INTRODUCTION TO QUANTUM MECHANICS

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Nov 14, 2013 (3 years and 11 months ago)

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1

Lecture
Notes


PH411/511

ECE 598

A. La Rosa

INTRODUCTION TO QUANTUM MECHANICS


PART
-
II

MAKING PREDICTIONS
in

QUANTUM
MECHANICS and
the

HEISENBERG’s
PRINCIPLE

________________________________________________________________

CHAPTER
-
4


WAVEPACKETS

DESCRIPTION OF
A

FREE
-
PARTICLE
’s

MOTION

4
.
1


S
pectral

D
ecomposition of a function

(relative to a basis
-
set)

4
.1
.
A


Analogy between the components of a vector V and spectral
components of a function


4

.
B


周攠sca污爠灲潤rc琠扥twee渠nw漠
灥物潤rc
畮捴楯湳

4
.
1
.
C


䡯w⁴漠晩湤⁴桥hspect牡氠c潭灯湥湴映⁦畮捴楯渠

?


4
.
1
.
䌮a


C
慳a
:

Pe物潤rc⁆畮c瑩潮献

周

e物es⁆潵物e爠r桥潲e

4
.
1⹃⹢

䍡Ce:⁎潮
-
pe物潤rc⁆畮瑩潮s


周攠䙯畲楥爠
䥮瑥杲l

4
⸱.
D


灥p瑲
al

c潭灯
s楴i潮



c潭灬⁶a物r扬b
.

周攠䙯畲楥爠呲rns景牭

4
.
1⹅

䍯牲C污瑩潮t扥bwee渠

汯la汩ed
-
晵湣瑩潮n(
f

)
and
its
spread
-
Fourier (spectral) transform
s

(
F
)

4
.1.
F


The scalar product
in complex variable

4
.
1.
G


Notation

in Terms of Brackets


4
.
2


P
has
e

V
elocity

and G
roup

V
elocity

4
.
2
.
A


Planes


4
.
2
.
B

Traveling Plane Waves and Phase
V
elocity

Traveling Plane Waves (propagation in one dimension)

Traveling Harmonic Waves

4
.
2
.
C

A
T
raveling
Wavepackage

and
its
Group
V
elocity

Wave
p
acket
composed
of two

harmon
ic

waves


Analytical description


Graphical description

Phasor method to analyze
a
wavepacket


Case: wavepacket composed of two waves

Case: A wavepacket composed of several harmonic waves


2


4
.
3

DESCR
IPTION of

a FREE PARTICLE

MOTION

4
.3
.
A

Trial
-
1: A wavefunction with a definite momentum

4
.
3.B



Trial
-
2: A wavepacket as a wavefunction


References:

R. Eisberg and R. Resni
c
k, “Quantum Physics,”

2nd Edition, Wiley,
1985

Chapter 3
.


D
.

Griffiths
,
"Introduction to Quantum Mechanics"
; 2
nd

Edition
, Pearson Prentice
Hall.
Chapter 2
.















3

CHAPTER
-
4


WAVEPACKET

MOTION
DESCRIPTION
of a

FREE
-
PARTICLE

In an effort to reach a better understanding of the wave
-
particle
duality, the motion of a
free
-
particle
will be described
by a
wave
packet




x,t



composed

o
f traveling harmonic waves [
the latter
have the

form
Sin
(
kx
-

t
),
Cos(kx
-

t)
]
.



x,t



]
[
)
(




)
(


k
t
k
kx
k
A
Sin



A wavepacket is a function

whose value
s
,
at a gi
ven time,
are

different from zero
only in a limited spatial region

of
extension


x
.

If a wavepacket of width

x
were to represent a particle,

x

is then
interpreted as the spread spatial location
where the particle may be located (i.e.
there is an uncerta
inty about its “exact”
location.)


