Download Lecture 3

2 Mathemical Excursion
2.1 Complex Numbers
In QM complex numbers play a key role. A complex number z looks like this
z = x + iy
where the imaginary unit i has the property
i2 =
1
p
so in other words i =
1. So, because i2 = 1 does not have a real solution, we call i
imaginary. A complex number has real and imaginary parts, x is the real part and y the
imaginary part. We can visualize a complex number as a point in a two-dimensional
space spanned by the x and y axis like this:
The set of complex number is often denoted by C. We can add complex number, if
z1 = x1 + iy1 and z2 = x2 + iy2 then
z1 + z2 = ( x1 + x2 ) + i ( y1 + y2 ).
So that’s very similar to adding 2d vectors. However, unlike vectors, we can multiply
complex numbers and get another complex number
z1 z2 = ( x1 + iy1 ) ( x2 + iy2 )
= x1 x2 + i 2 y1 y2 + i ( x2 y1 + y2 x1 )
= x1 x2 y1 y2 + i ( x2 y1 + y2 x1 )
An important notion is the complex conjugate of a complex number
z̄ = x
iy
so that’s like z with the opposite imaginary part. Geometrically complex conjugation
is reflection with respect to the real axis. Using the complex conjugate of a complex
number we can compute
zz̄ = z̄z = |z|2 = x2 + y2
So this is geometrically the length of the complex number, or the absolute value. Sometimes instead of the “bar” in z̄ we also use the “star” symbol for complex conjugation,
so z? .
19
Figure 2.1: The complex plane.
20
According to the above we use the pair of numbers ( x, y) to specify a complex number. Geometrically we could also use the length and angle variables in polar coordinates
(q, r ). Theta is calle the phase. We know that
r = x 2 + y2 ,
q = tan
1
y/x
We also know that x = r cos q and y = r sin q so we can write
z = r (cos q + i sin q )
Now it turns out that the series expansion of the cosine is given by
x2
x4
+
2!
4!
cos q = 1
and
sin q =
and because
eiq = 1 +
we can easily check that
x
1!
x6
+ ...
6!
x3
x5
+
+ ...
3!
5!
iq (iq )2
(iq )3 (iq )4
+
+
+
....
1!
2!
3!
4!
eiq = cos q + i sin q
(2.1)
so that any complex number can be written as
z = reiq
You can check for yourself that
z̄ = re
iq
This makes live easy, when you want to operate with complex numbers. For instance
multiplying two of them
z1 = r1 eiq1 z2 = r2 eiq2
then
z 1 z 2 = r 1 r 2 e i ( q1 + q2 ).
You can also use complex numbers and Eq. (2.1) to find an amazing identity
eip + 1 = 0
which relates the fundamental mathematical numbers, 0,1,e,p and i.
21
2.2 Vectorspaces
Let us repeat a bit of mathematics on vector spaces. We have learned that a vector r
is a quantity that has a magnitude and a direction and we visualize it typically as an
arrow in the vector space to which the vector belongs. Mostly we have seen the 2-d
and 3-d vector spaces R2 and R3 which we most readily identify with physical space,
so a vector has components that correspond to the coordinates of the vector space for
example in 2d we have
✓ ◆
x
r=
y
and in 3d we have
0
1
x
r=@ y A
z
Of course we can generalize the concept to more than three dimensions. It’s just no
longer impossible to draw well. In what follows we will discuss what we know using
2d vector spaces. We know that we can do stuff with vectors. For example we can add
them. Let’s say u and v are 2d vectors then the sum
w = u+v
is also a vector in the same space which we get by just adding the components of u and
v. Another thing we can do with vectors is multiply them by a numer a for instance
w = au
is also a vector. We just multiply all components of u with a. Geometrically this means
the new vector w points into the same direction as u it is just shorter or longer depending on the value of a. If a < 0 then w points into the opposite direction. Formally, to
have a nice proper vector space we also must have a 0 vector which, when we add it to
a vector, doesn’t change it
u+0 = u
For every vector u in the vector space we also must have a vector
to u we must get 0.
u + ( u) = 0.
u that if we add it
Some vector spaces are also equipped with a product which gives a number (not a
vector). We call this product a scalar product and denote it by
hu|vi
How do we compute this? We multiply the components and add the results, if u =
( x1 , y1 )t and v = ( x2 , y2 )t then
h u | v i = x1 x2 + y1 y2
22
Clearly
We can also compute
hu|vi = hv|ui .
hu|ui = x12 + y21
So the scalar product can be used to compute the length of the vector u
q
|u| = hu|ui
Geometrically we can also show that
hu|vi = |u||v| cos q
where q is the angle between both vectors. This also shows that
hu|vi = 0
if the two vectors are perpendicular to one another. There are a bunch of other things
that we require to be fullfilled by such a scalar product and you can check for the specific choise above that all of these are fullfilled, for example:
hu + w|vi = hw|vi + hu|vi
and
hau|vi = a hu|vi
2.2.1 Orthonormal basis
A basis of an N dimensional a vectorspace is a set of N vectors that can be used to
represent any vector in the space as a linear combination of vectors in the basis. In R3
for example a set of basis vectors are
0 1
0 1
0 1
1
0
0
@
A
@
A
@
0
1
0 A
e1 =
e2 =
e3 =
0
0
1
These are the vectors that point along the x,y,z directions and any vector v = ( x, y, z)t
can be trivially written as
v = xe1 + ye2 + ze3
Let’s say we have an N-dimensional vector space and a basis b1 , b2 , ..., b N we can write
any vector as
v = Â ai bi
i
where the coefficient ai are components of v along bi . Let’s now focus on bases that are
orthonormal. This means that all basis vectors are orthogonal, such that
⌦
↵
bi |b j = dij
23
Figure 2.2:
where dij is the Kronecker symbol. This thing is 1 when i = j and 0 otherwise. Now we
can compute
⌦
↵
⌦
↵
b j | v = Â ai b j | bi
i
= aj.
