Bacchelor Thesis: ”Harmonic...

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

page 2 of 24

Contents

1 Prologue............................................................................................................................................ 3 2 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


page 3 of 24

1 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 )(xf results. Figure 1 shows this behaviour.

)2(|)(||)(|)1()()(

xfxfxmamxkxfxf

−−===−=−−= &&

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. )(xf is a force and as such it Newton’s law:

.)( xmamxf &&== This force could be expanded by a Taylor series:

∑ =−+=

n

ii

i

xxi

xfxfxf

1 00

)(

0 )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 0)0( =f and for little extensions of the spring Hooke’s law xkxfxf −=−≈ |)0(|)( ' 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 )(xf 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.

Figure 1


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)5(2

)(

)4()()(

2xkxV

xmxVdxdxkxf

=

=−=−= &&

The generalization to higher dimensions is straightforward. With ),...,( 1 nxxx=r we have:

)7(2

)(

)6()()(xxkxV

xVxkxfrr

r

rrr

=

−∇=−=

Figure 2 shows the harmonic oscillator potential in one and two space dimensions with spring constant 1=k .

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 )(xV is the antiderivative of the force )(xf :

)8()()( ''∫−=x

dxxfxV

If we multiply (1) and (4) with x& we obtain:

)9()()( xxVdxdxxfxxm &&&&& −==

This we also can write as:

)10()()2

( 2 xVdtdxm

dtd

−=&

The integration of (10) gives us a new constant H, which by separation of the equation is called the Hamilton function:

)11()(2

2 xVxmH += &

The expression (11) reflects the total energy term. 2

2xm& or (expressed as a function of

the momentum) 2

2mp is the kinetic energy and the other one )(xV is the potential

energy. The harmonic oscillator is described usually by this Hamiltonian function:

)12(22

22

xkm

p+=Η

Figure 2


page 5 of 24

And classically we solve such a problem like we have, with the Hamiltonian equation:

)16(

)15(

)14(

)13(

xkxm

xkx

p

xmpmp

px

−==>

−=∂Η∂

−=

==>

=∂Η∂

=

&&

&

&&&

&

The result is a linear homogeneous differential equation with the oscillatory solution which expresses the motion (18) or the current amplitude (17):

)18()()()(

)17()()(

btCosamvmtxmtp

btSinatx

−===

−=

ωω

ωr&r

Where a and b have to be determined from given initial conditions 0)0( xx = and

0)0( pp = . Hence the classical motion is an oscillation with angular frequency ω . The spring constant k determines this oscillator frequency:

)20(

)19(

2

2

ω

ωω

mkmk

mk

=

=→=

which is independent from the amplitude.

Figure 3 shows the movement of a pointlike mass between two springs with extension rx =0 . 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

0xrx ±=±= .

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.

Figure 3


page 6 of 24

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:

)22(:

)21(2

.'....,:

xx

constsPlankhdxdhipp

=Π

→=)

)

to the classical Hamiltonian function. We obtain the quantum-mechanical Hamiltonian operator:

)23(22

22

xmm

pH ))) ω

+=

which acts on a square-integrable wave function Ψ . ),( txΨ 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 ppxx )) →→ , 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:

)24(),(),( txHtxdtdhi Ψ=Ψ

)


page 7 of 24

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:

)25()(),( )( tiextx ω−Ψ=Ψ

)26()(),()(

hiEt

extx−

Ψ=Ψ This will lead to the time-independent Schrödinger equation:

)27()()( xExH Ψ=Ψ))

The states (25) are called stationary states, because the corresponding probability densities 22 |)(||),(| xtx Ψ=Ψ are time-independent. The normalizing condition ( ∫ ∞<Ψ 2|)(| xdx ) 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:

)28(|| >Ψ>=Ψ EH))

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


page 8 of 24

vectors, e.g. from two ket vectors >A| and >B| we can form >>=+> CBcAc ||| 21 , say, where 1c and 2c 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 ∫ >=> Xdxx || 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 >+> '|| AA is the sum of the numbers corresponding to >A| and to >'| A , and the number corresponding to

>Ac | 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 >< AB | . 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 >< AB | 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


page 9 of 24

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 >< AA | . 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:

)29(0 AAcH))) ++=

The operator A

) and the adjoint operator +A

) will be determined like:

)||(||))((||

)(

)(

*2222*2

**

xppxpxpxpxAA

pxA

pxA

))))))))))))

)))

)))

βββαββα

βα

βα

+++=++=

+=

+=

+

+

with the commutator relation:

0

!*22*22222

*****

)(||||||||||

)(],[

cHpxipxAA

ipxipxpxxppxixppxpx

−=++−+=

−+=−+=+

=−=

+ )))h

))))

h))

h))))))))

h))))))

ββααββαα

ββββββββ


page 10 of 24

By comparison of coefficients with equation (23), we have the following solution:

)32(

)31(1||21||||

)30(22

||

*

22222

222

ωβββ

ωββα

ωαωα

mimm

mm

=⇒−=

=⇒=

=⇒=

The result of this is for A)

, +A)

and H)

:

)36(2

)35(2

)34()(2

)33()(2

2

2

ω

ωω

ω

ωω

h)))

h)))

)))

)))

+=

−=

−=

+=

+

+

+

AAH

HAA

mpixmA

mpixmA

With this the Hamiltonian operator takes shape:

)37()21( +=

+

ωω

h

))

h) AAH

Equation (33) and (34) represent the ladder operators. The commutator of the leader operators is:

0],[,0],[

]),[],([],)[)((],([2

)](),[(2

],[

==

−−−+=

−+=+

ppxx

xppxm

ippm

im

ixxmmpix

mpixmAA

))))

))))))))

h

))

))

h

))

ωωωω

ωωω

The commutator relations are:

)39(0],[],[

)38(1],[

==

=++

+

AAAA

AA))))

))

Now we define a new operator N)

. Which is defined:

)40(AAN))) +=

With this new operator we are able to simplify our Hamiltonian operator:


page 11 of 24

)41()21( += NH

)h

)ω

N)

and H)

are commutating and have therefore the same eigenvectors.

>+>=⇒>>= nnnHnnnN |)21(||| ωh

))

Here we can see that H)

has the eigenvalues )21( +nωh . The question is now, what

values n can have. To answer the question we have to consider the commutator relations of N

) with A

) and +A

):

AAAAAAAA

AAAAAAAAAAAAAAAAN

AAAAAAAA

AAAAAAAAAAAAAAAAN

))))))))

)))))))))))))))))

))))))))

)))))))))))))))))

−=−−=

=+−=−==

=−+=

=−+=−==

++

+++++

++++++

+++++++++++

)1()(],[],[

)1()(],[],[

Hence:

>+>=

>>=−>

>>=

++

+++

++

nAnnAN

nAnnAnAN

nAnAN

|)1(|

|||

||],[

)))

))))

)))

Analog:

>−>= nannaN |)1(|)

That means, if >n| is the eigenvector of N)

to the eigenvalue n then:

>+ nA |)

is eigenvector to the eigenvalue (n+1) and >nA |)

is eigenvector to the eigenvalue (n-1).

We call +A)

the generator operator and A)

the annihilator operator. We obtain:

)42(1|| >−>= nnA λ)

It applies:

)43(||1|1|| 2* λλλ >=−−<>=< + nnnAAn))

and

)44(||||| nnnnnNnnAAn >=<>=>=<< + )))

It means that the normalizing factor λ have to fulfil n=2|| λ . We choose n=λ and


page 12 of 24

insert a possible phase factor in definition of >n| . The result is:

)45(1|| >−>= nnnA)

Analogous reasoning for >+ nA |)

:

)46(1|1|

||1|1|||

||||

1||

2!

2

>++>=

=+>=+>=<<

>=<

>+>=

+

+

+

+

nnnA

nnNnnAAn

nAAn

nnA

)

)))

))

)

µ

µ

µ

We can now begin with any eigenstate >n| and use the operator A)

repeatedly.

