Schrödinger equation

The Schrödinger equation is the temporal evolution of the unperturbed non-relativistic quantum systems underlying differential equation. It describes the dynamics of the quantum mechanical state of a system, as long as no measurement is made on this. It is thus a fundamental equation of non-relativistic quantum mechanics.

The equation was first suggested in 1926 by Erwin Schrödinger ( 1887-1961 ) as a wave equation and successfully used already in its initial application to explain the spectra of the hydrogen atom.

The Schrödinger equation states that the temporal change of a state is determined by its energy. In the equation, the energy does not occur as a scalar value, but as the operator ( Hamiltonian ) which is applied to the state.

If the quantum system has a classic analogue ( eg, particles in three-dimensional space ), the Hamiltonian can be extracted according to a prescription -like rules from the classical Hamiltonian. For some systems Hamiltonians are also directly by quantum mechanical point of view constructed (example: Hubbard model).

As a special case of the temporal evolution of the Schrödinger equation describes the states of a quantum system, in which the absolute square of the wave function with time does not change ( stationary states, eigenstates of the Hamiltonian ), and allows the calculation of the defined by such states energy levels.

The Schrödinger equation is the basis for almost all practical applications of quantum mechanics. Since 1926, succeeded with it the explanation of many properties of atoms and molecules (where the electron wave functions are called orbitals ) and solids ( band model ).

General form of the Schrödinger equation

The Schrödinger equation in its most general form is

This denotes the imaginary unit, the reduced Planck's constant, the partial derivative with respect to time and the system Hamiltonian. The Hamiltonian acting in a Hilbert space, the quantity to be determined is a state vector in this space. This generic form of the Schrödinger equation in relativistic quantum mechanics and in quantum field theory. In the latter case, the Hilbert space is a Fock space.

Derivation of the Schrödinger equation in position representation

The eponymous equation was postulated by Schrödinger in 1926. The starting point here was going back to Louis de Broglie concept of matter waves and the Hamilton -Jacobi theory of classical mechanics. The action S of classical mechanics is identified with the phase of a matter wave. Once typical distances are smaller than the wavelength, diffraction phenomena play a role and no longer adequate classical mechanics must be replaced by a wave mechanics.

Formally created the Schrödinger equation in the coordinate representation by the correspondence principle from the Hamilton function ( expression for the energy ) of the considered problem.

By replacing the classical quantities energy, momentum and carried out by the corresponding quantum mechanical operators ( correspondence principle ):

Then applying to the unknown wave function yields

In the same way, the Hamiltonian can be transformed into a Hamiltonian.

Historically Schrödinger went out of Louis de Broglie's description of free particles and resulted in his work analogies between atomic physics, electromagnetic waves, in the form of de Broglie waves, a:

Where is a constant. This wave function is a solution of the Schrödinger equation with just mentioned. In the usual statistical interpretation of quantum mechanics ( founded by Max Born) gives her absolute square of the probability density of the particle.

Another way to set up the Schrödinger equation, used, introduced by Richard Feynman path integral of the term. This alternative derivation considers the probabilities of the different motions (paths ) of the particle to be analyzed from a location A to B and thus leads back to the same Schrödinger equation. Again, the classical action S plays a central role.

Particles in the three-dimensional space

The complex-valued wave function of a point particle in a potential is a solution of the Schrödinger equation

Wherein the mass of the particle, its location, the Laplace operator, and the time is. The potential is initially assumed here for simplicity as a scalar potential. For a free particle, on which no external forces act, shall apply.

The Schrödinger equation is a linear partial differential equation of second order. Due to the linearity of the superposition principle applies: if and are solutions, so is also a solution, where and are arbitrary complex constants.

With the Hamiltonian

Can the Schrödinger equation in its general form

. Write

Stationary solutions

For a system with Hamiltonian with no explicit time dependence is the approach

Obvious. In this case, the time dependence of the state vector is expressed by a factor ω e - iωt constant frequency. For the time-independent factor of the state vector, the Schrödinger equation is the eigenvalue equation

According to the Planck's formula, such a system has the power

Discrete eigenvalues ​​correspond to discrete energy levels of the system ( " Quantization as an eigenvalue problem ").

Normalization of the wave function

For the statistical interpretation of quantum mechanics, it is necessary to standardize the solutions of Schroedinger equation so that

Is. This so-called normalization condition states that the probability that the particle is to be found anywhere in the entire space is 1. The thus obtained solution then normalized corresponding to the probability density of the particle at the site at time t. However, not every solution of a Schrödinger equation can be standardized. If existent, this normalized solution up to a phase factor of the form of a real, but this is physically meaningless, uniquely determined.

From the fact that the Schrödinger equation is invariant under the phase transformation (U ( 1 ) symmetry ) follows by means of Noether 's theorem, the conservation of normalization, that is, the probability is conserved.

