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Institute of Theoretical Physics
Course: Quantum mechanics 05S.624.106
Prof. Dr. Reinhard ALKOFER
Karl-Franzens-University Graz - Austria
Bacchelor Thesis:
”Harmonic oscillator”
Huber Oliver
9811289
033 619 411
Field of Studies:
Environmental System Sciences with main focus on Physics
Graz, 07.07.2005
Harmonic oscillator
Huber Oliver
Contents
1
2
Prologue............................................................................................................................................ 3
The different views of a harmonic oscillator ........................................................................ 3
2.1 Classical mechanics ............................................................................................................. 3
2.2 Quantum mechanics ........................................................................................................... 5
2.2.1 Introduction .................................................................................................................... 5
2.2.2 Transition from classical to quantum mechanics .............................................. 6
2.2.3 The time-independent Schrödinger equation – stationary states ................. 7
2.2.4 Method of Dirac ............................................................................................................. 7
2.2.4.1
Bras and Kets ......................................................................................................... 7
2.2.4.2
Ladder operators and eigenvectors ................................................................. 9
2.2.5 Ground state in position space - Power Series Method ................................. 13
2.2.6 Excited states in position space ............................................................................. 15
2.2.7 Dynamics of the harmonic oscillator .................................................................... 17
3 Visualization with Mathematica ............................................................................................. 17
3.1 Visualization of the classical motion ............................................................................ 18
3.2 Visualization of harmonic oscillator probability density........................................ 18
3.3 Visualization of the oscillating state “0+1” ................................................................. 19
3.4 Visualization of Coherent state ...................................................................................... 20
3.5 Visualizations of an anharmonic oscillator ................................................................ 21
4 Summary and conclusions ...................................................................................................... 23
5 Sources ........................................................................................................................................... 24
5.1 Bibliography .......................................................................................................................... 24
5.2 Figures .................................................................................................................................... 24
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Harmonic oscillator
1
Huber Oliver
Prologue
This thesis is intended to provide an introduction to the reasoning and the formalism
of modern physics. This will be exemplified using the harmonic oscillator and
discussing especially the differences in its treatment at the level of classical physics
versus quantum mechanics. It shows the principles and the formalism. Attached to
this paper is an electronical version (qm.htm) with some extensions.
2
The different views of a harmonic oscillator
2.1 Classical mechanics
The harmonic oscillator is among the most important example of explicit solvable
problems, both in classical and quantum mechanics. An example is given by the
movement of atoms in a solid body. A harmonic oscillator describes this construct. If
the atoms are in equilibrium then no force acts. If one moves an atom out of this
stable position, a force f (x) results. Figure 1 shows this behaviour.
f ( x)   f ( x)  k x  m a  m x
(1)
| f ( x) ||  f ( x) |
(2)
Equation (1) relates the force exerted by a spring to the
distance it is stretched. k is the spring constant and x is
the extension of the spring. The negative sign indicates
that the force exerted by the spring is in direct opposition
to the direction of displacement. m is the mass. The
harmonic oscillator can be pictured as a pointlike mass
attached to a spring. The spring is idealized in the sense
that it has no mass and can be stretched infinitely in both
directions. f (x ) is a force and as such it Newton’s law:
f ( x)  m a  m x.
Figure 1
This force could be expanded by a Taylor series:
f ( x)  f ( x0 )  i 1
n
f (i ) ( x0 )
( x  x0 ) i
i!
(3)
The Taylor series of an infinitely often differentiable real (or complex) function f defined on an open
interval (x − x0, x + x0) is a power series.
In equilibrium position the force f (0)  0 and for little extensions of the spring
Hooke’s law f ( x)   | f ' (0) | x  k x holds. k describes the stiffness of the spring. The
harmonic oscillator is defined as a particle subject to a linear force field in a
potential. The force f (x) can be expressed in terms of a potential function V . A
potential function like equation (5) returns for example the potential energy an object
with a position x has.
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Harmonic oscillator
f ( x)  k x  
Huber Oliver
d
V ( x)  m x
dx
(4)
k x2
V ( x) 
(5)
2

The generalization to higher dimensions is straightforward. With x ( x1 ,..., xn ) we have:



f ( x)  k x V ( x)
(6)


xx
V ( x)  k
(7 )
2
Figure 2 shows the harmonic oscillator potential in one
and two space dimensions with spring constant k  1 .
Figure 2
Now we’d like to build out of (4) and (5) the expression of the total energy term. We
know and can see in (8) that the potential V (x ) is the antiderivative of the force f (x) :
V ( x)    f ( x ' ) dx '
(8)
x
If we multiply (1) and (4) with x we obtain:
m x x  f ( x) x  
d
V ( x) x
dx
(9)
This we also can write as:
d m 2
d
( x )   V ( x)
dt 2
dt
(10)
The integration of (10) gives us a new constant H, which by separation of the
equation is called the Hamilton function:
H
m 2
x  V ( x)
2
(11)
The expression (11) reflects the total energy term.
p
the momentum)
2m
energy.
m 2
x or (expressed as a function of
2
2
is the kinetic energy and the other one V (x ) is the potential
The harmonic oscillator is described usually by this Hamiltonian function:
p2 k 2

 x
(12)
2m 2
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And classically we solve such a problem like we have, with the Hamiltonian equation:
 p
x 

