Quantum harmonic oscillator

The harmonic oscillator in quantum mechanics analogous to the harmonic oscillator in classical physics describes the behavior of a particle in a potential of the form

Such a quadratic potential is also called a harmonic potential. Classic gives this potential for a system whose restoring force is proportional to the displacement from the equilibrium position.

Since an arbitrary potential in the neighborhood of a stable equilibrium can be described approximately as a harmonic potential, this is one of the most important model systems in quantum mechanics. It is also one of the few quantum mechanical systems for which an exact analytical solution is known.

Hamiltonian and eigenstates in the one-dimensional case

The Hamiltonian in quantum mechanics, the total energy ( kinetic energy potential energy) describes, is for the harmonic oscillator

Where m is the mass of particles, the natural angular frequency ω of the oscillator of the local operator, the operator in the pulse space representation and the Laplace operator.

The stationary Schrödinger equation

Is so for the one-dimensional harmonic oscillator in the coordinate representation

The eigenfunctions of the harmonic oscillator are obtained by solving these linear differential equation. There are the Hermite functions

Here are the Hermite polynomials:

So

The term describes the exponential decay outside the oscillator potential.

The ground state takes the form of a Gaussian curve

The adjacent chart above shows the first eight solutions. In addition to the wave functions including the amount of which is shown square, which. The probability of a particle at a given harmonic potential indicating ( respectively blue parabola in the graphs on the right)

The corresponding energy levels are obtained as

With positive integer or zero.

Zero-point energy

The state with the lowest energy is thus above the potential minimum. Thus the particle is in accordance with the heisenberg 's uncertainty relation is not exactly in localized, as one would expect from a classical oscillator. This is known as a zero-point energy or zero -point vibration.

The zero-point energy into the quantum field theories to more far-reaching consequences.

The ladder operator method

The problem of the harmonic oscillator in quantum mechanics can be treated favorably by the method of creation and annihilation operators. It was developed by Paul Dirac, based on work by Niels Bohr and Otto Wiener, developed. This approach is also called algebraic method.

For this approach we define two operators, and each drain a quantum of energy of an oscillator or add. They are called so annihilation and creation operators. Other common names are ladder operators and Aufsteige-/Absteigeoperator. The notation (see also Bra- Ket notation) is converted into the simpler notation for this. Such a state is called Fock state or occupation number state because it indicates the number n of the energy quanta in the oscillator. We define these operators so that they satisfy the following relations:

This formula makes the naming immediately plausible, because the application of a higher energy level leads to a lower level and vice versa. For these operators are still the so-called occupation number operator can be put together, the ( ie the number n ) returns the number of energy quanta in a state:

Now can the Hamiltonian with these new operators to rewrite to:

The operators and are represented by the canonical operators and represent:

• Annihilation operator:
• Creation operator:

This directly follows the presentation of and:

To determine the eigenfunctions can now be the Schrödinger equation for the lowest state explicitly solve ( this is a very simple differential equation) and so receives its local representation. All other states is then obtained via the recursive application of the creation operator on this ground state:

This method is a very elegant way to treat the harmonic oscillator. But she still has much broader applications. If you imagine about electromagnetic radiation of photons composed before, so easy to get to also establish creation and annihilation operators for photons. In fact, can even show that one can describe the electromagnetic field as a collection of harmonic oscillators. It is always a light wave oscillator for certain frequency ω. In this case, then n is the number of photons in this "mode" of the light field. General called such an approach second quantization. A detailed calculation of the eigenvalues ​​of the ( one-dimensional ) oscillator can be found in the article creation and annihilation operator with bosonic climbing operators.

Classical limit

In the limit of large quantum numbers n, the probability goes over into the classical probability. This classical probability density is proportional to the inverse rate 1 / v. The smaller the rate v of the particle in conventional potential, the longer it remains on a corresponding place. The speed can be derived directly from the energy conservation law. The following figure shows the classical and the quantum mechanical probability density. To be greater n is, the more similar the curves:

Quasi- classical states

Bring to a localized wave packet in a harmonic potential ( see figure at right ), so it behaves like a classical particle in this potential (hence quasi-classical state). Is it the potential edges, so it will turn around and run back. Effectively then feeds it from a vibration in potential.

