hello in this video we're going to take
a look at complex vector spaces
now there are various notations for
vectors the one i'll use in this course
which is particularly convenient for
complex spaces
is what's called dirac notation
denoting vectors like this
so this is some complex vector which we
can write out in full
as elements one two
three down to n where n is dimension of
the space
and a particular convenience of the
notation is that
we write what's called the hermitian
conjugate of a vector
which is the complex conjugate
transposed
in the following form
so in general the Hermitian conjugate is
indicated by a cross like this
and by definition this is
the complex conjugate transposed
and we'll see in the problems that
whether you take the complex conjugate
first or the transpose first doesn't
matter
and we denote this
with a backwards angle bracket like
this
so it's called dirac notation
dirac himself referred to it as bra ket
notation which is
also sometimes used where we define the
following quantities
the hermitian conjugate of the vector is
defined to be a bra
and the vector itself a ket
so that together when we put a bra
followed by a ket
we get a bracket
or bra-ket so the use of this
is that this object naturally defines an
inner product a dot product
between the vector and its complex its
Hermitian conjugate
so let's take a look at that on the next
board
so we have the ket or the vector
|v>; the bra
the Hermitian conjugate of |v> and let's
let's define a bra
<u| which written out component wise
is as follows
that is we've taken the transpose
so a vector is an n by one matrix
like this so its transpose must be a one
by n matrix
so it has one row and n entries
whereas this has n
entries within sorry
n rows and one column and so when we
place
we have the
following
the sum of u_n*.v_n so the elements
we're taking the complex conjugate this
one and the result here must be
a one by one matrix which is just a
complex
scalar so the bracket forms
complex scalars
so compare this to our usual or the
perhaps more familiar vector notation
where we just underline or we could use
bold and so on
but if we take u dagger dot v
where the v dot is the dot product or
the inner product
then we would also get this complex
scalar we get the same thing so that's
all we're doing here it's just a
different notation
a particular convenience of this
notation though is that if we want to
look at the complex conjugate of this
so remember this bracket is a
complex number
a complex scalar if we take the
complex conjugate of that we just get
the following
so the complex conjugate of u inner
product v
or bracket is equal to v inner
product u
and you can just check this explicitly
in terms of the elements
so that's a nice convenience of this
notation this also tells us that if
we take
the inner product of v with itself we
must get
the sum over v_n*.v_n and that's
nothing other than
we just get the norm of the vector v
squared where norm is just a
slight generalization of the
length of the vector so this makes
sense if we take v
dot v for for real vectors we expect
to get the length squared or the
modulus squared
okay so we have
a complex vector space with an inner
product on it or a dot product
and actually we have the following
definition so our definition
a complex linear vector space endowed
with an inner product
in which all vectors are normalizable is
an example of what's called a Hilbert
space
so when we say that the vectors are
normalizable this means that their
norms squared are all finite
so this is the relevant to these complex
vector spaces to quantum mechanics
although we've dealt with wave functions
so far we'll see in an upcoming video
how those fit into this scheme but in
complete generality we can say that
in quantum mechanics
the states live in
Hilbert spaces
okay all right so let's take a look at
matrices acting on our vectors
okay so if we have a matrix M acting on
a vector
|u> in general we expect to get
some other vector let's call it |v>.