So we can write
v=
 h bi | v i bi .
i
2.2.2 Operators
An operator  is like a machine that transforms vectors into other vectors in the vectorspace, so say we have a vector v and and operator  then
w = Âv
is the vector that we get when we apply the operator Â. Think of the operator as a
box where you throw in a vector in the top and another comes out at the bottom. For
example this is an operator
Â
v ! hv|vi (v + a)
(2.2)
where a is some constant vector. Let’s say we are in R2 then the operator R̂q that rotates
a vector by q is also an operator
w = R̂q v
Another example is are projection operator P̂x and P̂y that project vectors onto the x
or y axis, see Fig. 2.2. Or maybe a general projection operator P̂a that projects onto a
fixed operator a. Another interesting operator is the operator Sx for example that takes
a vector v and maps it onto the vector that is the mirror image reflected from the x axis.
Clearly there are a lot of operators one can think of in R2 or also R3 .
2.2.2.1 Linear Operators
One special class that will be important for quantum theory are linear operators for
which we must have.
Â(v + w) = Âv + Âw
Âav = a Âv
24
Let’s see if the operator defined in Eq. (2.2) is linear
Â(v + w) = hv + w|v + wi (v + w + a)
=
=
=
6=
[hv|vi + hw|wi + 2 hv|wi] (v + w + a)
hv|vi (v + a) + hv|vi w + hw|wi (v + a) + hw|wi v + 2 hv|wi (v + w + a)
Âv + Âw + hv|vi w + hw|wi v + 2 hv|wi (v + w + a)
Âv + Âw
so this is a non-linear operator.
How about the rotation R̂q . We an really see that
R̂q (w + v) = R̂q w + R̂q v
geometrically. Rotating the sum of two vectors is the same as rotating them individually
and then adding them. Rotation is linear. The same is true for projection operators P̂a .
2.2.3 Eigenvectors and Eigenvales
An important notion in quantum theory are eigenvectors and eigenvalues of linear operators. Given a linear operator  those are defined by the equation
Âv = lv
with
v 6= 0
A vector v that is a solution to such an equation is called an eigenvector and the number
l is called an eigenvalue. Note that v has to be non-zero but l can be zero. Operators in
R2 can have from zero to an infinite number of eigenvectors. Let’s look at rotation first.
The general rotation operator R̂q has no eigenvector, unless q has certain values. An
eigenvector only gets shrunk or expanded when operated on but still points in the same
direction. Note also that if v is an eigenvector then so is av so it’s really a eigen-”ray”.
How about the reflection operator Ŝx that reflects vectors across the x-axis. Clearly all
vectors that lie along the x-axis are eigenvectors, for example if we have v = a(1, 0)t
we have
Ŝx v = v
so their eigenvalue l = 1. But also all vectors along the y-axis are eigenvectors, if
v = a(0, 1)t we have
Ŝx v = v
so for those vectors l = 1.
How about the projection operator P̂x . This one also has the two eigenvectors v1 =
(1, 0)t and v2 = (0, 1)t with
P̂x v1 = v1 P̂x v2 = 0,
so, very similar to the reflection operator except that one eigenvalue is zero. This is in
general the case. All projection operators have eigenvalues that are either 0 or 1. The
25
Figure 2.3:
also have another interesting property. If I take a general projection operator Pa and
apply it to a vector v getting
w = Pa v
then w is parallel to a, see Fig. 2.3. Which means that if I compute
Pa w = w
the new vector is an eigenvector of P̂a . That also means that
Pa w = Pa Pa v = Pa v
so that
Pa2 = Pa
which we write as a multiplication of operators but at this stage we mean the successive
application of Pa .
This is a general feature of projectors.
2.2.4 Multiplication of operators
If we have two operators  and B̂, we can investigate what happens if we first apply
one than the other and compare to the reverse situation. So let’s compute
 B̂v = w1
and
B̂ Âv = w2
In general we have
w1 6 = w2
For example if we rotate a vector and then project to the x-axis we get something that is
different from when we first project and then rotate. For example if we have the rotation
operator R̂90 and the projection operator P̂x and start with the vector v = (1, 1)t we
have
R̂90
P̂x v = (1, 0)t
p
p
P̂x v = (1/ 2, 1/ 2)t
R̂90 v = ( 1, 1)t
P̂x R̂90 v = ( 1, 1)t
So clearly the combined operations aren’t equal
R̂90 P̂x 6= P̂x R̂90
26