4
.
1


SPECTRAL
DECOMPOSITION OF A FUNCTION
(relative to
a basis
-
set of functions)

The approach

of describing a
wave
-
profile

(
x
) as the sum of
harmonic waves

is formally know
n

spectral
Fourier
analysis
.


The Fourier analys
is (based on harmonic waves) is, however, a
particular case of a broader mathematical approach that describes a
given function


as a linear combination o
f

a well defined set of



basis
-
functions

{


1

,


2
,

3

,

}
.


In the partic
ular case that

the basis
-
set is chosen to be composed of
harmonic function
s

then the
Fourier analysis results.
But, in general
,

different
types of basis

sets

do
exist
.
In what follows we will provide a
view of th
is

more general description

since it will al
low us to provide
different
optional
descriptions of quantum mechanics phenomena
.





Spatially

localized pulse

x



(
x
)


Fig.1
Accelerometer for automotive

applications
.


MEMS


4

4
.
1
.
A


Analogy between
the

components

of a vector V

and
spectral
components

of a function


Let’s
consider the

analogy
between the components of a three
dimensional vector, and the
spectral decomposition

of a
n

arbitrary
function

.



Vector
v



Function


v

=
v
1

ê
1
+
v
2

ê
2
+
v
3

ê
3



=
c
1


1
+
c
2


2
+


Spectral
components

Vector components

(1)

The latter means



x


=
c
1


1

x


+
c
2


2

x



+



w
here
{

ê
1
,

ê
2
,

ê
3
}
is a
particular
basis
-
set

w
here
{


1
,


2
,

}
is a
particular
basis
-
set




Vector components

In the expression
(1)
above,



1

,

ê
2

,
ê
3

}

(
2
)

is a set of un
it vectors

perpendicular to each other; that is,

ê
i

ê
j

=

ij

(
3
)


5


ê
2

ê
1

ê
3

v

where

ij


0

if j ≠ i




1




if j = i



How to find the components of
a

vector
v
?



If
,

for example

a vector
v

were expressed as



v

=
3

ê
1

+
7

ê
2
-

2

ê
3
,

then, its component would be given by




ê
1

v


=
3

;
ê
2

v


=
7

; and
ê
3

v


=
-
2





In
a more

general case,


if

v

=
v
1

ê
1

+
v
2

ê
2
+
v
3

ê
3

its comp
onents
v
j

are obtained by



v
j


=

ê
j

v


=
v
j

; for j=1, 2,3




Thus,
v

=


3
1
j
(

ê
j

v
)

ê
j


(4)

Notice the involvement of the scalar product to obtain the
components of a vector. For

the effect of describing the spectral
components of a function we
similarly
introduce
in the following
section

a
type of
scalar product between functions.


4
.
1
.
B

T
he scalar product between two
periodic

functions

Set of base functions
.
In the expression (
1) above, we assume that

{


1

,


2
,

3

,

}

(
5
)

is a
infinite
basis
-
set of
giv
en functions perpendicular to each other.

That is,


i


j

=

ij




But what
would


i


j

mean?



6

To answer this question, let’s consider the particular cas
e where
all

the functions under consideration
are periodic

and real
; let


be the
periodicity of the function
s
;



(
x
+

) =


(
x
)


(
6
)

Definition.

A scalar product between two
periodic
(
but otherwise
arbitrary
)

real
functions


and



is defined as

follows
,










0
)
(

)
(
dx
x
Φ
x
ψ


(Throughout these lecture notes, the symbol








means

“definition

)
.

Rather than
using





a

more common notation is





,
Φ
ψ







0
)
(

)
(
,

dx
x
Φ
x
ψ
Φ
ψ




(
7
)


definition of ―scalar product‖

(
for the
case of real functions
)


In Sec
tion
4
.1
.F below,

we extend th
is

definition to
include

functions
whose values lie in the
complex
variable

domain
.