Expected values ​​of measured variables

From the wave function thus found now found all this physical properties of the particle. For example, the classical value for the position of the particle by the mean position of the particle is at the time, so

Replaced while the classical value for the momentum of the particle is replaced by the following average:

In principle, then every classical measure is replaced by an averaging of the corresponding operator on the space in which the particle is, :

The term is also referred to as the expected value of. The expectation value of the energy is the same.

Different representations

Position and momentum representation

The most famous and important representation of the Schrödinger equation is the coordinate representation:

Another relevant representation is the momentum representation:

Position and momentum representation go by means of Fourier transform into each other:

Representation -independent form

In the display -independent form, a quantum state is described by a vector of a complex unitary Hilbert space. Usually the Dirac notation is used with Bra and Ket. The presentation independent notation facilitates the transition to the mathematical treatment of quantum mechanics, where concepts of functional analysis are used. The structure of the Hilbert space is determined by the system under consideration. For the description of the spin of a particle with spin 1 /2 of the Hilbert space is two-dimensional example, for a particle in a box with infinitely high walls or for a harmonic oscillator its dimension is countably infinite. A free particle is described in an ( improper ) Hilbert space with uncountably infinite dimension.

The representation -independent form of the Schrödinger equation for a Hilbert space vector ( ket )

And for a dual Hilbert space vector ( Bra)

The Hamiltonian in the absence of magnetic fields

Which here are also to be regarded as independent basis the operators and.

The time evolution of the states will be described as in the above equation by the application of a Hamiltonian to the states. " Integrated out " we obtain the time evolution operator:

The time evolution operator has for time-independent Hamiltonians the simple form:

The standard of a state is equal to the L2 norm induced by the inner product:

The probability conservation ( conservation of the norm of the state ) is expressed by the unitarity of the time evolution operator, which in turn due to the fact that is self-adjoint. With and follows:

Substituting the conservation of probability density in the theory requires that the time evolution operator must therefore be unitary. The change in a time-dependent state is therefore determined by an anti - Hermitian operator, which you can already start before knowledge of the Schrödinger equation without loss of generality. This reduces the postulation of the Schrödinger equation to the determination of the shape of the Hermitian operator.

The hermeticity is a requirement which is imposed on all operators of quantum mechanics, which represent measurement results according to the correspondence principle, moreover. Since measurement results must always be real, come as assigned to operators only Hermitian operators in question. Such operators are called observables.

Notes

With the Schrödinger equation, the ad hoc construction of the Bohr model of the atom was overcome ( as previously with the more cumbersome Heisenberg matrix mechanics ). The discrete energy levels of the hydrogen atom, which must be assigned in the Bohr model certain classical orbits of an electron in the Coulomb potential of the nucleus, resulting in the framework of the Schrödinger equation as the eigenvalues ​​of the Schrödinger equation for an electron in the potential of the atomic nucleus.

While the path of a particle is determined in classical mechanics by Newton's equation of motion in quantum mechanics, the Schrödinger equation instead delivers a probability distribution for the location of the particle. One speaks sometimes illustrative assumption that the particle is delocalized over the room. As a more comprehensive theory of quantum mechanics, classical mechanics, however, must contain. One form of this correspondence is prepared by the Ehrenfest theorem. The theorem states, inter alia, that the mean of Teilchenkoordinate satisfies the classical equation of motion. Relevant and evident is the correspondence in localized coherent wave packets. Such wave packets can be at higher quantum numbers, eg construct at higher excited states of the hydrogen atom.

In the Schrödinger equation, the wave function and the operators in the so-called Schrödinger picture occur. In the Heisenberg picture instead equations of motion for the operators themselves are considered. These equations of motion are referred to as the Heisenberg equation of motion. The two formulations are mathematically equivalent.

The Schrödinger equation is deterministic, meaning that their solutions are clearly in default of initial conditions. On the other hand, the solutions of the Schrödinger equation, after the Copenhagen interpretation of statistical variables from which follow only statements about the mean values ​​of measurement results in similar experimental setups. According to the Copenhagen interpretation of quantum mechanics this is not due to a shortage of the measurement arrangement, but this is due to the nature itself.

Solutions

Even for systems with only one particle can be solved in closed form, the Schrödinger equation only in a few cases. Potentials for which this is the case, for example, a finite or infinitely deep potential well, the three-dimensional Coulomb potential ( hydrogen atom), the potential barrier are (results tunnel effect), the harmonic potential ( harmonic oscillator ) and the Morse potential. In other cases one has to rely on series expansions, other approximation methods and numerical techniques. are important

• Perturbation theory
• Calculus of variations
• WKB approximation and semi- classical development
• Hartree- Fock approximation and extensions
• Density functional theory

In the case of fewer particles can solve the Schrödinger equation numerically with computer programs. In Vielteilchenfall such as in most atoms and almost all molecules ( eg in the field of theoretical chemistry ) are encountered, however, soon to a quasi- insurmountable hurdle: In the N-particle case is a wave function in the 3N -dimensional configuration space to be determined. Using Q ( base or variation ) values ​​for each dimension, then Q3N values ​​to determine - in the case of this carbon atom with N = 6 and q = 5 for example, a large expense. Walter Kohn has referred to this exponential growth resources as " Exponentialbarriere ". The density functional theory is a highly developed method to successfully circumvent the Exponentialbarriere. Another option for this would be a quantum computer.