(13)
p m
 p  m x
(14)

 k x
x
 m x  k x
p  
(15)
(16)
The result is a linear homogeneous differential equation with the oscillatory solution
which expresses the motion (18) or the current amplitude (17):
x(t )  a Sin ( t  b)


p(t )  m x (t )  m v  m  a Cos( t  b)
(17)
(18)
Where a and b have to be determined from given initial conditions x(0)  x0 and
p(0)  p0 . Hence the classical motion is an oscillation with angular frequency  . The
spring constant k determines this oscillator frequency:
k
k

 2 
(19)
m
m
k  m 2
(20)
which is independent from the amplitude.
Figure 3 shows the movement of a pointlike mass between two
springs with extension x0  r . As we can see the oscillating
sphere follows the vertical projection of the circular motion in
direction x . The motion takes place between the turning points
x   r   x0 .
Figure 3
2.2 Quantum mechanics
2.2.1 Introduction
Quantum mechanics emerged in the beginning of the twentieth century as a new
discipline because of the need to describe phenomena, which could not be explained
using Newtonian mechanics or classical electromagnetic theory.
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Like our introducing example of an atom in a solid body vibrates somewhat like a
mass on a spring with a potential energy that depends upon the square of the
displacement from equilibrium. The energy levels are quantized at equally spaced
values.
The harmonic oscillator is the prototypical case of a system that has only bound
states: All states remain under the influence of the force field for all times; no state
can escape toward infinity. Although such a system does not exist in nature, the
harmonic oscillator is often used to approximate the motion of more realistic systems
in the neighbourhood of a stable equilibrium point. The quantum harmonic oscillator
is the foundation for the understanding of complex modes of vibration also in larger
molecules, the motion of atoms in a solid lattice, the theory of heat capacity, etc. In
real systems, energy spacings are equal only for the lowest levels where the potential
is a good approximation of the "mass on a spring" type harmonic potential. The
anharmonic terms which appear in the potential for a diatomic molecule are useful
for mapping the detailed potential of such systems.
In quantum mechanics eigenvalues and eigenfunctions of operators are relevant.
Following we derive the eigenfunction and eigenvalue of the Hamiltonian operator.
2.2.2 Transition from classical to quantum mechanics


We build now an momentum operator p and a position operator x and apply the
substitution rules:
Plank ' s const.
d

p : p  i h , h....
dx
2

x : x
to the classical Hamiltonian
Hamiltonian operator:
 p 2  m  2
H

x
2m
2
(21)
(22)
function.
We
obtain
the
quantum-mechanical
(23)
which acts on a square-integrable wave function  .  ( x, t ) represents a wave
function and associates the statistical description of an experimental output.
The transition to quantum mechanics takes place by substitution of the dynamic


variables x  x , p  p to operators.
In our experimental observation which led to the concepts of quantization, the
fundamental equation describing quantum mechanics, is the Schrödinger equation
(24). The time evolution of a state of a quantum harmonic oscillator is then described
by a solution of the (time-dependent) Schrödinger equation:
ih

d
 ( x, t )  H  ( x, t )
dt
(24)
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2.2.3 The time-independent Schrödinger equation – stationary states
The time–independent Schrödinger equation is the eigenvalue equation of the

Hamiltonian operator H . In (24) we insert the ansatz of the wave function:
( x, t )  ( x) e ( it )
(25)
( x, t )  ( x) e
(
 iEt
)
h
(26)
This will lead to the time-independent Schrödinger equation:


H( x)  E( x)
(27)
The states (25) are called stationary states, because the corresponding probability
densities | ( x, t ) | 2 | ( x) | 2 are time-independent. The normalizing condition
(  dx |  ( x) | 2   ) will restrict the possible values of energy E .
2.2.4 Method of Dirac
2.2.4.1
Bras and Kets
Expression (28) is apparently not explicit time-dependent and the solution of the
time-dependent Schrödinger equation for the state vector is:


H |   E |  
(28)
In (28) we can see a notation which was established by P. A. M. Dirac and is called
the bra-ket notation and is a popular scheme to describe quantum mechanic
phenomena. I will describe now the basics to understand the following mathematical
expressions.
We shall begin to set up the scheme by dealing with mathematical relations between
the states of a dynamical system at a fixed time, which relations will come from the
mathematical formulation of the principle of superposition. The superposition
process is a kind of additive process and implies that states can in some way be
added to give new states. The states must therefore be connected with mathematical
quantities of a kind which can be added together to give other quantities of the same
kind. The most obvious of such quantities are vectors. Ordinary vectors, existing in
space of a finite number of dimensions, are not sufficiently general for most of the
dynamical systems in quantum mechanics. We have to make a generalization to
vectors in a space of an infinite number of dimensions, and the mathematical
treatment becomes complicated by question of convergence. For the present,
however, we shall deal merely with some general properties of the vectors, properties
which can be deduced on the basis of a simple scheme of axioms, and questions of
convergence and related topics will not be gone into until the need arises.
It is desirable to have a special name for describing the vectors which are connected
with the states of a system in quantum mechanics, whether they are in a space of a
finite or an infinite number of dimensions. We shall call them ket vectors, or simple
kets, and denote a general one of them by a special symbol |  . If we want to specify
a particular one by a label, A say, we insert it in the middle, thus | A  . Ket vectors
may be multiplied by complex numbers and may be added together to give other ket
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vectors, e.g. from two ket vectors | A  and | B  we can form c1 | A  c2 | B | C  , say,
where c1 and c2 are any two complex numbers. We may also perform more general
linear processes with them, such as adding an infinite sequence of them, and if we
have a ket vector | x  , depending on and labelled by a parameter x which can take
on all values in a certain range, we may integrate it with respect to x, to get another
ket vector  | x  dx | X  say. A ket vector which is expressible linearly in terms of
certain others is said dependent on them. A set of ket vectors are called independent
if no one of them is expressible linearly in terms of the others.
We now assume that each state of a dynamical system at a particular time
corresponds to a ket vector, the correspondence being such that if a state results
from the superposition of certain other states, it’s corresponding ket vector is
expressible linearly in terms of the corresponding ket vectors of the other states, and
conversely. Thus the state X results from a superposition of the states A and B .
Whenever we have a set of vectors in any mathematical theory, we can always set up
a second set of vectors, which mathematicians call dual vectors. The procedure will
be described for the case when the original vectors are our ket vectors. Suppose we
have a number N which is a function of a ket vector | A  , then to each ket vector
| A  there corresponds one number N , and suppose further that the function is a
linear one, which means that the number corresponding to | A   | A'  is the sum of
the numbers corresponding to | A  and to | A '  , and the number corresponding to
c | A  is c times the number corresponding to | A  , c being any numerical factor.
The number N corresponding to any | A  may be looked upon as a scalar product of
that | A  with some new vector, there being one of these new vectors for each linear
function of the ket vectors | A  . The new vectors are of course, defined only to the
extent that their scalar products with the original ket vectors are given numbers. We
shall call the new vectors bra vectors, or simple bras, and denote a general one of
them by the symbol  | , the mirror figure of the symbol for a ket vector. The
specification of a particular one is the same as with kets. The scalar product of a bra
 B | and a ket vector | A  will be written  B | A  . As a juxtaposition of the symbols
for the bra and ket vectors, that for the bra vector being on the left, and the two
vertical lines being contracted to one for brevity. One may look upon the symbols 
and  as a distinctive kind of brackets. A scalar product  B | A  now appears as a
complete bracket expression and a bra vector  B | or a ket vector | A  as an
incomplete bracket expression. We have the rules that any complete bracket
expression denotes a number and any incomplete bracket expression denotes a
vector, of the bra or ket kind according to whether it contains the first or second part
of the brackets. A bra vector is considered to be completely defined when its scalar
product with every ket vector is given, so that if a bra vector has its scalar product
with every ket vector vanishing, the bra vector itself must be considered as vanishing.
The bra vectors, as they have been here introduced, are quite a different kind of
vectors from the kets, and so far there is no connection between them except for the
existence of a scalar product of a bra and a ket. The relationship between a ket and
the corresponding bra makes it reasonable to call one of them the conjugate
imaginary of the other. Our bra and ket vectors are complex quantities, since they
can multiplied by complex numbers and are then of the same nature as before, but
they are complex quantities of a special kind which cannot be split up into real and
pure imaginary parts. The usual method of getting the real part of a complex
quantity, by taking half the sum of the quantity itself and its conjugate, cannot be
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applied since a bra and a ket vector are of different natures and cannot be added
together. To call attention to this distinction, we shall use the words ‘conjugate
complex’ to refer to numbers and other complex quantities which can be split up into
real and pure imaginary parts.
In ordinary space, from any two vectors one can construct a number – their scalar
product – which is a real number and is symmetrical between them. In the space of
bra vectors or the space of ket vectors, from any two vectors one can again construct
a number – the scalar product of one with the conjugate imaginary of the other – but
this number is complex and goes over into the conjugate complex number when the
two vectors are interchanged. We shall call a bra and a ket vector orthogonal if their
scalar product is zero, and two bras or two kets will be called orthogonal if the scalar
product of one with the conjugate imaginary of the other is zero. Further, we shall
say that two states of our dynamical system are orthogonal if the vectors
corresponding to these states are orthogonal. The length of a bra  A | or of the
conjugate imaginary ket vector | A  is defined as the square root of the positive
number  A | A  . When we are given a state and wish to set up a bra or ket vector to
correspond to it, only the direction of the vector is given and the vector itself is
undetermined to the extent of an arbitrary numerical factor. It is often convenient to
choose this numerical factor so that the vector is of length unity. This procedure is
called normalization and the vector so chosen is said to be normalized. The foregoing
assumptions give the scheme of relations between the states of a dynamical system
at a particular time. The relations appear in mathematical form, but they imply
physical conditions, which will lead to results expressible in terms of observations.
For instance, if two states are orthogonal, it means at present simply a certain
equation in our formalism, but this implies a definite physical relationship between
the states, which soon we will enable us to interpret in terms of observational results.
2.2.4.2
Ladder operators and eigenvectors
In (28) we can see that the solution of the state vector of the time-dependent
 
Schrödinger equation is determined by the eigenvalues and eigenvectors of H . H is




selfadjoint and quadratic in p and x . Also x and p are selfadjoint operators. Sure

we can bring H into the form:
 

H  c0  A A
(29)


The operator A and the adjoint operator A  will be determined like:



A   ( x   p)



A    * ( x   * p)


 





A A   |  | 2 ( x   * p)( x   p) |  | 2 ( x 2  |  | 2 p 2   xp   * px )
with the commutator relation:
 
  
[ x , p]  xp  px  i





 xp   * px   xp   * xp  i * (    * ) xp  i *



 ! 
A A  |  | 2 x 2  |  | 2 |  | 2 p 2  i * |  | 2  |  | 2 (    * ) xp  H  c0
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By comparison of coefficients with equation (23), we have the following solution:
|  |2 
 2m
2
|  |2 |  |2 
1
2m
   *
 2m



|  |2 


(30)
2
1
(31)
m 2
2
i
m
(32)
 

The result of this is for A , A  and H :