Orthogonally property
.
As mentioned above, t
he set of
base
function
s

indicated

in (5)

are typically

chosen in such a way as to
have

the follo
wing property,



ij
j
i
j
i
dx
x
x









0
)
(
)
(
,


(
8
)


E
xercise
:
Given the functions
,




(

x
)
=

Cos(
x
) and

(

x
)
=

Sin(
x
)
,


defined over the range (0,2

),

evaluate the scalar product (


).

Answer
:



2
0
)
(


)
(
dx
x
Sin
x
Cos

0





7

Bracket notat
ion
.
Dirac introduced a bracket notation, where the
scalar product is denoted by


Φ
ψ


instead of




,

Φ
ψ

Still t
he parenthesis notation is much more clear and straightforward.
The bracket notation however offers (as
we will see
in the next
chapters
) great flexibility and simplification
to (when properly used)
represent both states and operators (as far as the distinction between
states and operators is implicitly understood)
. But occasionally the
bracket notation will

present difficulties
o
n how to use it. When such
cases arise, we will resort back to the parenthesis notation for
clarification. Since the bracket
notat
ion is so
broad spread

in quantum
mechanics w
e

will frequently use it in this course.


4
.
1
.
C

How to fi
nd the
spectral
components

of a function


?

Given an
arbitra
ry

periodic

function



we wish to express it as a
linear combination of the
periodic
base
functions
j





=
c
1


1


+
c
2



2
+


(
9
)

Using the
scalar product definition given

in (9) we
can
obtain

the
correspond
ing values of the coefficients
c
j
j

in the following manner

(adopting the bracket notation for the scalar product)
,







0
)
(

)
(
dx
x
ψ
x
ψ
c
j
j
j

for j= 1,2,3, …


(1
0
)

Still one question remains: How do the functions


j

look like?

An
swer:



There exist different types of set
s.

T
hey are even defined with very
much generality

in quantum mechanics
, as we will see
when

descri
bing

an

electron traveling in a lattice of atoms

(Chapter
7
).



O
ne particular set is the one composed by
harmonic fun
ctions


8

B
ASE SET

{

Cos

o

,
Cos

1
,

Sin
1
,



Cos
2
,

Sin
2

, …
}




w
here





Cos

o




x


λ
1






Cos

n



x


λ
2
Cos
)
(

/
2

x
n






for
n
=1,

2
,
...







Sin

n


x


λ
2
Sin
)
(
/
λ
2
x
n






for

n

=
1
,

2
,

..


(1
1
)


which are useful to describe any
periodic
function



of period
equal to

.




x

Sin

1

Sin

2

Sin

3

Sin

4

Cos

1

Cos

2

Cos

3

Cos

4

x




Fig. 4.1

Set of harmonic functions defined in expression (11). They are
used as a base to express any periodic function of periodicity equal to

.


It can be directly verified
, using the definition of
sca
la
r product given
in (1
0
),
that
the harmonic functions defined in (11) satisfy


Cos

n

Sin
m

=



0
)
(


)
(
dx
x
Sin
x
Cos
m
n

0








More generally,


9


Cos

n

Sin
m

0





for arbitrary
integ
er
s
m , n;


Cos

n

Cos

m
m
n







for arbitrary
integ
er
s
m , n;


Sin

n


Sin
m

m
n








for arbitrary
integ
er
s
m , n
.

(
1
2
)



4
.1.C.a

Spectral decomposition of periodic functions



The

Series
Fourier Theorem

Using

the base
-
set of harmonic functions defined in (1
1
)

above,
the following theorem results:


A function

of period

can be expressed as,








x


=

A
o

Cos

o

x







1
n
A
n

Cos

n

x







1
n
B
n

Sin

n

x


or simply







=


A
o

Cos

o






1
n
A
n

Cos

n





1
n
B
n

Sin

n


(1
3
)


w
here
the coefficient are given by



A
n

=

Cos

n





dx
x
x
Cos
ψ
)
(

)
(
0
n



n
= 0,1,2, ...