The value N = 6 as a barrier is of course an oversimplification. Because of symmetries, new approaches and growing computational power are also, for example Systems with N = 10 without DFT calculated. In contrast to the DFT, this procedure gives any accurate results. Can be described with appropriate approaches with Hartree- Fock methods and extensions also systems with many atoms.

Schrödinger equation for charged particles in an electromagnetic field

Note: Electrodynamic sizes are given here in the CGS system of units

If the particles, as in the case of an electron or proton, an electric charge has, generalized to the presence of an external electromagnetic field, the one-particle Hamiltonian in the local representation to

In which case q is the electric charge of the particle ( q = -e for electrons), c is the speed of light in vacuum, A is the vector potential, and Φ designate the scalar potential. The location resulting Schrödinger equation occurs in place of the classical equation with the Lorentz force. The potentials are linked by the following relations with the electric field E and the magnetic field B:

The Hamiltonian of a many-body system is the sum of one-particle Hamiltonians and the interaction energies (eg, the Coulomb interactions between the particles ).

A simple model for chemical bonding

This example describes a simple model for chemical bonding (See Feynman Lectures. ). An electron is bound to an atomic nucleus 1 and is located in the state, or to a nucleus 2 and is located in the state. If no transitions are possible, in each case the stationary Schrödinger equation. If transitions are possible after, the Hamiltonian must produce when applying an admixture of state to state, and for transitions to analog. A parameter determines the transition rate. The system is modeled as follows:

By adding and subtracting these equations we see that there are new stationary states in the form of superpositions of and:

Because for this can be found with elementary algebra

The pre-factors of the stationary states are again interpreted as measurable energies. One of the two energies ( depending on the sign of ) is smaller than the original one. The corresponding superposition state is the bonding state of the molecule.

Schrödinger equation in mathematics

In mathematics, the Schrödinger equation is investigated in a Hilbert space and it must be shown first that the Hamiltonian is self-adjoint. Then it follows from the theorem of Stone the existence of a unitary group and thus the unique solvability of the initial value problem. It is important from a mathematical point of view, to distinguish self-adjointness of the weaker property of symmetry. The latter is easy to show, as a rule by a partial integration, for the self-adjoint a detailed investigation of the domain of the adjoint operator is necessary. For bounded operators, both terms coincide, but Schrödinger operators are unbounded in general and can not be defined on the whole Hilbert space by the theorem of Hellinger - Toeplitz. Then it applies the spectrum of to investigate the dynamics to understand.

Free Schrödinger equation

The free Schrödinger equation

Can be treated by means of Fourier transformation and the free Schrödinger operator is self-adjoint on the Sobolev space. The spectrum is the same.

Preservation of the Hs - norms

Easy to see by Fourier transformation. This is reflected in the case of the conservation of probabilities.

Dispersion

It is

This property expresses the deliquescence of the wave packets.

Schrödinger equation with potential

The Schrödinger equation with a potential

Can be treated with methods of the perturbation theory. For example, it follows from the theorem of Kato - Rellich: Applies in three (or fewer) dimensions, is bounded and vanishes at infinity and is square- integrable, then to self-adjoint and the essential spectrum. Under the essential spectrum, it can give maximum countable number of eigenvalues ​​can accumulate only at zero. These conditions cover in particular the Coulomb potential, and thus the hydrogen atom,

Which is explicitly solvable by separation in spherical coordinates. Considering atoms of more than one electron or molecules, the self-adjoint was later proven by Tosio Kato. The structure of the essential spectrum is described in this case by the HVZ theorem ( according to W. Hunziker, C. van Winter and GM Zhislin ). Such models can only be usually solved numerically and even that is extremely time-consuming due to the large number of equations.

The one-dimensional Schrödinger equation is a special case of a Sturm-Liouville equation.

Nonlinear Schrödinger equation

A number of problems in physics leads to a generalization that the non-linear Schrödinger equation

Even with a non-linear interaction term. The explicit dependence of the solution function of time and place was omitted. Especially in the case of the cubic nonlinear Schrödinger equation, and one dimension is an integrable wave equations with Solitonenlösungen. In dimension one has in the cubic case, the Gross- Pitaevskii equation, the Bose -Einstein condensate describes.

View

The interaction of the spin or intrinsic angular momentum of the particle with an external magnetic field is not considered in the above form of the Schrödinger equation. If this interaction is not to be ignored is to use for an electron in the presence of an external magnetic field, the Pauli equation.

Unfortunately, however, the Pauli equation still has some fundamental flaws. It is for example not Lorentz invariant, but "only" Galilean - invariant (non- relativistic ). The correct relativistic generalization of the Schrödinger and also the more general Pauli equation, for electrons is the lorentzinvariante Dirac equation, a partial differential equation of first order, in contrast to the Schrödinger equation.