 2 m  ip
A
(x 
)
2
m


 2 m  ip
A 
(x 
)
2
m
   
A A  H 
2
    
H  A A
2
(33)
(34)
(35)
(36)
With this the Hamiltonian operator takes shape:
 

A A 1
H   (
 )
 2
(37)
Equation (33) and (34) represent the ladder operators. The commutator of the leader
operators is:


 
m 
p
p

[ A, A ] 
[( x  i
), ( x  i
)]
2
m
m
m  
i
i
i
 
 
 

([ x , x ]  (
)( 
)[ p, p] 
([ x , p ]  [ p, x ])
2
m
m
m
 
 
[ x , x ]  0, [ p, p ]  0
The commutator relations are:
 
[ A, A  ]  1
 
 
[ A, A]  [ A  , A  ]  0
(38)
(39)

Now we define a new operator N . Which is defined:
  
N  A A
(40)
With this new operator we are able to simplify our Hamiltonian operator:
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
 1
H   ( N  )
(41)
2


N and H are commutating and have therefore the same eigenvectors.

N | n  n | n 


1
H | n   (n  ) | n 
2

1
Here we can see that H has the eigenvalues  ( n  ) . The question is now, what
2
values n can have. To answer the question we have to consider the commutator



relations of N with A and A  :
  
 
     
  
 
[ N , A  ]  [ A  A, A  ]  A  ( AA  )  A  A  A  A  ( A  A  1)  A  A  A 
   
   
 A A A  A  A A A  A
  
 
   
 

 
[ N , A]  [ A  A, A]  A  AA  ( AA  ) A  A  AA  ( A  A  1) A 
    

 A  AA  A  AA  A   A
Hence:

 
[ N , A  ] | n  A  | n 



NA  | n   A  n | n  A  | n 


NA  | n  (n  1) A  | n 
Analog:

Na | n  (n  1)a | n 

That means, if | n  is the eigenvector of N to the eigenvalue n then:


A  | n  is eigenvector to the eigenvalue (n+1) and A | n  is eigenvector to the
eigenvalue (n-1).


We call A  the generator operator and A the annihilator operator. We obtain:

A | n   | n  1 
(42)
It applies:
 
 n | A  A | n  *   n  1 | n  1 |  | 2
(43)
and
 

 n | A  A | n  n | N | n  n  n | n  n
(44)
It means that the normalizing factor  have to fulfil |  | 2  n . We choose   n and
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insert a possible phase factor in definition of | n  . The result is:

A | n  n | n  1 
(45)

Analogous reasoning for A  | n  :

A  | n   | n  1 

 n | AA  | n |  | 2
!


 n | AA  | n  n | 1  N | n  n  1  |  | 2

A  | n  n  1 | n  1 
(46)

We can now begin with any eigenstate | n  and use the operator A repeatedly.

A | n  n | n  1 

AA | n  n(n  1) | n  2 

A m | n  n(n  1)( n  2)...( n  m  1) | n  m 
(47)

So we get the eigenstates | n  m  to decreasing eigenvalues ( n  m) of N . This means
that we can generate negative eigenvalues, but they are:
 

m  m | N | m  m | A  A | m   |  ||  || 2  0
(48)
If n is an integer, then the series (45) aborts with a | n  0  0 . If there would exist
non integer values in the considered space the annihilator operator would generate
negative eigenvalues, which are not allowed. That’s why just integer values can be
used.

The eigenvalues of N are:

N | n  n | n 
n  N 0
(49)
And finally we find our eigenvalues of the harmonic oscillator:
1
En   (n  )
2
n  N0
Figure 4
In ground state ( n  0 ) the particle has the

zero point energy
. For the quantum case
2
is the so-called zero-point vibration of the
n  0 ground state. This implies that
molecules or atoms are not completely at rest,
page 12 of 24
(50)
Harmonic oscillator
Huber Oliver
even at absolute zero temperature. The explanation is offered by Heisenberg’s
principle of uncertainty. The energy of a harmonic oscillator is quantized in units of
 . The energy levels of the quantum harmonic oscillator are given by (50).
The oscillator transition is given by n  1. The interval width between the close-by
energy level has the same value. The system has thus an equidistant spectrum.
2.2.5 Ground state in position space - Power Series Method
The probability amplitude to find a quantum mechanical particle at a position x if it
is in eigenstate | n  is:
 x | n  n ( x)
(51)
The wave function in ground state 0 ( x) can be calculated as follows. We know that

A | 0  0 hence:



 x | A | 0  0    x | A | x '  x ' | 0  dx '    x | A | x '  0 ( x ' )dx '
We need the matrix elements of a :

m
i


 x | A | x ' 
( x | x | x '  
 x | p | x ' )
2
m
 '
'
 x | x | x   ( x  x ) x
 d

 x | p | x ' 
 (x  x' )
i dx
This gives us:

0  x | A | 0 



m
i  d
( ( x  x ' ) x 
( )  ( x  x ' )) 0 ( x ' )dx '

2
m i dx
m

d
( x0 ( x) 
 ( x  x ' )0 ( x ' )dx ' )

2
m dx
m
 d
( x0 ( x) 
 ( x  x ' )0 ( x ' )dx ' )

2
m dx
m
 d
( x0 ( x) 
 ( x))
2
m dx 0
d
m

0 ( x)  
x0 ( x)
dx


Ordinary differential equations first orders have explicit solutions. The solution of
this differential equation can be calculated via a power series ansatz:

0 ( x)   c n x n
n 0
The insertion in our differential equation gives us:
page 13 of 24
Harmonic oscillator
Huber Oliver
m 
c n x n 1  0

 n 0
n 1
m

 c1  l 1 cl 1 (l  1) 
cl 1 ) x l  0


 cn nx n1 
This equation has to be individual fulfilled for every power, because the equation has
to be right for any x . {x l } forms a linear independent base. We follow:
c1  0
cl  
m 1
cl  2 ;
 l
l2
odd terms:
m 1
c3  
c1  0
 3
c5  c3  0
even terms:
m 1
c2  
c0
 2
m 1
m 2 1 1
c 2  (
)
c0
c4  
 4

42
m n 1
1
1
m n
1
m n 1
c 2 n  (
)
... c0  (
) n
c0  (
)
c0

2n 2n  2 2


n!
2 n(n  1)( n  2) ...1
 c 2 n 1  0
The solution of the differential equation is thus:
m n
)

 (
2n
2

0 ( x)   x c2 n 
( x 2 ) n c0
n
!
n 0
n 0
With the characteristically length x0 :

of the
m
oscillator we can transform:
0 ( x)  c0 e

x2
2 x02
When we normalize 0 ( x) we get:
2

 | 0 ( x) | dx  c

2
0

 c0 
e

x2
x02
!
dx  c 02 x 0   1

1
 x 02
4
and finally we obtain to the ground state:
0 ( x) 

1
4
x
2
0
e
x2
2 x02
(52)
Figure 5
Figure 5 above shows a comparison of the probability density of the harmonic oscillator.
The dashed line represents the classical solution. The other one shows the quantum mechanic result
for: n  1, n  2, n  20
In chapter 3.1 you will see a Mathematica code which plots this behaviour.
page 14 of 24
Harmonic oscillator
Huber Oliver
2.2.6 Excited states in position space
The nth stimulated state can be generated by n-time appliance of the generator
operator (34) to the ground state (52). We will use this to get other stimulated states
in position space.
With:
1  n
|n
(A ) | 0 
(53)
n!
we build:

1
 x | ( A ) n | 0 
n!

n
1 m 2
p n


(
)  x | (x  i
) |0
m
n! 2


n
n
1 m 2
p n '
1 m 2
p n '


'
'

(
)   x | (( x  i
) | x  x | 0  dx 
(
)   x | (( x  i
) | x  0 ( x ' )dx '
2

m

2

m

n!
n!
( 53)
n ( x) : x | n  
with the matrix elements of position operator (54) and momentum operator (55) we
can go on:

 x'

 x'
 
 
| x | x  x  ( x '  x )
 
 d
 
| p | x)  
 ( x '  x)
i dx
(54)
(55)
m 2
 d n
) (x 
)   ( x  x ' )0 ( x ' )dx '
m dx
n! 2

1

1

1
n
(
m 2
 d n
(
) (x 
) 0 ( x)
m dx
n! 2
n!
n
(2)

n
2
n
x0 ( x  x02
d n
) 0 ( x)
dx
Now we insert the function for the ground state (52) and substitute z 
get what we want – stimulated states in position space (56):
page 15 of 24
x
and finally
x0
Harmonic oscillator



1
(2 (
n!
1
n!
n
2
2

n
2
d
)n
x
d( )
x0

1
4
x
2
0
e
x2
2 x02
z
x
x0

z2
d 
[( z  )e 2 ] x
z
dz
x02
x0
1
4
n ( x) 
x

x0
Huber Oliver
1
n!
2

n
2
1
4
x02
[e

z2
2
hn ( z ) ]
z
(56)
x
x0
The position probability densities | n ( x) | of the
2
lowest oscillator eigenstates drawn at a height
corresponding to their energy. The horizontal
lines show the values of the energy of the
eigenstates. The part of a horizontal line inside
the potential curve is the classically allowed
region for a classical particle with that energy.
Figure 6
eigenfunctions of the harmonic oscillator for
y
x

x0
n  0...5 with:
mh
x
2
Figure 7
Where hn (z ) are real polynomial from order n in z . They are called hermit polynomial
(57) and have even and odd parity: hn ( z )  (1) n hn ( z ) .
x2
2
2
x
 n

hn ( x)  e ( 2 A ) | x0 1 e 2
e e
x2

x2
2
x2
2
d
(x  ) n e 2 ex
dx
(57)
Concluding we have:
 The eigenvalue problem can be directly



 2 d 2 m 2
H  E  (

x ) ( x)  E ( x)
2
2m dx
2
page 16 of 24
solved
in
position
space:
Harmonic oscillator


Huber Oliver
The probability density has n zeros.
The Maxima are twice as high as in classical case.
2.2.7 Dynamics of the harmonic oscillator
In this chapter we will like to analyze the time series of the wave function in the
potential of a harmonic oscillator. At time t  0 the state is given by: |   . And at
later moment t  0 the state is given by:
( x, t ) : x |  t  x | e
i
t

| 0 
(58) .
The Hamiltonian operator is not explicit time-dependent. We expand the ground state
|  0  in series of the eigenstates of the harmonic oscillator.

|  0   c n | n 
n 0
c n  n |  0   n ( x) 0 ( x)dx
After insertion in (58):