B
n

=


Sin

n






dx
x
x
Sin
ψ
)
(
)
(

0
n



n
= 1,2, .
..

(1
4
)




1
2

3

4

5

n

A
n



x



1
2

3

4

5

n

B
n


Fig. 4.
2

Periodic function


and its corresponding Fourier
spectrum fingerprint


10

Notice



in expression
(1
3
) above

the explicit dependence of the functions
on the variable
x

can be omitted. That is, we can work simply and
directly with the functions
Cos
n

instead of working with the
nu
mbers
Cos
n

x



whenever convenient
.



In the notation for the scalar product we use

Cos

n





and not

Cos

n
(
x
)


(
x
)

. This is to emphasize that the
scalar product is
b
etween functions and not between numbers.



U
sing explicitly the

expression for
Cos

o
=
λ
1

a
nd, according to
(13),
A
0


Cos
o




dx
x
x
Cos
ψ
)
(

)
(
0
o





dx
x
ψ
)
(

1
0




we realize
tha
t
the first term in the Fourier series expansion is
nothing but
the
average value (average taken over one period) of the function

.

That is,







x


=

'
)
'
(
λ
1
0



dx
x
ψ




1
n
A
n

Cos

n

x





1
n
B
n

Sin

n

x



(1
5)



4
.1.C.b

Spectral decomposition of

Non
-
periodic

Functions.

The Fourier Integral

The series Fourier expansion allows the analysis of periodic
functions
, where


specifies the periodicity
.
For

t
he case of non
-
periodic functions a similar analysis is pur
sued by taking the limit
when



.

For an arbitrary function




of period


we have the Fourier series
expansion,






x


=

A
o

Cos

o


x







1
n
A
n

Cos

n


x







1
n
B
n

Sin

n


x



Writing the
ba
se
-
functions

in a more explicit form

(using expression
(11))
,
one obtains
,


11






x


=


1



0
)
(
dx
x




1


n

A
n

2
Cos
)
(
n
/
2
x






1


n

B
n

2
Sin
)
(
n
/
2
x




Cos

n
(
x
)

Sin

n
(
x
)



Since


and all the harmonic functions have period

, we c
an
change the interval of interest (0,

) to (
-


/

2,


/

2) and thus re
-
write,






x


=

1




2
/
2
/
'
)
'
(
dx
x
ψ





1


n

A
n

[
)
n
2

(
x



Cos
2
]





1


n
B
n

[
)
n
2

(
x



Sin
2
]



where


A
n


Cos

n









2
/
2
/
[
)
'
n

2
(
x



Cos
2
]
dx
x
ψ
)
'
(


n
=
1,2, ...


B
n


Sin

n








2
/
2
/
[
)
'
n

2
(
x



Sin
2
]
dx
x
ψ
)
'
(

;

n
= 1,2, ...

Cos

n
(
x
)

Sin

n
(
x
)

Let’s define





2
o
k

and

o
nk
k



(1
6
)

In terms of which the previous
expression

adopts the form,




x

=

1




2
/
2
/
'
)
'
(
dx
x
ψ





1


n

A
n
[
)
n
0


(
x
k
Cos
2

]




1


n

B
n

[
)
n
0


(
x
k
Sin
2

]

where


A
n


Cos

n









2
/
2
/
[
)
n
'
0


(
x
k
Cos
2

]
dx
x
ψ
)
'
(


n
=
1,2, ...


B
n


Sin

n








2
/
2
/
[
)
n
'
0


(
x
k
Sin
2

]
dx
x
ψ
)
'
(

;

n
= 1,2, ...




12

From the (discrete) variable

n

to the (continuum)
k

variable

When




,
the value
0
2




o
k

and
t
he summation
over

n

becomes an Integral of th
e variable
o
nk
k

.
In
a range
k


there
will be

an inte
ger number
o
k
k
/
)
(


of terms
that
in the summation
above
w
ill

have a similar coefficient
A
n
.