M is an n by n matrix
|u> being a vector must be an n by one
matrix
and an n by n matrix acting on n by one
matrix gives us an n by one matrix
we just cancelled it out to the middle
and so this works out so a matrix acting
on a vector
gives us another vector in this
complex space so that's good news
now consider the inner product let's
take another vector |w>
act it on M|u>
well this thing must by definition
then equal
|w> inner product |v> because M|u> is just |v>
and so this thing is a complex scalar
so that's also good news we can act
matrices on our vectors and
we can take inner products and
everything works out as we'd expect
okay so we've taken a look at the inner
product we saw
that so we write the complex
conjugate row vector here conjugate
transpose
multiplied by the vector gives us a
complex scalar
which is the inner product how about
this object
so i've just written the ket |v> on the
left
of the bra <u| well this is an n by one
matrix this is a one by n matrix
and so the result
middle should be an n by n matrix
and it comes out as follows so we have
this
vector to multiply well this matrix to
multiply by this matrix
n by 1 times 1 by n we'll get an n by
n it looks like this
so it's an n-by-n matrix where the first
entry is v1.u1*
second well element one two is
v1.u2*
and so on so this is what we call the
outer product
just as the inner product may be
familiar as the dot product between
two vectors
the outer product may or may not be
familiar as what's called the tensor
product between two vectors
okay so these two objects are extremely
useful in dealing with our complex
vector spaces let's take a look at some
bases
so we can define an orthonormal basis as
follows
it's the set of vectors which we'll call
|e_i> where i
ranges from 1 to n in the n-dimensional
space
and they're defined by the following
fact
the inner product between |e_i> and |e_j>
where remember this is defined as the
hermitian conjugate
of vector |e_i> or ket |e_i>
this inner product is the kronecker
delta, \delta_{ij}
which is defined to be 0 if i doesn't
equal j and 1 if i does equal j
so we can for example sandwich a matrix
between two basis vectors
and we'll simply select out element i j
of the matrix
similarly with vectors we can take any
vector we like and decompose it
into any complete orthonormal basis such
as this
so for example we can write this that is
we can write any vector
as a sum over the basis vectors |e_i>
multiplied by coefficients
where |v_i> is the projection of the
vector v
along the basis direction |e_i> so
that's completely general but then if
you
think about what we're saying
this |v_i> is we want it to be the
projection
of vector v along direction |e_i>
but in our Dirac notation that's
nothing other than the following
that is |v_i> is given by take the
vector |v> and we project it along
the direction |e_i> remember that an inner
product is the projection of one vector
along another
so another way to rewrite this which
looks even neater
in Dirac notation is this
that is all i've brought is i've brought
the ket |e_i> over to the left
this remember is a complex scalar so it
can just pull through here
and so you see that you have the
outer product of the |e_i> with itself
acting on the vector |v> just as a
quick mention of notation
when i draw the outer products between
say u and v
really it's ket
bra like this but it's much easier to
actually write it down as follows
just in terms of how you actually write
it if you draw a cross (X) I think it's
indistinguishable
but i'm not writing a cross here i'm
doing a ket followed by
a bra okay so
in order to prove a very useful
relation
we need to use the following theorem
if we have two matrices A and B and
the inner product well if a acts on v
and we take the inner product with u
and that thing is equal to u inner
product B acting on v
for all u and v arbitrarily then
that means that the matrix A is equal to
the matrix B
this should hopefully make some
intuitive sense
so if we use this theorem we can
prove the following very nice result
rewriting our vector v written out in
its basis |e_i>
rewriting the vector |v> projected into
the basis |e_i>
we can act from the left with some bra <u|
but between the bra and the ket we
can always insert the identity matrix
because the identity matrix acting on
any vector just gives the vector back
again so we can say this is completely
equivalent to the following
well we've just inserted the identity
matrix but then that means that for
arbitrary u and v
we haven't said anything about them so
for u inner product the identity
matrix acting on v
must equal this u inner product product
let me take the u and v outside the sum
here because they're not
they don't have the index i on them
so for any u and v we have that
the identity matrix sat between
u and v
is equal to the sum of i equals one to n
of the outer products of the basis
vectors e i for an orthonormal basis
and so therefore from our theorem we
have the following result
the identity matrix is equal to the
sum from i equals 1 to n the outer
product
of e i where e i are the
basic basis vectors of a complete orthonormal
basis
and this is called the resolution of the
identity and it's a very useful
result it's a simple result but it turns
out to be extremely useful
in quantum mechanics let's just take a
quick look at a very simple example
so for a two dimensional vector space we
can take as our basis vectors these
vectors we don't have to of course this
is a particularly
simple choice and note that this also
works for a complex
two-dimensional vector space provided
that we can
multiply these by complex
scalars which we can and so
we can form this object as follows
so the i just runs from one to two
because it's a two dimensional space
and we just have e1 outer product e1
plus e2 outer product e2
and sticking in the forms we have up
here we get this result
and multiplying these out this is a 2 by
1 matrix
times a 1 by 2 matrix so it's going to
be a 2 by 2 matrix
same for that one and we add those two
matrices together
to find which equals our two-dimensional
identity matrix which is what we
wanted to show all right thank you for
your time