, from Hermicity we know:
•This means that μn is real.
•And that suggests that such an operator would be useful for representing a quantity that is real, like a measurable quantity.
The second statement claims that the eigenfunctions of a Hermetian operator corresponding to different eigenvalues are orthogonal.
•This can easily be proven using our Dirac notation:
Start with the trivial equation:
Use the associative rule for adjoints:
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
Now apply what we know about Hermicity to the problem at hand:
In particular, we use:
1. The definition of Hermicity (i.e. that the operator is self-adjoint).
2. The Hermitian adjoint of a complex number is its complex conjugate.
3. The eigenvalues of a Hermitian operator are real.
This gives us:
We’ve been assuming that the two eigenvalues are different, so:
And we have proved that the eigenfunctions associated with different eigenvalues of a Hermitian operator are orthogonal.
But what happens if we have more than one eigenfunction associated with a given eigenvalue? It’s called degeneracy. In fact, it’s provable that the number of such degenerate solutions for a given finite eigenvalue is itself finite.
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
What about the third point? What about the completeness of sets of eigenfunctions?
•A very important result for compact Hermitian operators is that, provided that the operator is bounded or results in a vector of finite length when it operates on a finite input vector, the resulting set of eigenfunctions is complete.
•Or it spans the space on which the operator is compact.
•We can prove this, but just take my word for it. It would require that we set up a mathematical framework for functional analysis beyond what we need here. It’s not worth it.
•This result means in practice that we can use the eigenfunctions of any bounded Hamiltonian operator to expand functions.
•This greatly increases the available basis sets beyond the simple spatial or Fourier transform sets.
•Practically, it means that we can simplify the description of them.
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
But we still have no idea why we would care if the operator is bounded or not… The fact is that they do have some very attractive properties:
1. Real eigenvalues 2. Orthogonal eigenfunctions 3. Complete sets of eigenfunctions 4. They represent physical quantities
Some Hermitian operators we have already encountered:
•Momentum •Energy
We are going to encounter many more as we move through our analysis of quantum mechanical systems.
•This means we’d better learn what we’re doing and how to handle these operators now or we’re screwed.
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
A good place to start is with simple differential operators…
•These are operators such as d2/dx2 or d/dx.
•We have seen them in the momentum operator in the schrodinger equation but it seems kind of odd that we can represent them as matrices, but we can.
•It’s usually easier to use the integral form of the inner products and matrix elements.
•But it may be useful, so let’s think about how to write them out…
The matrix form looks something like this..
Assuming as is usual with the derivative that we can take the limit as δx goes to zero.
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
Now let’s multiply it by a column vector whose elements are the values of the function, f(x), at a set of values spaced by δx…
So now we have a way of representing the derivative as a matrix…
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III
Alright, some final things to note…
•We assumed that the matrix is symmetric about the diagonal but in this case it is anti-symmetric about the diagonal.
•This matrix is not Hermitian which reflects the fact the derivative is not a Hermitian operator.
•However, it is Hermitian if we multiply by ± i, just like we do in the momentum operator.
•One can also show the second derivative is also Hermitian when represented in matrix form:
M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1