Introduction to quantum mechanics, Chemia, Spektroskopia, NMR lecturer
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1 Introduction to quantum mechanics
Quantum mechanics is the basic tool needed to describe, understand and
devise NMR experiments. Fortunately for NMR spectroscopists, the
quantum mechanics of nuclear spins is quite straightforward and many
useful calculations can be done by hand, quite literally "on the back of an
envelope". This simplicity comes about from the fact that although there are
a very large number of molecules in an NMR sample they are interacting
very weakly with one another. Therefore, it is usually adequate to think
about only one molecule at a time. Even in one molecule, the number of
spins which are interacting significantly with one another (
i.e.
are coupled)
is relatively small, so the number of possible quantum states is quite limited.
The discussion will begin with revision of some mathematical concepts
frequently encountered in quantum mechanics and NMR.
0DWKHPDWLFDOFRQFHSWV
1.1.1 Complex numbers
An ordinary number can be thought of as a point on a line which extends
from minus infinity through zero to plus infinity. A
complex number
can be
thought of as a point in a plane; the
x
-coordinate of the point is the
real part
of the complex number and the
y
-coordinate is the
imaginary part
.
If the real part is
a
and the imaginary part is
b
, the complex number is
written as (
a
+
ib
) where
i
is the square root of –1. The idea that
b
a
real
1 (or in
general the square root of any negative number) might have a "meaning" is
one of the origins of complex numbers, but it will be seen that they have
many more uses than simply expressing the square root of a negative
number.
i
appears often and it is important to get used to its properties:
-
A complex number can be
though of as a point in the
complex plane with a real part
(a) and an imaginary part (b).
i
i i i
2
=-´-=-
=´ =-
=´=+
=
æ
1
1
1
3
2
i
i
4
i i
2
2
1
11
ç
÷
æ
ç
i
i
ö
÷
{multiplying top and bottom by }
i
i
i
==
-
i
i
=-
i
i
2
1
The
complex conjugate
of a complex number is formed by changing the
sign of the imaginary part; it is denoted by a *
(
aib aib
+=-
) (
*
)
The square magnitude of a complex number
C
is denoted
C
2
and is
1–1
ö
found by multiplying
C
by its complex conjugate;
C
2
is always real
if
Caib
=+
(
)
CCC
aibaib
ab
2
=´
=+ -
=+
*
(
)(
)
2
2
These various properties are used when manipulating complex numbers:
addition:
multiplication:
division:
(
aib cid ac ibd
a ib
+++ =+++
+´+ = - + +
) (
) (
) (
)
(
) (
c id
) (
ac bd i ad bc
) (
)
(
aib
cid
+
+
)
=
(
aib
cid
+
+
)
´
(
cid
cid
+
+
)
*
*
{multiplying top and bottom by
(
cid
)
*}
(
)
(
)
(
)
=
(
aibcid
cd
+
)(
+
)
*
=
(
aibcid
cd
+
)(
-
)
=
(
ac bd i bc ad
cd
++ -
+
) (
)
(
)
(
)
(
)
2
+
2
2
+
2
2
2
Using these relationships it is possible to show that
(
CDE
´´´ = ´ ´ ´
K
) (
*
C D E
*
*
*
K
)
b
, between the real
axis and the vector joining the origin to the point (see opposite). By simple
geometry it follows that
q
r
a
Re
Re
[
(
aib a
+=
)
]
Im
[
(
aib b
+=
)
]
[1.1]
An alternative representation of
a complex number is to specify
a distance, r, and an angle,
=
r
cos
q
=
r
sin
q
q
.
Where Re and Im mean "take the real part" and "take the imaginary part",
respectively.
In this representation the square amplitude is
(
aib a b
+
)
2
=
2
+
2
=
r
2
(
cos
2
q
+
sin
2
q
)
=
r
2
where the identity cos
2
q
+ sin
2
q
= 1 has been used.
1–2
+
The position of a number in the complex plane can also be indicated by
the distance,
r
, of the point from the origin and the angle,
1.1.2 Exponentials and complex exponentials
The exponential function, e
x
or exp(
x
), is defined as the power series
exp
()
=+ + + +
1
1
2
x
2
1
3
x
3
1
4
x
4
K
!
!
!
The number e is the base of natural logarithms, so that ln(e) = 1.
Exponentials have the following properties
exp
01
=
exp
( )
A
´
exp
( )
B
=
exp
(
A B
+
)
[
exp
( )
A
]
2
=
exp
( )
2
exp
() ( )
´
exp –
A
=
exp
( )
A A
-
=
exp
()
01
=
exp
-=
A
1
exp
exp
()
()
A
B
=
exp
() ( )
A
´ -
exp
B
exp
()
A
The
complex exponential
is also defined in terms of a power series:
exp
()
i
q
=+
1
1
2
() () ()
i
q
2
+
1
3
i
q
3
+
1
4
i
q
4
+
K
!
!
!