)47(|)1)...(2)(1(|

2|)1(|

1||

>−+−−−>=

>−−>=

>−>=

mnmnnnnnA

nnnnAA

nnnA

m)

))

)

So we get the eigenstates >− mn| to decreasing eigenvalues )( mn − of N)

. This means that we can generate negative eigenvalues, but they are:

)48(0||||||||| 2 ≥>=>=<>=<=< + ψψψmAAmmNmm)))

If n is an integer, then the series (45) aborts with 00| >==na . 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:

)49(|| 0NnnnnN ∈∀>>=)

And finally we find our eigenvalues of the harmonic oscillator:

0

)50()21(

Nn

nEn

∈

+= ωh

In ground state ( 0=n ) the particle has the zero point energy 2ωh . For the quantum

case is the so-called zero-point vibration of the 0=n ground state. This implies that molecules or atoms are not completely at rest,

Figure 4


page 13 of 24

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

ωh . The energy levels of the quantum harmonic oscillator are given by (50).

The oscillator transition is given by 1±∆n . 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:

)51()(| xnx nΨ>=< The wave function in ground state )(0 xΨ can be calculated as follows. We know that

00| >=A)

hence:

∫∫ Ψ><=>><<=>=< ''0

'''' )(||0|||00|| dxxxAxdxxxAxAx)))

We need the matrix elements of a :

)(||

)(||

)||||(2

||

''

''

'''

xxdxd

ixpx

xxxxxx

xpxm

ixxxmxAx

−>=<

−>=<

><+><>=<

δ

δω

ω

h)

)

))

h

)

This gives us:

)()(

))()((2

))()()((2

))()()((2

)())()()((2

0||0

00

00

''0

'0

''0

'0

''0

''

xxmxdxd

xdxd

mxxm

dxxxxdxd

mxxm

dxxxxdxd

mxxm

dxxxxdxd

imixxxm

Ax

Ψ−=Ψ⇒

Ψ+Ψ=

Ψ−+Ψ=

Ψ−+Ψ=

Ψ−+−=

>=<

∫

∫

∫

h

h

h

h

h

h

h

h

h

)

ωω

ω

δω

ω

δω

ω

δω

δω

Ordinary differential equations first orders have explicit solutions. The solution of this differential equation can be calculated via a power series ansatz:

∑∞

=

=Ψ0

0 )(n

nn xcx

The insertion in our differential equation gives us:


page 14 of 24

0))1(

0

11 11

1 0

11

=+++⇒

=+

−∞

= +

∞

=

∞

=

+−

∑

∑ ∑l

ll l

n n

nn

nn

xcmlcc

xcmnxc

h

h

ω

ω

This equation has to be individual fulfilled for every power, because the equation has to be right for any x . }{ lx forms a linear independent base. We follow:

2;10

2

1

≥−=

=

− lcl

mc

c

llh

ω

odd terms:

00

031

12

35

13

=⇒=∝

=−=

+nccc

cmch

ω

even terms:

0002

02

24

02

!1)(

1...)2)(1(21)(

21...

221

21)(

21

41)(

4121

cn

mcnnn

mcnn

mc

cmcmc

cmc

nn

nnn

hhh

hh

h

ωωω

ωω

ω

−=−−

−=−

−=

−=−=

−=

The solution of the differential equation is thus:

02

002

20 )(

!

)2

()( cx

n

m

cxx n

n

n

nn

n ∑∑∞

=

∞

=

−==Ψ h

ω

With the characteristically length ωm

x h=:0 of the

oscillator we can transform:

20

2

200 )( x

x

ecx−

=Ψ When we normalize )(0 xΨ we get:

4 20

0

!

020

20

2

0

1

1|)(|20

2

xc

xcdxecdxx xx

Π=⇒

=Π==Ψ ∫∫∞

∞−

−∞

∞−

and finally we obtain to the ground state:

)52(1)(20

2

2

4 20

0x

x

ex

x−

Π=Ψ

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: 20,2,1 === nnn In chapter 3.1 you will see a Mathematica code which plots this behaviour.

Figure 5


page 15 of 24

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:

)53(0|)(!

1| >> + nAn

n)

we build:

><=>=<Ψ + 0|)(|!

1|:)()53(

nn Ax

nnxx

)

''0

'2'''2

2

)(|)((|)2

(!

10||)((|)2

(!

1

0|)(|)2

(!

