Harmonic oscillatorHuber Oliver
Institute of Theoretical Physics
Course: Quantummechanics 05S.624.106
Prof. Dr. Reinhard ALKOFER
Karl-Franzens-University Graz - Austria
BacchelorThesis:
”Harmonic oscillator”
Huber Oliver
9811289
033 619 411
Field of Studies:
Environmental System Sciences with main focus on Physics
Graz, 07.07.2005
Contents
1Prologue
2The different views of a harmonic oscillator
2.1Classical mechanics
2.2Quantum mechanics
2.2.1Introduction
2.2.2Transition from classical to quantum mechanics
2.2.3The time-independent Schrödinger equation – stationary states
2.2.4Method of Dirac
2.2.4.1Bras and Kets
2.2.4.2Ladder operators and eigenvectors
2.2.5Ground state in position space - Power Series Method
2.2.6Excited states in position space
2.2.7Dynamics of the harmonic oscillator
3Visualization with Mathematica
3.1Visualization of the classical motion
3.2Visualization of harmonic oscillator probability density
3.3Visualization of the oscillating state “0+1”
3.4Visualization of Coherent state
3.5Visualizations of an anharmonic oscillator
4Summary and conclusions
5Sources
5.1Bibliography
5.2Figures
1Prologue
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.
2The different views of a harmonic oscillator
2.1Classical mechanics
The harmonic oscillator is among the most important example of explicitsolvable 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 movesan atom out of this stable position,aforce results.Figure 1 shows this behaviour.
Equation (1) relates the force exerted by a spring to the distance it is stretched. is the spring constant and 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. 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.is a force and as such it Newton’s law:
This force could be expanded by a Taylor series:
The Taylor series of an infinitely often differentiable real (or complex) functionf defined on an open interval (x − x0, x + x0) is apower series.
In equilibrium position the force and for little extensions of the spring Hooke’s law holds. 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 can be expressed in terms of a potential function .A potential function like equation (5) returns for example the potential energy an object with a position has.
The generalization to higher dimensions is straightforward. With we have:
Figure 2 shows the harmonic oscillator potential in one and two space dimensions with spring constant .
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 is the antiderivative of the force :
If we multiply (1) and (4) with we obtain:
This we also can write as:
The integration of (10) gives us a new constant H, which by separation of the equation is called the Hamilton function:
The expression (11) reflects the total energy term. or (expressed as a function of the momentum)is the kinetic energy and the other one is the potential energy.
The harmonic oscillator is described usually by this Hamiltonian function:
And classicallywe solve such a problem like we have,with the Hamiltonian equation:
The result is a linear homogeneous differential equation with theoscillatory solution which expresses the motion (18) or the current amplitude (17):
Where and have to be determined from given initial conditions and . Hence the classical motion is an oscillation with angular frequency. The spring constant determines this oscillator frequency:
which is independent from the amplitude.
Figure 3 shows the movement of a pointlike mass between two springs with extension. As we can see the oscillating sphere follows the vertical projection of the circular motion in direction . The motion takes place between the turning points .
2.2Quantum mechanics
2.2.1Introduction
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.
Like our introducing example of an atomin 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 isthe 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.
2.2.2Transition from classical to quantum mechanics
We build now an momentum operator and a position operator and apply the substitution rules:
to the classical Hamiltonian function. We obtain the quantum-mechanical Hamiltonian operator:
which acts on a square-integrable wave function. representsa wave function and associates the statistical description of anexperimental output.
The transition to quantum mechanics takes place by substitution of the dynamic variables 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:
2.2.3The time-independent Schrödinger equation – stationary states
The time–independent Schrödinger equation is the eigenvalue equation of the Hamiltonian operator. In (24) we insert theansatzof the wave function:
This will lead to the time-independent Schrödinger equation:
The states (25) are called stationary states, because the corresponding probability densities are time-independent. The normalizing condition () will restrict the possible values of energy .
2.2.4Method of Dirac
2.2.4.1Bras and Kets
Expression (28) is apparently not explicit time-dependent and the solution of the time-dependent Schrödinger equation for the state vector is:
In (28) we can see a notation which was established by P. A. M. Dirac and is called the bra-ket notation and is apopular 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 . Ket vectors may be multiplied by complex numbers and may be added together to give other ket vectors, e.g. from two ket vectors and we can form , say, where and 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 , depending on and labelled by a parameter which can take on all values in a certain range, we may integrate it with respect to x, to get another ket vector 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 results from a superposition of the states and .
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 which is a function of a ket vector , then to each ket vector there corresponds one number , and suppose further that the function is a linear one, which means that the number corresponding to isthe sum of the numbers corresponding to and to , and the number corresponding to is times the number corresponding to , being any numerical factor. The number corresponding to any may be looked upon as a scalar product of that with some new vector, there being one of these new vectors for each linear function of the ket vectors . 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 and a ket vector will be written . 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 now appears as a complete bracket expression and a bra vector or a ket vector 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 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 or of the conjugate imaginary ket vector is defined as the square root of the positive number . 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.
2.2.4.2Ladder 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 .is selfadjoint and quadratic in and . Also and are selfadjoint operators. Sure we can bring into the form:
The operator and the adjoint operator will be determined like:
with the commutator relation:
By comparison of coefficients with equation (23), we have the following solution:The result of this is for , and :
With this the Hamiltonian operator takes shape:
Equation (33) and (34) represent the ladder operators. The commutator of the leader operators is:
The commutator relations are:
Now we define a new operator . Which is defined:
With this new operator we are able to simplify our Hamiltonian operator:
and are commutating and have therefore the same eigenvectors.
Here we can see that has the eigenvalues . The question is now, what values can have. To answer the question we have to consider the commutator relations of with and :
Hence:
Analog:
That means, if is the eigenvector of to the eigenvalue then:
is eigenvector to the eigenvalue (n+1) and is eigenvector to the eigenvalue (n-1).
We call the generator operator and theannihilator operator. We obtain:
It applies:
and
It means that the normalizing factor have to fulfil . We choose and insert a possible phase factor in definition of . The result is:
Analogousreasoningfor:
We can now begin with any eigenstate and use the operator repeatedly.
So we get the eigenstates to decreasing eigenvalues of . This means that we can generate negative eigenvalues, but they are:
If is an integer, then the series (45) aborts with . 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 are:
And finally we find our eigenvalues of the harmonic oscillator:
In ground state () the particle has the zero point energy . For the quantum case is the so-called zero-point vibration of the ground state. This implies that molecules or atoms are not completely at rest, 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 . The interval width between the close-by energy level has the same value. The system has thus an equidistant spectrum.
2.2.5Groundstate inposition space- Power Series Method
The probability amplitude to find a quantum mechanical particle at a position if it is in eigenstate is: