So now, we will continue putting together the basic conditional principles of
quantum mechanics, and I'm also going to introduce the so-called Dirac notations,
which are mathematical symbols Dirac introduced back in the early days.
And these symbols turned out to be very convenient in doing calculations in
quantum mechanics. So, let me mention actually that as a
practicing theoretical physicist I noticed that an important component of a new,
putting together a new physical theory is to come up with the convenient and elegant
notation. So, these are not some minor unimportant
questions. So, it's actually quite important, and
having awkward notations could make our lives very difficult.
So, Dirac notations certainly satisfy the criterion of being convenient, and also
they are widely used in the literature. So, if you don't know them already, it's a
good idea to familiarize yourselves with the, with them, and this is what we're
going to do today. But before introducing the Dirac notations
and going further with the development of sort of more abstract quantum theory, I
would like to mention a relevant superposition principle in quantum
mechanics. So, this so called superposition principle
simply states that if we have 2 solutions to the Schrodinger equation, psi 1 of r
and t and psi 2 of r and t, they are arbitrary linear combination such as here
with some arbitrary complex constant, c1 and c2, is also a solution to this
equation. So actually, this so-called superposition
principle is not specific to the Schrodinger equation.
The superposition principle holds for any linear differential equation, which the
Schrodinger equation happens to be. So, there are no terms between that.
There are no terms like psi squared psi cubed, etc.
So well, to prove this, one can just simply plug it into the Schrodinger
equation and cancel the corresponding pieces, term by term.
So, what this superposition principle motivates is the notion of so-called
Hilbert space, which is a linear vector space where quantum states sort of live.
To remind you, a linear vector space is a basic concept of Linear Algebra, which
implies a set of elements that we call vectors such that we can define a sum of
any two vectors and a sum of any two vectors itself belongs to this set, to
this vector space. And where we can also define a
multiplication by a constant, either real or complex, depending on the
circumstances, and this sort of stretching of a vector, also gives rise to another
element of the set. So, in other words, this multiplication
and addition of vectors are closed operations.
So, through this superposition principle, we can suspect that whatever the
mathematical objects are the described quantum states, they also in some sense
belong to a closed linear vector space that is what we call the Hilbert space.
But what about the wave-function that we have become already familiar with.
So, this wave-function psi of r should be thought of in this abstract language and
approached as a specific representation of a quantum state.
We're going to discuss the meaning of it a little later.
But here, I just would like to mention that it's much like coordinates of a
vector. So, let's say, if we have in the simplest
example we could have, let's say, vector in two dimensions with some coordinates ax
and ay. So, these particular coordinates are
specific to a choice of coordinate axis. See, we are free to choose another
coordinate system, in which case, the coordinates will change, but the vector
itself in some sense, in some absolute sense will remain the same.
So likewise, in quantum theory, one can think about an existing quantum state,
which is sort of independent of our way to describe it.
And if we choose to do so using the wave-function, it, sort of, corresponds to
the choice of particular basis. But there are circumstances, however,
where we don't want to specify a representation.
In this case, we could have called our vector, state vector as psi with a vector
sign on top which would have been the usual notation in Linear Algebra, but
instead of using this notation, Dirac found useful to use another notation
presented here which is so-called ket vector by Dirac.
And also, there is a bra vector which corresponds essentially to psi dagger and
I, to tell you the truth, there is no profound reason for the choice of this
particular notation so they just turned out to be convenient to a couple metric
elements, etc., we will see. And, not that there is a profound reason
to call them this way, bra and ket vectors basically are called this way because when
put together, they sound like a bracket, which sort of they look like.
So now, let me present a few embarrassingly simple facts from the basic
Linear Algebra that I do expect most of you to know but however, I'm going to, I'm
still going to remind you of those because they're going to be connected to slightly
more complicated structures that arise in the abstract mathematical formulas of
quantum mechanics. So, what I have here is simply a planar
vector in two dimensions, some vector e and if I have a coordinate system, x and
y, with some unit vectors ex and ey, so I can write my vector as a linear
combination of this unit vectors. So, ex here is 1, 0 and ey is 0, 1.
Or I can write it as so. So, the vector and its conjugate.
