Harmonic oscillator

A harmonic oscillator is an oscillatory system, which is characterized by a linear restoring size. For a mechanical system, this means that there is a force which counteracts a deflection proportional. After an impulse from outside a harmonic oscillator oscillates sinusoidally ( = harmonic ) to its rest position. Examples of harmonic oscillators are spring pendulum, electrical resonant circuits and tuning forks.

The harmonic oscillator is an important model system of physics. It is completely described by only two parameters, the natural frequency and damping. Many complex systems behave with small deflections as harmonic oscillators, such as the pendulum. The harmonic oscillator in quantum mechanics is one of the few quantum mechanical systems that can be solved without approximations.

The term harmonic oscillator is also used for damped harmonic oscillators, even if it strictly enforce no harmonic oscillation.

• 4.1 One-dimensional undamped oscillator 4.1.1 Description of the swing operation
• 4.1.2 Derivation of the wave equation
• 4.1.3 energy

• 4.2.1 Linear Damping

• 4.5.1 Two-dimensional oscillator

• 5.1 Electrical resonant circuit
• 5.2 The harmonic oscillator in quantum mechanics
• 5.3 Lorentz oscillator in the optical

• 6.1 Forced vibration
• 6.2 Self-excited vibration
• 6.3 Parameter -excited vibration

• 7.1 continuum transition

Differential equation of the harmonic oscillator

Mathematically describe each free harmonic oscillator by the following differential equation. Exceptions are oscillators in quantum mechanics and related theories in which uncertainty relations must be considered.

In this case, the displacement of the system and the resonance frequency. It is an ordinary, linear, homogeneous second order differential equation, which therefore easily can be solved analytically. The solution of such an equation is a sinusoidal function.

Potential of the harmonic oscillator

The undamped harmonic oscillator is a conservative system. This means that the energy of the vibration is preserved. Therefore, there is a potential for each oscillator force field.

One-dimensional oscillator

The graphic representation of the potential of the harmonic oscillator is a square parabola. It is also called harmonic potential.

In mechanics, the force is applied to a particle in such a potential by the negative derivative of the potential.

Multi-dimensional oscillator

This concept can be extended to multiple dimensions. The potential here has the shape of an elliptical paraboloid. In n dimensions, it can be a suitable choice of coordinates written as follows:

Because no mixed terms occur between different directions in the individual summands, can be attributed to the problem of an n -dimensional harmonic oscillator in n -dimensional oscillators. In quantum mechanics, such a property is called separability. It can be concluded that for a harmonic oscillator is not only the energy but also the energy for the components of each direction are conserved quantities.

Depends on the value of the potential only on the distance to the origin, but not on the direction, it is called the isotropic oscillator, otherwise anisotropic. So are In an isotropic oscillator have all the constants the same value:

In mechanics, the force is applied to a particle in such a potential by the negative gradient of the potential.

Minimum of the potential

The minimum of this potential is a stable fixed point of the system. In mechanics we call this point also the rest position and the force experienced by the particle, restoring or restoring force. In particular, one lying at rest particle experiences no force, from which the name " rest position " is derived. The name is somewhat misleading, however, in this respect: While acting on a particle at rest, no power, the particles must be there, but not at rest. In general, it takes even there at its maximum speed.

Importance in physics

An ideal harmonic oscillator, in which the restoring force for arbitrarily large displacements increase linearly with the deflection, does not exist in nature. Nevertheless, the concept for the physics of fundamental importance, because often only small displacements of an object viewed from the position of rest. If you restrict yourself out so you can potentials, which exhibit a local minimum, a good approximation is replaced by a harmonic potential and the whole problem can be described as a harmonic oscillator. The advantage of such a harmonic approximation is that the problem with standard methods of theoretical physics is to handle and provides easy -to-interpret, analytical solutions. In the adjacent figure, this was for a Lennard-Jones (12,6 ) potential (blue curve ) is performed. The result ( red curve) as can be seen only for small distances from the minimum a useful approximation.