 ( x, t )   c n  x | n  e
t
i En

n 0

1
 i t ( n  )
2
  c n n ( x)e
n 0
e
i
t 
2
 c  ( x )e
n 0
n
itn
(59)
n
The matter of fact that the energy difference E n  E n' is an integer factor of our energy
unit  will lead that the wave function is time periodic. The period T 
2

is
equivalent to the classical oscillation period.
i
( x, t  T )  e e
i
t 
2
 c  ( x) e
n 0
n
n
itn i 2 n
e
 ( x, t )
(60)
In expression (60) we can see that the balance point of the wave function periodically
oscillates.
3 Visualization with Mathematica
The following examples should graphically illustrate quantum mechanic behaviours.
Some of the Source codes is adapted from the book: Visual Quantum mechanics [5].
The code is surely executable with Mathematica 4.x. Please take note that you have
to install some packages if you want to execute 3.3 and 3.4 or you would like to make
some own illustrations using some prepared functions. The VQM packages contain
tools for the visualization of complex valued functions (wave functions) and the
numerical solution of the Schrödinger and Dirac equation, etc.
page 17 of 24
Harmonic oscillator
Huber Oliver
The VQM packages encompass the functionality of the packages distributed with
Visual Quantum Mechanics [5] from Bernd Thaler or you also will find it on the
attached electronic version on CD (VQM_packages.zip).
Further instructions are given at http://vqm.uni-graz.at.
3.1 Visualization of the classical motion
The code shows the oscillating motion of the particle. In the attached electronically
version of this publication you will see the different colours. The blue curve describes
the position x  x (t ) of the mass point as a function of time t . The green curve is the
velocity v (t ) . As I showed in chapter 2.1 the force can be described by a potential
function V (x ) which is shown here as a black parabola.
The classical particle in the field of the harmonic oscillator can have positive energy.
In quantum mechanics, the energy is a discrete quantity.
(* This command generates a movie
showing the motion of the classical
harmonic oscillator
*)
6
5
Do[
Show[
{ParametricPlot[{{-Cos[t],t},
{ Sin[t],t}},{t,0,2*Pi},
PlotStyle->{RGBColor[0,0,1],
RGBColor[0,1,0]},
DisplayFunction->Identity],
Plot[x^2/2-1/2,{x,-1.3,1.3},
DisplayFunction->Identity]},
Graphics[{{PointSize[0.05],Point[{-Cos[s],0}]},
{PointSize[0.05],Point[{-Cos[s],s}]}}],
AspectRatio->1.5,
Ticks->{None, Automatic},
DisplayFunction->$DisplayFunction],{s,0,2*Pi-Pi/24,Pi/24}]
4
3
2
1
Figure 8
3.2 Visualization of harmonic oscillator
probability density
The probability density of the harmonic oscillator can be plotted using the commands
listed below. The quantity n is the order of the eigenfunctions and represents the
number of energy quanta the oscillator contains, 'ulimit' determines the horizontal
range of the plot, and 'A' is the normalization factor. With the following Mathematica
commands below we can plot the probability density.
n 1
0.5
0.4
0.3
P
n=1;
ulimit = 10;
f[u_] := Exp[-u^2/2]*HermiteH[n,u]
A = N[Integrate[f[u]^2, {u,-Infinity, Infinity}]];
Plot[(1/A)*f[u]^2, {u,-ulimit,ulimit},
PlotRange->{0,0.5},
Frame->True,
FrameLabel->{"u","P",StringForm["n=``",n],""}]
0.2
0.1
-10
-5
0
u
Figure 9
page 18 of 24
5
10
Harmonic oscillator
Huber Oliver
3.3 Visualization of the oscillating state “0+1”
Unlike the stationary states which cannot be easily interpreted in classical terms,
this state behaves truly oscillatory: The preferred position of the particle moves
periodically from one side to the other.
This movie shows the time evolution of a superposition of two stationary eigenstates
of the harmonic oscillator (ground state + 1st excited state). For the graphical
representation, the harmonic oscillator potential in the background is shifted down
by the mean energy.
(* Time evolution of
superpositions of eigenstates *)
(* Input files: *)
<<Graphics`ArgColorPlot`
(* Definitions: *)
V[x_] := x^2/2;
Eosc[n_] := n + 1/2;
phi[n_, x_] := 1/Sqrt[2^n n!]/Pi^(1/4) HermiteH[n, x] Exp[-x^2/2];
Psi[n_, x_, t_] := phi[n, x] Exp[-I Eosc[n] t];
(* Input parameters: *)
ind = {0, 1};
(* =indices of wave functions to be superposed *)
coeff = {1/Sqrt[2], 1/Sqrt[2]};
(* =coefficients of wave functions *)
meanE = Abs[coeff]^2 . Eosc[ind];
(* =energy expectation value *)
wavefunc[x_, t_] := Simplify[coeff . Psi[ind, x, t]]
xleft = -4; xright = 4;
lower = -0.19; upper = 0.95;
(* Graphics: *)
pot[x_] := Which[Abs[x] > Sqrt[2*(upper + meanE)], upper,
Abs[x] < Sqrt[2*(lower + meanE)], lower, True, V[x] - meanE];
doplot[t_] :=
Show[{
FilledPlot[{pot[x], lower},
{x, xleft, xright},
Fills -> GrayLevel[0.8],
PlotPoints -> 150,
DisplayFunction -> Identity],
ArgColorPlot[Evaluate[wavefunc[x, t]],
{x, xleft, xright},
PlotPoints -> 120,
DisplayFunction -> Identity],
Plot[pot[x], {x, xleft, xright},
PlotPoints -> 150,
PlotStyle -> Dashing[{0.005, 0.025}],
DisplayFunction -> Identity]},
PlotRange -> {lower, upper},
Frame -> True,
Axes -> {True, None},
PlotLabel -> StringForm["t =`1`", PaddedForm[N[t], {3, 2}]],
DisplayFunction -> $DisplayFunction]
(* Animation: *)
Do[doplot[t];,{t, 0., 95 Pi/24, N[Pi/24]};
page 19 of 24
Figure 10
Harmonic oscillator