1

dxxxmpixxm

ndxxx

mpixxm

n

mpixxm

n

nn

nn

nn

Ψ>−<=>><−<=

>−<=

∫∫ ωω

ωω

ωω

))

h

))

h

))

h

with the matrix elements of position operator (54) and momentum operator (55) we can go on:

)55()()||

)54()(||

''

''

xxdxd

ixpx

xxxxxxrrhr)r

rrr)r

−−=<

−>=<

δ

δ

)()()2(!

1

)()()2

(!

1

)()()()2

(!

1

0200

2

02

''0

'2

xdxdxxx

n

xdxd

mxm

n

dxxxxdxd

mxm

n

nnn

nn

nn

Ψ−=

Ψ−=

Ψ−−=

−−

∫

ωω

δω

ω

h

h

h

h

Now we insert the function for the ground state (52) and substitute 0xxz = and finally

get what we want – stimulated states in position space (56):


page 16 of 24

)56(][12!

1)(

])[(12!

1

1))(

(2(!

1

0

2

0

2

020

2

)(2

4 20

2

2

4 20

2

2

4 20

0

0

2

xxzzn

zn

n

xxz

zn

xxz

xx

nn

hexn

x

edzdz

xn

ex

xxd

dxx

n

=

−−

=

−−

=−−

Π=Ψ

−Π

=

=Π

−=

The position probability densities 2|)(| xnΨ of the 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.

eigenfunctions of the harmonic oscillator for 5...0=n with:

xhmxxy

Π==

20

ω

Where )(zhn are real polynomial from order n in z . They are called hermit polynomial (57) and have even and odd parity: )()1()( zhzh n

nn −=− .

)57()(

|)2()(

2

22

2

2

0

2

22

21

2

xx

nx

x

x

xn

x

n

eedxdxee

eAexh

−−

−

=+

−=

=)

Concluding we have:

• The eigenvalue problem can be directly solved in position space:

)()()22

( 22

22

xExxmdxd

mEH Ψ=Ψ+−⇒Ψ=Ψ

)h)) ω

Figure 6

Figure 7


page 17 of 24

• 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 0=t the state is given by: >Φ| . And at later moment 0>t the state is given by:

)58(|||:),( 0 >Φ>=<Φ=<Φ−h

ti

t exxtx . The Hamiltonian operator is not explicit time-dependent. We expand the ground state

>Φ 0| in series of the eigenstates of the harmonic oscillator.

∫

∑ΦΨ>=Φ=<

>>=Φ∞

=

dxxxnc

nc

nn

nn

)()(|

||

00

00

After insertion in (58):

)59()(

)(

|),(

0

2

0

)21(

0

∑

∑

∑

∞

=

−−

∞

=

+−

∞

=

−

Ψ=

Ψ=

><=Φ

n

tninn

ti

n

nti

nn

n

Eti

n

exce

exc

enxctx n

ωω

ω

h

The matter of fact that the energy difference 'nn EE − is an integer factor of our energy

unit ωh will lead that the wave function is time periodic. The period ωΠ

=2T is

equivalent to the classical oscillation period.

)60(),()(),( 2

0

2 txeexceeTtx nitni

nnn

tii Φ−=Ψ=+Φ Π−−∞

=

Π ∑ ωω

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 18 of 24

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 )(txx = of the mass point as a function of time t . The green curve is the velocity )(tv . As I showed in chapter 2.1 the force can be described by a potential function )(xV 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 *) 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}]

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; 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],""}]

-10 -5 0 5 10u

0.1

0.2

0.3

0.4

0.5

P

n=1

Figure 9

1

2

3

4

5

6

Figure 8


page 19 of 24

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]};

Figure 10


page 20 of 24

3.4 Visualization of Coherent state This state is really remarkable for several reasons. At time 0=t 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

Figure 11


page 21 of 24

]}, 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 4x , like:

)61(40 xHH λ+=

It is trivial to generate the Hamiltonian matrix of the simple harmonic oscillator, since it is diagonal. We create the matrix: h0[basissize_] := DiagonalMatrix [ Table[n + 1/2, {n, 0, basissize - 1} ] h0[4]

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_, λ_]:= h0[basissize] + λ x[basissize] . x[basissize] . x[basissize] . x[basissize]