It is also convenient to use the index

k


instead of
n
:
A
n


becomes

A
k


k
0

k
=
n
k
0

A
k


k

The
# of terms in this
interval
is
equal to


k / k
0




Fig. 4.
3

Transition of the Fourier component from discrete variable
n

to a

continuum variable
k
.


Thus, as




the last
expression becomes,






x












0
o
k
k

A
k
[

2
Cos
(
kx
)
]








0
o
k
k

B
k

[

2
Sin
(
kx
)

]

(17)



where



A
n









A
k






2
/
2
/
[

2
)
'

(
x
k
Cos
]
'
)
'
(
dx
x



B
n








B
k






2
/
2
/
[

2
)
'

(
x
k
Sin
]
'
)
'
(
dx
x





Replacing the coefficients
A
k

and
B
k

i
n (1
7
)

itself
,


13



(

x

)











0
o
k
k

[

2
'
)
'
(
'
2
/
2
/
)
(
dx
x
kx





Cos
]
[

2
Cos
(
kx
)
]










0
o
k
k

[

2
'
)
'
(
'
2
/
2
/
)
(
dx
x
kx





Sin
]
[

2
Sin
(
kx
)
]



A
k

B
k




Since

o
k
1

2

2


1
,

a further simplification is obtained,



x











0
k

[

1
'
)
'
(
'
2
/
2
/
)
(
dx
x
kx





Cos
]
Cos
(
kx
)







0
k

[

1
'
)
'
(
'
2
/
2
/
)
(
dx
x
kx





Sin
]
Sin
(
kx
)





(

x

)

=


0
dk
[

1
'
)
'
(
'
)
(
dx
x
kx





Cos
]

Cos
(
kx
)








0
dk
[

1
'
)
'
(
'
)
(
dx
x
kx





Sin
]

Sin
(
kx
)


(18)



Or, equivalently



(

x

)

=


1


0
dk
[

1
'
)
'
(
'
)
(
dx
x
kx





Cos
]

Cos
(
kx
)








1


0
dk
[

1
'
)
'
(
'
)
(
dx
x
kx





Sin
]

Sin (
kx
)



A(k)

B(k)



14


Thus,
using

the
BASIS
-
SET


{
Cos
k
,
Sin
k

;





k
0
}





(19)

where

Cos

k
(

x
)


Cos(k
x
)

and

Sin
k
(

x
)


Cos(k
x
)



we have demonstrated that

an arbitrary
periodic
function

(
x
)

can be
expressed as a linear combination of such basis
-
set functions,



(

x

)
=

1


0
A(k)
Cos
k
(
x
)

dk


1


0
B(k)
Sin
k
(
x
)

dk




where the amplitude coefficients of the
harmonic functions components are given by,

A(k)

=


1




'
)
'
(
Cos

)
'
(
dx
kx
x

, and

B(k)

=


1




'
)
'
(
Sin

)
'
(
dx
kx
x



Fourier
coefficients

(
20
)




4
.1.D

Spect
ral
decomposition

in

complex variable
.

The Fourier Transform
.

In expression (
1
8
) above we have







x


=




0
dk
[

1
'
)
'
(
'
)
(
dx
x
kx





Cos
]

Cos
(
kx
)








0
dk
[

1
'
)
'
(
'
)
(
dx
x
kx





Sin
]

Sin
(
kx
)



15


which can be expressed as,






x


=



0
dk

1




{

Cos

)
'

(
x
k
Cos
)
(

x
k
}

'
)
'
(
dx
x










0
dk

1





{
Sin
)
'

(
x
k
Sin
)
(

x
k
}

'
)
'
(
dx
x






(

x

)

=



0
dk

1




{
Cos
)

'

(
)
(
x
x
k

}

'
)
'
(
dx
x





In the expression above
o
ne can identify a
n even function in the
variable
k
,





(

x

)

=



0
dk
[

1




Cos
)

'

(
)
(
x
x
k

'
)
'
(
dx
x

]






(
21
)


Even function in the
variable


k

accordingly
,

we have the following equality







0
dk

[

1




Cos
)

'

(
)
(
x
x
k

'
)
'
(
dx
x

]












0
dk

[

1




Cos
)

'

(
)
(
x
x
k

'
)
'
(
dx
x

]

(
22
)


(notice the different range of integration in each integral).