So, another things I want to emphasize is that if we consider the full length
structure, ex, ex dagger plus ey, ey dagger, on this two, there's a metrics
multiplication of these vectors. So, we're going to have 1, 0, 0, 0 in the
first room plus 0, 0, 0, 1 in the second room, which when put together, becomes an
identity matrix. But very importantly, this result is not
unique to the particular choice of these orthonormal bases.
The result holds for any such orthonormal bases in any dimension and in any linear
vector space. Now, the reason I'm telling you all these
is because in quantum mechanics, so as we just discussed, so the state vectors, they
represent elements in some Hilbert space, in some linear vector space, which we call
Hilbert space. But in order to be able to work with this
vector, just like here, we need introduce a coordinate system so that we have
something concrete to work with. So, in quantum mechanics, we have to
define a basis. Since the Hilbert space is much more
complicated in, generally than this two dimensional vector space, and most often,
we're dealing actually with infinite dimensional Hilbert spaces, so the
structure of the basis is generally more complicated.
But nevertheless, the properties that we just discussed, the simple properties of
Linear Algebra, so are, still remain the same.
So, for example, this identity that we just derived for this basis ex and ey also
is, holds for the basis in a, in the Hilbert space.
So, let's see if we have a set of the x as either a discrete set or a continuum set.
We will see both of them. So the sum of this ket-bra products over
all basis vectors, gives rise to an Identity operation.
And this resolution of identity is going to be extremely useful for us in a number
of derivations that we're going to see throughout the course.
So, for example, for this q, I haven't really specified the physical meaning of
this q, it can be anything. So, one example we're going to see pretty
often is when we're dealing with the integral over volume of product like this,
where this vectors respond to eigenstates of the position operator.
So, another thing I want to mention here is that any wave-function, so what,what
basis really means is that any vector in a linear vector space can be represented as
a linear combination of the basis vectors. Just like here, I represented my vector a
as a linear combination of ex and ey. So, the fact that I have a basis here
implies that I can represent my wave-function or my state vector as a
linear combination of these q's. And the matrix elements between q and psi
is what we actually call the wave-function in the q representation.
So, this is much like these coordinates. So, these coordinates, for instance here,
ax is can be written as ex dagger times a, and here psi of q can be written as a
bracket product of q and psi. So, and again, in principle, we can have
two situations, either we have a discrete spectrum or we have a continuum spectrum
in the former case, we have a sum over q, in the latter case we have an integral
over q. But now, a question remains of how to
actually choose a basis or representation to describe our state vector or
wave-function. Just like in the usual Linear Algebra
there is no absolute or correct answer. The choice of a basis is a question of
convenience. But there is some standard or natural
choices that can be constructed, and so I'm going to discuss now how it's how it's
done. So, we already know that physical
observables in Quantum Mechanics are associated with linear Hermitian or
self-adjoined operators. For a generic such operator, one can
define the eigenvalue problem as here, which basically is the problem of finding
a vector or vectors, set of vectors, which under the action of the operator don't
really change but just are stretched with a certain eigenvalue a.
And for the Hermitian and self-adjoined operator, these values of a are real
numbers. So what's essential is that, in many
cases, this set of eigenvalues of such operators form a basis in the Hilbert
space. This means that we can expand our
wave-function in terms of this basis vectors.
And the corresponding matrix elements, so let's say for this vector a, whatever it
is, is called the wave-function in the a representation.
So, one can construct an infinite number of such representations, just like there
exist an infinite number of observables and corresponding operators.
But sort of a natural way to choose a basis is to use operators that are
irrelevant to our problem, such as for instance, the coordinate operator, it's
just here in one dimensional quantum mechanics, so the momentum operator.
And the corresponding matrix elements, psi of x or psi of p, are called the
coordinate representation and the momentum representation correspondingly.
So for example, psi of x or psi of r, we have already seen many times in our
lectures. Now, the meaning of this of this basis
vector is, is essentially this tightly localized wave packets.
We shall locate it in a precise point and space.
So in contrast, the vectors p correspond to states with the well-defined momentum,
but which are completely delocalized in space.
So, these are completely different vectors, completely different basis, and
which one to choose is entirely the question of convenience.