Mathematically, these states correspond to the so-called coherent states. They are characterized by a complex number α and can be represented as a linear combination of the states:

Important are those states in the description of coherent radiation, since one can show that the light field in the quantum field theory on harmonic oscillators (one for each mode of the field) can be traced back (see also coherent radiation ). The probability distribution of the occupation numbers of coherent states follows (like the photon statistics of coherent light ) of the Poisson distribution:

A classical state of the quasi -like state is produced when stimulating a diatomic molecule ( such as hydrogen H2 ) using intense femtosecond lasers. It has already been explained, that can be used as an approximation of the harmonic oscillator, the oscillation of diatomic molecules. The following figure shows the events is shown:

First, a low-lying narrow wave function is raised to a higher energy state with a laser pulse. There she remains localized and begins as a " quasi-classical state" in the potential move. A second pulse is emitted for the measurement then ionizes the molecule. The position of the wave function gives the distance of the atoms in the molecule. From the kinetic energy of the fragments can be deduced that distance, and the shape of the wave packet.

N- Dimensional harmonic oscillator

The one-dimensional harmonic oscillator can be easily extended to the N- dimensional case. The Hamiltonian in N dimensions is

It is clear that the N -dimensional harmonic oscillator exactly N independent one-dimensional harmonic oscillators corresponds with the same mass and restoring force, as a sum of N independent eigenfunctions for each coordinate according to the above one-dimensional scheme solves the Schrödinger equation. This is an advantageous feature of the potential ( cf. Theorem of Pythagoras ), which allows the potential energy to separate into terms that depend only on one coordinate each.

The possible energy levels resulting in correspondingly

Note, however, that the energy levels of the N-dimensional oscillator according to the combinatorial possibilities for the realization of the level - fold degenerate are as follows:

Coupled harmonic oscillators

Considering the simplest case, a system of two -dimensional particles which are coupled to each other only by a harmonic force, we obtain for the Hamiltonian

As a simple separation of the Schrödinger equation initially prevented a transformation offers in gravity coordinates:

To succeed on a separation:

This corresponds to a single harmonic oscillation with respect to the vibration of the differential 2 particles ( double reduced mass ), wherein the system further moved as a whole as a free particles (). The solution of the Schrödinger equation leads according to the harmonic energy levels

For a chain of N such pairs harmoniously consecutively coupled particles ( one-dimensional lattice ) can be found similarly a coordinate transformation such that independent collective harmonic oscillations (plus a local center of mass motion ) result.

For 3- dimensional crystal lattices in solid state physics, this observation leads to the phonons.

Applications

The harmonic oscillator is an important model system of quantum physics, as it is one of the few closed ( ie without approximations and numerical methods ) detachable systems of quantum mechanics. With it, a number of physical issues can be approximately described:

• In molecular physics, it allows an approximation of the bonding between atoms and thus allows, for example, a prediction of vibrational spectra. This can be illustrated by a bond using a spring ( harmonic potential ) in interconnected mass points (the atoms) that vibrate against each other shown:

• Another example is the torsional vibration of Ethenmoleküls shown in the following diagram:

• In the modern atomic physics to be investigated atoms and ions are trapped and cooled, for example, to obtain a higher resolution when making measurements in optical traps or ion traps. In addition, one can in such cases, new states of matter study (eg Bose -Einstein condensates, Fermi condensates). Such traps have, to a first approximation, parabolic potential. Thus, particles can be described with the model of the quantum harmonic oscillator in these cases also.
• In solid-state physics, the Einstein model ( Albert Einstein) describes a method to calculate the contribution of the lattice vibrations ( phonons) to the heat capacity of a crystalline solid. It is based on the description of the solid consisting as quantum harmonic oscillators of N that can oscillate independently of each in three directions. Also phonons may also be described by a collection of coupled harmonic oscillators. In this case, each atom in the crystal lattice of an oscillator which is coupled to its neighboring atoms.

Swell

• . Claude Cohen- Tannoudji, Bernard Diu, Franck Laloë, Franck: Quantum Mechanics 1/2, 2nd edition, Walter de Gruyter, Berlin - New York 1999
• Jun John Sakurai: Modern Quantum Mechanics. Addison-Wesley

Left

• Complete solution of the harmonic oscillator using the algebraic method ( starting on page 63; pdf file; 4.26 MB )
• Representation of the direct solution of the harmonic oscillator in the coordinate representation
• Harmonic oscillator, quantum mechanically

Individual sources

• Quantum mechanics