Huber Oliver
3.4 Visualization of Coherent state
This state is really remarkable for several reasons. At time t  0 it’s just the ground
state shifted to the left. The Gaussian shape does not change with time. The wave
packet oscillates back and forth very much like a classical particle. (This behaviour
doesn’t depend on the initial amplitude). Somehow the oscillator potential prevents
the Gaussian from spreading like in the case of free particles. This explains the name
“coherent state”. At the turning points the wave function appears as a typical
“Gaussian at the rest”. At the origin, the phase has the shortest wave length. This
means that the momentum has a maxima. The analogy with classical mechanics goes
even further: For a harmonic oscillator the expectation values of position and
momentum obey the classical equation of motion. We also note that for the coherent
state the product of the uncertainties in position and momentum has the minimal
possible value for all times.
(* Generate a movie of a squeezed
state and its Fourier transform *)
(* Packages needed: *)
<<Graphics`ArgColorPlot`;
(* Definition of a coherent state *)
psi[x_,t_] :=
(1/Pi)^(1/4) Exp[-x^2/2 - 2 E^(-I t) Cos[t] - 2 x E^(-I t) - I t/2];
(* Parameters *)
lower = -.2; upper = 1.;
xleft= -6.; xright = 6.;
(* Auxiliary quantities: *)
integrand[x_] =
Simplify[
psi[x,0]*
(-1/2*D[psi[x,0],{x,2}]
+1/2*(x^2)*psi[x,0])
];
meanE = Integrate[integrand[x],{x,-Infinity,Infinity}];
(* Plot the potential shifted down
by the amount of the mean energy: *)
V[x_] = x^2/2;
pot[x_] := Which[Abs[x] > Sqrt[2*(upper + meanE)], upper,
Abs[x] < Sqrt[2*(lower + meanE)], lower, True, V[x] - meanE];
(* Graphics commands *)
doplot[t_] :=
Show[{
FilledPlot[{pot[x],lower},
{x,xleft,xright},
Fills -> GrayLevel[0.8],
PlotPoints -> 60,
PlotStyle -> GrayLevel[.5],
DisplayFunction -> Identity
],
ArgColorPlot[Evaluate[psi[x,t]],
{x,xleft,xright},
PlotPoints -> 120,
DisplayFunction -> Identity
],
Plot[pot[x],{x,xleft,xright},
PlotPoints -> 60,
PlotStyle -> GrayLevel[0.5],
DisplayFunction -> Identity
page 20 of 24
Figure 11
Harmonic oscillator
Huber Oliver
]},
PlotRange -> {lower,upper},
Frame -> True,
PlotLabel -> StringForm["t =`1`",PaddedForm[N[t], {4, 2}]],
Axes -> {True,None},
DisplayFunction->$DisplayFunction
];
(* Animation: *)
Do[doplot[t];,{t,0.00001,4 Pi,N[Pi/24]}];
3.5 Visualizations of an anharmonic oscillator
In some situations it is not possible to find analytic solutions. For this reason we
have to use some approximation methods.
In this chapter we study some problems in quantum mechanics using matrix
methods. We know that we can solve quantum mechanics in any complete set of
basis functions. If we choose a particular basis, the Hamiltonian will not, in general,
be diagonal, so the task is to diagonalize it to find the eigenvalues (which are the
possible results of a measurement of the energy) and the eigenvectors.
In many cases this can not be done exactly and some numerical approximation is
needed. A common approach is to take a finite basis set and diagonalize it
numerically. The ground state of this reduced basis state will not be the exact ground
state, but by increasing the size of the basis we can improve the accuracy and check
if the energy converges as we increase the basis size. We will apply this approach
here for an anharmonic oscillator with some examples written in Mathematica.
Attached to this paper is a Mathematica Notebook (oscillator.nb) describing the
anharmonic oscillator.
Now we make the problem non-trivial by adding an anharmonic term to the
Hamiltonian operator. We will take it to be proportional to x 4 , like:
H  H 0  x 4
(61)
It is trivial to generate the Hamiltonian matrix of the simple harmonic oscillator,
since it is diagonal. We create the matrix:
1
0 0 0
2
h0[basissize_] := DiagonalMatrix [ Table[n + 1/2, {n, 0, basissize - 1} ]
h0[4]
0
3
2
0
0
0
0
5
2
0
0
0
0
7
2
It is easy to generate the matrix for H using the matrix obtained above for x and the
convenient "dot" notation in Mathematica for performing matrix products:
h[basissize_,
]:=
h0[basissize] +
x[basissize] . x[basissize] . x[basissize] . x[basissize]
For example, with a basis size of 4 we get:
h[4,
1
2
3
4
0
]
3
0
3
2
3
15
4
0
2
0
page 21 of 24
0
2
3
3
2
0
5
2
3
27
4
0
3
2
0
7
2
15
4
Harmonic oscillator
Huber Oliver
The eigenvalues can also be obtained numerically and then sorted. Here we give a
function (with delayed assignment) for doing this:
evals[basissize_,
]:=
Sort [ Eigenvalues [ N[ h[basissize,
]
] ] ]
Now we get some numbers. We start with a basis of size 15 and plot the eigenvalues
for a range of .
basissize = 15;
p1 = Plot [ Evaluate [ evals[basissize,
]
PlotStyle -> {AbsoluteThickness[2]} ,
], {, 0, 1},PlotRange -> {0, 11} ,
AxesLabel -> {"", "E"}];
E
10
8
We see that the energy levels and their spacing
increase as  increases.
The interval width at the harmonic oscillator between
the close-by energy level has the same value.
6
4
2
0.2
0.4
0.6
0.8
1
Figure 12
Next we use matrix methods to calculate the lowest energy levels in a double well
potential. The Hamiltonian is given by:
p2
H
 V ( x)
2
(62)
where
V ( x)  
x2
x4