By comparing this series expansion with those for sinq and cosq it can
easily be shown that
exp
()
i
q
=
cos
q
+
i
sin
q
[1.2]
This is a very important relation which will be used frequently. For
negative exponents there is a similar result
exp
(
-= -+ -
=
i
q
)
cos
( )
q
i
sin
( )
q
[1.3]
cos
q
-
i
sin
q
q q have been used.
By comparison of Eqns. [1.1] and [1.2] it can be seen that the complex
number (
a + ib
) can be written
cos
( )
-=
q
cos
q and
sin
( )
-=-
sin
(
aib r i
+=
)
exp
( )
q
= (
b
/
a
).
In the complex exponential form, the complex conjugate is found by
changing the sign of the term in
i
q
if
then
Cr i
Cr
=
*
q
exp
()
(
=
-
i
q
)
1–3
x
()
A
A
( )
where the identities
where
r
=
a
2
+
b
2
and tan
It follows that
CC
r
2
=
=
*
exp
() ( )
(
i r
q
exp
-
i
q
=
r
2
exp
i
qq
-
i
)
=
r
2
exp
( )
0
=
r
2
Multiplication and division of complex numbers in the
(
r
,
q
)
format is
straightforward
let
Cr i
=
exp
()
q
and
Ds i
=
exp
()
f
then
1
=
1
=
1
exp
(
-
i
q
)
C Drs
´ =
exp
(
i
( )
q f
+
)
()
Cr i
exp
q
r
C
D
=
r
exp
exp
()
()
i
q
f
=
r
s
exp
()
i
q
exp
( )
-
i
f
=
r
s
exp
(
i
( )
q f
-
)
s
i
1.1.2.1 Relation to trigonometric functions
Starting from the relation
exp
()
i
q
=
cos
q
+
i
sin
q
it follows that, as cos(–
q
) = cos
q
and sin(–
q
) = – sin
q
,
exp
(
-= -
i
q
)
cos
q
i
sin
q
From these two relationships the following can easily be shown
exp
() ( )
q
+-=
exp
i
q
2
cos
q
or
cos
q
=
1
2
1
2
[
exp
() ( )
i
q
+-
exp
i
q
]
[
]
exp
() ( )
q
--=
exp
i
q
2
i
sin
q
or
sin
q
=
i
exp
() ( )
i
q
--
exp
i
q
1.1.3 Circular motion
In NMR basic form of motion is for magnetization to precess about a
magnetic field. Viewed looking down the magnetic field, the tip of the
magnetization vector describes a circular path. It turns out that complex
exponentials are a very convenient and natural way of describing such
motion.
1–4
i
i
Consider a point
p
moving in the
xy
-plane in a circular path, radius
r
,
centred at the origin. The position of the particle can be expressed in terms
of the distance
r
and an angle
y
p
r
and the
y
-
component is
r
×
sinq. The analogy with complex numbers is very
compelling (see section 1.1.1); if the
x
- and
y
-axes are treated as the real and
imaginary parts, then the position can be specified as the complex number
r
×
q
: The
x
–component is
r
×
cos
x
).
In this complex notation the angle
q
A point p moving on a circular
path in the xy-plane.
is called the
phase
. Points with
different angles q are said to have different phases and the difference
between the two angles is called the
phase difference
or
phase shift
between
the two points.
If the point is moving around the circular path with a constant speed then
the phase becomes a function of time. In fact for a constant speed, q
is
simply proportional to time, and the constant of proportion is the angular
speed (or frequency)
q
w
q
=
w
t
is in radians s
–1
. Sometimes it is
convenient to work in Hz (that is, revolutions per second) rather than rad
q
is in radians,
t
is in seconds and
w
×
s
–1
;
) can be thought of as a phase, it is seen that
there is a strong connection between phase and frequency. For example, a
phase shift of q = w
t
will come about due to precession at frequency w for
time
t
.
Rotation of the point
p
in the opposite sense is simply represented by
changing the sign of
q
y
w
:
r
exp(–
i
w
t
). Suppose that there are two particles,
p
p
; assuming that they both start
on the
x
-axis, their motion can be described by exp(+
i
w
t
) and exp(–
i
w
t
)
respectively. Thus, the
x
- and
y
-components are:
w
and the other at –
w
x
p’
x
- comp.
y
- comp.
The x-components of two
counter-rotating points add, but
the y-components cancel. The
resultant simply oscillates along
the x-axis.
p
cos
w
t
sin
w
t
p
’ cos
w
t
-
sin
w
t
It is clear that the
x
-components add, and the
y
-components cancel. All that
is left is a component along the
x
-axis which is oscillating back and forth at
frequency w. In the complex notation this result is easy to see as by Eqns.
[1.2] and [1.3], exp(
i
w
t
) + exp(
- i
w
t
) = 2cos
w
1–5
q
exp(
i
where
the frequency in Hz, n, is related to w by w = 2 pn.
The position of the point can now be expressed as
r
exp(
i
w
t
), an
expression which occurs very frequently in the mathematical description of
NMR. Recalling that exp(
i
and
p
', one rotating at +
t
. In words, a point
oscillating along a line can be represented as two counter-rotating points.
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