T
hus
, expression (21) can be re
-
written as,




(

x

)

=
2
1




dk
[

1




Cos
)

'

(
)
(
x
x
k

'
)
'
(
dx
x

]


(
2
3
)


On the other hand, notice the following identity




=

-

i
2
1




dk

[

1




Sin
)

'

(
)
(
x
x
k

'
)
'
(
dx
x

]


(
2
4
)


16

where
i

is the complex number

satisfying
i
2
=

-
1
.

This follows from the
fact that the function within the bracket is an odd function with respect
to the variable
k
.

From (2
3
)

and (2
4
) we obtain





(

x

)

=

2
1




dk




[
Cos
)

'

(
)
(
x
x
k


i

Sin
)

'

(
)
(
x
x
k

]
'
)
'
(
dx
x






e
)
'
(
x
x
k
i







)
(
x


=

2
1




dk




[
e
)
'
(
x
x
k
i


]
'
)
'
(
dx
x



(
2
5
)


Rearranging the terms,





(

x

)



=

2
1




dk

e
kx
i




[
e
'
-
ikx
'
)
'
(
dx
x


]



(

x

)

=


2
1






2
1




[
e
'
-
ikx
'
)
'
(
dx
x


]

e
x
k

i
dk




F(k)



Thus,
using

the
infinite
basis
-
set

of complex func
tions

BASIS
-
SET {
e
k
,






k
}

where
e
k
(

x

)



e
ikx

(26)

an arbitrary function



can be expressed as a linear
combination of such basis
-
set functions,




(

x

)

=


2
1





F(k)

e
i

k

x
dk



Fourier
coefficients

(
2
7
)

Base
-
function
s



17

where the
weight

coefficients

F(k)

of the
complex
harmonic
functions components are given by,

F(k)

=

2
1




e
'

-


x
k
i
'
)
'
(
dx
x



(2
8
)

which is
typically referre
d

to as the
Fourier transform

of the
function

.


In essence, the Fourier formalism associates to a given function

f


its
Fourier
transform

F
.


F

f


4
.1.E

Correlation between localize
d
-
functions

f

= f(x)

and
sprea
d
-
Fourier
(spectral) transform
s


F

=
F(
k
)

A fundamental characteristic in the Fourier formalism is that, it turns
out,


the more localized
is
the function
f,
the broader its

spectral
Fourier transform

F
(k)
; and vice

versa.

(2
9
)





Spatially

corresponding

Fourier

localized pulse



spatial
spectra

Wider
pulse

narrower
Fourier spectra


x

f

(x)

g

(x)

k

k

F

(
k
)

G
(
k
)

x



18

Fig. 4.
4

Reciprocity between a function and i
ts spectral Fourier
transform

Due to its important application in quantum mechanics, this
property will be described in greater detail in the
S
ection
4
.2 below.
It
is worth to emphasize here
, however
,

that the property expressed in
(
2
9
) has nothing to do w
ith quantum mechanics. It is

rather an
intrinsic property of the Fourier analysis of waves. However, by
identifying
(via the de Broglie hypothesis)
some variables in the
mathematical Fourier

description

of waves

with
corresponding
physical variables

charac
terizing a particle (
i.e
.

position, linear
momentum, etc)
, a better understanding of
the
quantum mechanic
al
description of the world

at the atomic level

can be obtained
.


4
.1.
F