Your mathematical reasoning finds the harmonic approximation in the fact that the potentials can be developed in a Taylor series. If a potential given and this is sufficiently differentiable, then by the theorem of Taylor:

Whereby, the so-called residual limb, only terms of third order contains. For small distances, it is therefore negligible. As a development point we choose a minimum of the potential, so that is true. Thus, the first order term is omitted. For better handling can be obtained by an appropriate mathematical coordinate transformation of the apex are specified in the origin of coordinates in order to apply. Furthermore, it is always possible to set. We then obtain approximately the harmonic potential:

That is, for a sufficiently small deflection, the oscillator behaves harmonious. One example of oscillators that are already in anharmonic mean amplitudes, the pendulum.

An approximate solution method, in which a complicated problem is first reduced to a solvable analytically to the solution ignoring influences in the form of interference then add above is referred to as interference theory.

The harmonic oscillator in classical mechanics

One-dimensional undamped oscillator

A mechanical oscillator consists of a mass of the body and out of a force which drives back this, when it is deflected from its rest position. Thus, an oscillator, a harmonic, the restoring force must be proportional to the deflection, thus removing the body from its rest position to be. In practice, such a force is usually realized by means of springs, as in a pendulum spring, or by the weight of the body, as is the case for example with a water pendulum.

Description of the swing operation

A harmonic oscillator is moved from its rest position. The farther away it is, the greater is the force that tries to move back him. Due to the deflection of the oscillator is added potential energy. Potentially indicates that the energy is used, for example, to tension a spring, and thus, this energy is stored in the position of the oscillator.

If the oscillator then released, so it is accelerated due to the tension of the spring. He therefore moves with increasing speed back to the rest position. If he has arrived there, the oscillator has reached its maximum speed. The spring is relaxed and there is no force acting on the oscillator more. The potential energy which has been supplied to it, has now been completely converted into kinetic energy. This means it is now no longer stored in the position, but the speed of the oscillator.

Due to the inertia of the oscillator moves on, however. This causes the spring, this time in the other direction, is excited again. For clamping the spring of the oscillator must expend its kinetic energy to move against the force of the spring can. He becomes slower until it reaches the point where he no longer moves and the total energy is again present in the form of potential energy. The motion sequence then starts over again.

Derivation of the wave equation

We take as exemplified above a spring pendulum. The mass of the body. The rest position we take as the zero point and denote the deflection with. The force acting on the body, is described by the Hooke's law:

The constant is a spring constant that depends on the strength of the restoring force in a fixed deflection. It is also known that the acceleration of a body is proportional to the force acting on it. The acceleration can be written as the second derivative of position with respect to time. A time derivative is often referred to in physics as a point on the variable:

If, now, these two expressions for the force equal, one obtains a differential equation:

To simplify the following calculations, and writes the equation is substituted for:

This equation can be solved for example by means of a Exponentialansatzes. The result is a sinusoidal function, also called harmonic oscillation:

The solution contains two constant amplitude, and the phase shift angle. They are obtained according to the initial conditions. The amplitude is the maximum deflection of the oscillator and thus the energy of the vibration. The phase shift angle is determined, the position and at the same time so that the speed at which the body has at the moment.

The sine function is a periodic function, because their values ​​repeat at regular intervals (). Therefore, the oscillator performs a periodic motion. denotes the natural angular frequency and the resonant frequency of the oscillator. It determines the frequency at which the oscillator oscillates. In a harmonic oscillator that frequency is independent of the amplitude of vibration.

Energy

In free undamped oscillator energy is conserved because it is a closed system and only conservative forces. In the equilibrium position, the potential energy vanishes. Therefore, the total energy is equal to the maximum kinetic energy:

To the same result is obtained if one calculates the total energy above the maximum value of the potential energy:

Represents here in each case the maximum value of deflection, so for the amplitude. Is expected to be complex numbers, so takes the place of the squared modulus of the complex amplitude, if appropriate.

One-dimensional damped oscillator

A mechanical vibration is not without friction in general. That is, the vibration loses frictional energy and therefore decreases its amplitude. One speaks of a damping of the oscillation, whereby it is no longer in harmony in general. Such a system is no longer conservative but dissipative. In the differential equation meets to accelerating force F added a frictional force FR.

