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.