For example, with a basis size of 4 we get: h[4, λ]

ik

12

0 0 0

0 32

0 0

0 0 52

0

0 0 0 72

y{

i

k

12

+ 3l4

0 3 l

20

0 32

+ 15l4

0 3$32

l

3 l"2 0 52

+ 27 l4

0

0 3$32

l 0 72

+ 15 l4

y

{


page 22 of 24

The eigenvalues can also be obtained numerically and then sorted. Here we give a function (with delayed assignment) for doing this: evals[basissize_, λ]:= Sort [ Eigenvalues [ N[ h[basissize, λ] ] ] ]

Now we get some numbers. We start with a basis of size 15 and plot the eigenvalues for a range of λ. basissize = 15;

p1 = Plot [ Evaluate [ evals[basissize, λ] ], {λ, 0, 1},PlotRange -> {0, 11} , PlotStyle -> {AbsoluteThickness[2]} , AxesLabel -> {"λ", "E"}];

We see that the energy levels and their spacing increase as λ increases. The interval width at the harmonic oscillator between the close-by energy level has the same value.

Next we use matrix methods to calculate the lowest energy levels in a double well potential. The Hamiltonian is given by:

)62()(2

pH2

xV+=

where

)63(42

)(42 xxxV λ+−=

Note that the coeficient of 2x is negative. We plot the potential for the case of 2.0=λ Plot[-x^2/2 + λ x^4 /4 /. λ -> 0.2, {x, -4, 4}];

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.

0.2 0.4 0.6 0.8 1l

2

4

6

8

10

E

Figure 12

-4 -2 2 4

-1

-0.5

0.5

1

1.5

Figure 13


page 23 of 24

Next we consider a smaller value, 1.0=λ , for which the minima are deeper. Plot[-x^2/2 + λ x^4 /4 /. λ -> 0.1, {x, -4.7, 4.7}];

minpot = FindMinimum [ -x^2/2 + λ x^4 /4 /. λ -> 0.1, {x, 3.5}] {-2.5, {x -> 3.16228}}

When the potential is very deep the two the particle will only tunnel very slowly from one well to the other. The two lowest energy levels are split by this small tunneling splitting, and the eigenstates are the symmetric and anti-symmetric combinations of the wavefunction 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.

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.

-4 -2 2 4

-2.5

-2

-1.5

-1

-0.5

0.5

1

Figure 14


page 24 of 24

5 Sources

5.1 Bibliography [1] Quantenmechanik – QM I, Franz Schwabl, Springer, ISBN: 3-540-43106-3 [2] Grundkurs Theoretische Physik, Wolfgang Nolting, vieweg, ISBN: 3-540-41533-5 [3] Mathematische Methoden in der Physik, C. B. Lang, N. Pucker, Spektrum, ISBN: 3-8274-0225-5 [4] Quantenmechanik, Torsten Fließbach, Spektrum, ISBN: 3-8274-0996-9 [5] Visual Quantum Mechanics, Bernd Thaller, Telos, ISBN: 0-387-98929-3

5.2 Figures Figure 1, harmonic oscillator, Source: rugth30.phys.rug.nl/.../ figures/potent22.gif Figure 2, potential of an harmonic oscillator, Source: Visual Quantum Mechanics, Bernd Thaller Figure 3, sphere between two springs, Source: Repetitorium Experimentalphysik, E. W. Otten Figure 4, eigenvalues of harm. oscillator, Source: www.vsc.de/.../oszillatoren_m19ht0502.vscml.html Figure 5, comparison position probability density of classic and harmonic oscillator Figure 6, position probability density 2|)(| xnΦ , Source: Visual Quantum Mechanics, Bernd Thaller

Figure 7, eigenfunctions of the harmonic oscillator , Source: QM I, Franz Schwabl Figure 8, classical particle motion, Source: Visual Quantum Mechanics, Bernd Thaller Figure 9, position probability density, Source: Mathematica code see above Figure 10, oscillating state “0+1”, Source: Visual Quantum Mechanics, Bernd Thaller Figure 11, coherent states, Source: Visual Quantum Mechanics, Bernd Thaller Figure 12, energy levels and their spacing, Source: oscillator.nb (electronic version) Figure 13, anharmonic potential plot with 2.0=λ , Source: oscillator.nb (electronic version) Figure 13, anharmonic potential plot with 1.0=λ , Source: oscillator.nb (electronic version)

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