The sign of the force is opposite to the velocity. The exact expression for FR depends on the type of friction. The amount of F may be constant, or, for example, have a linear or quadratic function of the speed.

In the case of sliding friction, the amount of FR is constant:

An example for a linear dependency, the air friction at low speeds. There, the air flow can be considered to be laminar. Thus, it is according to the law of Stokes proportional to the velocity, so the first time derivative of the displacement.

In the case of such a linear damping factor of proportionality is called the damping constant.

Linear Damping

With linear damping, the friction can be generally provided by a damping term, which is proportional to and aligned opposite to the speed. The constant is referred to as decay. This gives the equation of motion of a linearly damped oscillation as an ordinary linear differential equation of second order:

Denotes the undamped natural angular frequency of the oscillator. With a Exponentialansatz leads to the general solution

Here, and complex-valued constants that are determined from the initial conditions.

• In the case of weak damping (), this results as in the undamped case, a sinusoidal oscillation whose amplitude but decays exponentially. The strength of this waste is determined by the exponent of the exponential envelope: The amplitude falls in the period on the original amplitude from. The angular frequency of the oscillation is reduced to itself.
• In the case of strong damping (), called Kriechfall, no real vibration more forms. Rather, the deflection creeps towards the rest position.
• In the aperiodic limiting case (), the vibration still achieves maximum deflection (), but then falls faster than in the case of strong damping to the rest position. Whether a zero crossing occurs is dependent on the initial conditions.

Are you in front of the initial conditions and the time, then you get in the resonant case, the particular solution

With

For the special case, ie without damping, the solution simplifies to

For the aperiodic limiting case results

Variant: torsion oscillator

A variant of the classical harmonic oscillator, the torsion oscillator dar. Instead of a coil spring is here a torsion spring or a torsion wire used. Then instead of translational movements to rotational movements occur. The basic calculation on the same path. It simply replaces the mass by the moment of inertia and the speed by the angular velocity.

Description in the Hamiltonian dynamics

The equation of motion of the harmonic oscillator can be derived with the Hamiltonian mechanics. As above, we consider a mass on a spring with spring constant. As a generalized coordinate is used. The Hamiltonian consists of potential and kinetic energy as follows:

With the canonical equations

You get to the above described equation of motion.

Since the total energy is conserved (), form, position and momentum an ellipse with semi-axes and off.

The total energy of the system is proportional to the area enclosed by the ellipse

In the case of a damped oscillator, the trajectory is an ellipse instead of a spiral that moves toward the origin.

In a multi-dimensional harmonic oscillator can dial in using the main axis transformation and along the main axes of the potential. With such a choice decouple the equations of motion of the individual directions.

Multi-dimensional oscillator

For example, with a Hamiltonian approach as explained in the previous section, can the force law for an n-dimensional harmonic oscillator formulated as:

It is seen that the differential equations are decoupled, that is, the component of force in one dimension depending only on the displacement in that dimension. Therefore, the solutions for the individual components of the position vector are the solutions of the corresponding one-dimensional problem:

The eigenvalues ​​correspond to the natural angular frequencies. Can be written as an integer multiple of a constant all, the oscillation of the harmonic oscillator is periodic. An isotropic harmonic oscillator is therefore always periodic.

Two-dimensional oscillator

The trajectories on which such particles are moved Lissajous curves. An example of an isotropic harmonic oscillator in two dimensions is the Foucault pendulum with small deflections. In an isotropic oscillator, the solution can be written as:

Where the constants are determined by the initial conditions.

The harmonic oscillator outside the mechanics

Electrical resonant circuit

The electrical resonant circuit is a harmonic oscillator in electrodynamics. Whereas in the periodic mechanical potential and kinetic energy can be converted into one another, the electric energy stored and interchanged in a capacitor with a capacity in a coil having the inductance of the magnetic energy stored in the resonant circuit. This results in a differential equation for the current:

The similarity with the equation of motion of the mechanical oscillator is obvious. The following table is to make analogies between the mechanical and electrical oscillator clear:

The harmonic oscillator in quantum mechanics

As an arbitrary potential can be developed to provide a stable position of equilibrium of the harmonic oscillator in quantum mechanics is a standard model for particles within a periodic potential. It is one of the few systems for which an analytical solution is known.