2
4
(63)
Note that the coeficient of x 2 is negative. We plot the potential for the case of   0.2
Plot[-x^2/2 +
x^4
/4 /.
->
0.2, {x, -4, 4}];
1.5
The new physics in this example is the possibility of
tunneling between the two minima. The reader is
referred to QM I [1] from Franz Schwabl for more
details on tunneling.
1
0.5
-4
-2
2
4
-0.5
-1
Figure 13
page 22 of 24
Harmonic oscillator
Huber Oliver
Next we consider a smaller value,   0.1 , for which the minima are deeper.
Plot[-x^2/2 +

x^4 /4 /.

-> 0.1, {x, -4.7, 4.7}];
1
minpot = FindMinimum [ -x^2/2 +
{x, 3.5}]
0.5
-4
-2
2
4

x^4 /4 /.
->
0.1,
{-2.5, {x -> 3.16228}}
-0.5
When the potential is very deep the two the
particle will only tunnel very slowly from one
-1.5
well to the other. The two lowest energy levels
-2
are split by this small tunneling splitting, and
the eigenstates are the symmetric and anti-2.5
symmetric combinations of the wavefunction
Figure 14
for the particle localized in the ground state of
the left well and the right well. If we represent
the potential at the bottom of each well as a parabola then the lowest energy of each
of these states is given by the simple harmonic ground state energy.
-1
4 Summary and conclusions
Within the framework of this thesis we saw the principle of the harmonic oscillator
model in two different views. The first disquisition was about the classical harmonic
oscillator followed by the quantum oscillator perspective. An example was given by
the movement of atoms in a solid body pictured as a pointlike mass attached to a
spring. Then we saw that the transition to quantum mechanics takes place by
substitution of the dynamic variables to operators. With the classical Hamiltonian
function we obtained the quantum mechanical Hamiltonian operator and saw that
the eigenvalues of this operator provides us quantized energy levels at equally spaced
values. To solve the problem of time evolution of a state of a quantum harmonic
oscillator we used the time dependent Schrödinger equation which is a differential
equation. We discussed this Schrödinger equation in position space and saw that
general solutions are given by linear combination of the eigenvalues. We learned
about Dirac’s notation which is called the bra-ket notation. With this we are able to
describe different states with vectors. It is a generalization to vectors in a space of an
infinite number of dimensions. A scalar product appears as a complete bracket
expression. With this new notation we saw that the solution of the state vector of the
time dependent Schrödinger equation is determined by the eigenvalues and
eigenvectors of the Hamiltonian operator. Finally we built the ladder operators. With
these we are able to find the energy eigenvalues of the harmonic oscillator for
different states. After that we analyzed the different states especially the ground and
excited states in position space. Last but not least we determined the time series of
the wave function in the potential of a harmonic oscillator. In the end we illustrate
some Mathematica examples adapted to the harmonic and anharmonic oscillator.
The attached Mathematica Notebook (oscillator.nb) presents also a summary to this
topic.
page 23 of 24
Harmonic oscillator
Huber Oliver
5 Sources
5.1 Bibliography
[1]
[2]
[3]
[4]
[5]
Quantenmechanik – QM I, Franz Schwabl, Springer, ISBN: 3-540-43106-3
Grundkurs Theoretische Physik, Wolfgang Nolting, vieweg, ISBN: 3-540-41533-5
Mathematische Methoden in der Physik, C. B. Lang, N. Pucker, Spektrum, ISBN: 3-8274-0225-5
Quantenmechanik, Torsten Fließbach, Spektrum, ISBN: 3-8274-0996-9
Visual Quantum Mechanics, Bernd Thaller, Telos, ISBN: 0-387-98929-3
5.2 Figures
Figure
Figure
Figure
Figure
Figure
1,
2,
3,
4,
5,
harmonic oscillator, Source: rugth30.phys.rug.nl/.../ figures/potent22.gif
potential of an harmonic oscillator, Source: Visual Quantum Mechanics, Bernd Thaller
sphere between two springs, Source: Repetitorium Experimentalphysik, E. W. Otten
eigenvalues of harm. oscillator, Source: www.vsc.de/.../oszillatoren_m19ht0502.vscml.html
comparison position probability density of classic and harmonic oscillator
Figure 6, position probability density |  n ( x ) | , Source: Visual Quantum Mechanics, Bernd Thaller
2
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
,
7, eigenfunctions of the harmonic oscillator Source: QM I, Franz Schwabl
8, classical particle motion, Source: Visual Quantum Mechanics, Bernd Thaller
9, position probability density, Source: Mathematica code see above
10, oscillating state “0+1”, Source: Visual Quantum Mechanics, Bernd Thaller
11, coherent states, Source: Visual Quantum Mechanics, Bernd Thaller
12, energy levels and their spacing, Source: oscillator.nb (electronic version)
13, anharmonic potential plot with   0.2 , Source: oscillator.nb (electronic version)
13, anharmonic potential plot with   0.1 , Source: oscillator.nb (electronic version)
page 24 of 24