In quantum mechanics, position and momentum of a particle are replaced by operators. The wave functions, which can calculate the probabilities of a particle, are eigenfunctions of the Hamiltonian. The energy levels correspond to the eigenvalues. The Hamiltonian of a harmonic oscillator is given by

Lorentz oscillator in the optical

The Lorentz oscillator is used in optics as a model for the behavior of the atoms of a solid under the influence of an electromagnetic wave to describe. For example, then it is the susceptibility, which counteracts the development of the field, the analogue of the damping due to friction in the mechanism. With the help of the Lorentz oscillator can be in Drudemodell optical phenomena such as birefringence or the complex refractive index explain.

Excitation harmonic oscillators

If an oscillator energy is added, it is called excitation. For the mechanical oscillator so that either acts an external force is, or parameters of the oscillator will change as the natural frequency. The excitation of quantum-mechanical oscillators is done as in the article Harmonic Oscillator ( quantum mechanics) described, by means of conductor Opara gates. The discharge of energy, also called de-excitation occurs analogously.

Forced oscillation

A forced oscillation is excited by an independent, usually periodic force or voltage. An example is a dipole antenna. The differential equation, here the example of the damped oscillator is inhomogeneous by:

Self excited vibration

From a self-excited vibration occurs when the power supply is controlled by a suitable control and the vibration process itself. Mathematically, such a power supply, for example, realized by a special damping term, in which the damping can be negative. Such a system is usually non-linear. An example is of the van der Pol oscillator.

Parameter -excited vibration

If varies over time due to changes in parameters such as the length of a pendulum, the natural frequency of a harmonic oscillator, one speaks of a parametrically excited vibration. An example is a swing with small deflections.

Coupled harmonic oscillators

A multi-dimensional harmonic oscillator in which the individual components, ie the harmonic oscillators along the main axes of the potential, are not independent but interact with each other is referred to as being coupled. This means that the energy of the vibration of the individual components does not have to be obtained because they can be transferred by the interaction of one component to another.

Coupled mechanical oscillators are also called coupled pendulums. A mechanical interaction between two pendulums is generated for example by connecting the masses of two separate pendulum with a spring. If more of the same pendulum, arranged in a row, each with its immediate neighbors are connected by springs, refers to the arrangement as Schwinger chain. An interesting embodiment in which the power between a translational motion and a rotary motion changes, the Wilberforce pendulum.

With the help of coupled oscillators and lattice vibrations in crystals, for example, can be modeled. Here, the electrical interaction between the ions provides, molecules or atoms of the crystal lattice for the necessary coupling. The quantum-mechanical consideration in the article Harmonic Oscillator ( quantum mechanics) then leads to the phonons.

Continuum transition

Vibrations of a continuum, such as a string vibration can be described by means of an infinite dimensional coupled harmonic oscillator or an infinite number of one-dimensional coupled harmonic oscillators. The transition to an infinite number of oscillators is subsequently performed for a longitudinal wave. The method can be analogously carried out for transverse waves.

We take the example of coupled oscillators on the ground, which are connected by springs with spring constants. The deflection of the i-th oscillator is denoted by. The spacing of the individual masses. The Lagrangian of this system is then:

The equation of motion of the system can be derived from it as:

We share this equation by and get:

Through a continuum transition of the discrete index by a continuous coordinate, and the discrete function is replaced by the wave function. For such a continuum limit is the same limit, and taken, so here following variables are held constant:

• The total length
• The total mass and therefore the density
• The product of the spring constant and spring length

The factor to the left of the equation is constant. One can therefore write this page as

The right side of the equation can be rewritten as:

This is just the difference quotient of the second derivative. Namely obtained using a Taylor expansion around

This gives the wave equation