To what extent is the Schroedinger equation deterministic



The Schrödinger equation is a, resp. the, central basic equation of non-relativistic quantum mechanics. The solutions to this equation are also called wave functions. These wave functions describe the spatial and temporal development of the state of a quantum system. The equation was first established in 1926 by Erwin Schrödinger (1887-1961) as a wave equation. As the "equation of motion in quantum mechanics", it still forms the basis for almost all practical applications of quantum mechanics.

introduction

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

,

in which m the mass of the particle, its location, Δ the Laplace operator and t the time are. For the sake of simplicity, the potential is initially assumed here as a scalar potential. The constant , pronounced “h-across”, is the quantum of action and i the imaginary unit. In the case of a free particle on which no external forces act, the following applies .

If the potential does not depend on the wave function, which is true in most applications, the Schrödinger equation is linear, so that the superposition principle applies to the solutions: If ψ1 and ψ2 Solutions are, so is a solution, where α and β are arbitrary complex constants. From a mathematical point of view, the Schrödinger equation is then a linear, partial differential equation of the second order, which, like the heat conduction equation, is of the parabolic type.

With the Hamilton operator

the Schrödinger equation in its general form

write.

Heuristic derivation

The Schrödinger equation is, like Newton's axioms in classical physics, a postulate and therefore cannot be derived strictly mathematically. Rather, the equation was postulated by Schrödinger in 1926 as a new theory, taking into account certain physical principles and based on the quantum mechanical phenomena already known at the time. Some authors also emphasize the strong reliance on the principles of optics.

From a mathematical point of view, the Schrödinger equation arises in the position representation according to the correspondence principle from the Hamilton function (expression for the energy) of the problem under consideration

by replacing the classical quantities energy, momentum and location with the corresponding quantum mechanical operators (correspondence principle):

Then apply it to the unknown wave function results

.

The Hamilton function can be converted into a Hamilton operator in the same way.

Historically, Schrödinger started from Louis de Broglie's description of free particles and in his work introduced analogies between atomic physics and electromagnetic waves in the form of De Broglie waves:

,

where A is a constant. This wave function is a solution of the Schrödinger equation just mentioned . The physical meaning of the wave function remains open for the time being. In the usual statistical interpretation of quantum mechanics (founded by Max Born), its absolute square gives | ψ | 2 the probability density of the particle.

A completely different way of setting up the Schrödinger equation uses the method developed by the American physicist and Nobel Prize winner R.P. Feynman introduced the concept of the path integral. This alternative derivation considers the probabilities for the different movements (paths) of the particle to be examined from a location A to B and thus comes back to exactly the same Schrödinger equation.

Normalization of the wave function

For the statistical interpretation of quantum mechanics it is necessary to normalize the solutions of the Schrödinger 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 100%. For the so obtained normalized Solutions then corresponds the probability density of the particle at the location at the time t. However, not every solution of a Schrödinger equation can be normalized. If it exists, this standardized solution is the form except for a phase factor e(iK) for a real one K, which, however, is physically meaningless, clearly determined.

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

Expected values ​​of measured quantities

From the wave function found in this way, all the sought-after physical properties of the particle result. For example, the classic value for the location of the particle is by the mean position of the particle at the time t, so

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

.

In principle, every classic measured variable is used f (r, p, t) replaced by an averaging of the associated operator over the space in which the particle is located:

.

The expression is also called the expected value of f designated. The expected value of the energy is the same.

Explanations

While the orbit of a classical particle via the classical Newtonian equation of motion

is precisely defined, in quantum mechanics the position of a particle appears only as an expected value with distribution functions ψ, which are given by the Schrödinger equation. It is therefore often said that the particle is delocalized over space. If the width of the wave packet that the particle describes is sufficiently small, Newton's equation of motion can be derived as a limiting case with the help of the Schrödinger equation. The same result also results from the formation of the limit and is known in physics as the Ehrenfest theorem. This theorem contributed significantly to establishing the Schrödinger equation as a serious physical theory.

The Schrödinger equation given above is the one in the so-called position representation. In the form independent of a specific representation, the Schrödinger equation reads with the “ket” vector :

,

where here the operators and are also to be regarded as independent of the base. This notation facilitates the transition to the mathematical treatment of quantum mechanics, whereby terms of functional analysis are used. The states of a quantum mechanical system are described there by vectors in infinitely dimensional vector spaces (Hilbert spaces) - usually also written in the specified Ket notation - and their time evolution is described, as in the above equation, by applying a Hamilton operator H to the states. "Integrated" then results for time-independent H:

and the conservation of probability (conservation of the norm of the state) is expressed by the unitarity of the time evolution operator U, which in turn is based on the fact that H is self-adjoint and thus has only real eigenvalues. Incidentally, this is a requirement that is placed on all operators who represent measurement results according to the correspondence principle. Since measurement results are always assumed to be real, only Hermitian or self-adjoint operators can be used as assigned operators. Operators that are self-adjoint and represent measurement results are also called observables.

In the Schrödinger equation, the wave function and the operators appear in the so-called Schrödinger picture. In the Heisenberg picture, equations of motion for the operators themselves are considered instead. These equations of motion are then also referred to as Heisenberg's equation of motion. Both equations are physically equivalent.

On the one hand, the Schrödinger equation is deterministic, which means that the solutions are precisely determined if sufficient initial and boundary value conditions are specified. On the other hand is their solution ψ However, according to the Copenhagen interpretation, it is a statistical variable and consequently only makes statements about the mean values ​​of measurement results in similar test arrangements. According to the Copenhagen interpretation of quantum mechanics, this is not due to a lack of the measuring arrangement or the experimenter, but is due to the physics and the nature of the atomic particles themselves.

With the formulation of the Schrödinger equation, the ad hoc construction of Bohr's atomic model was overcome. The only postulated discrete energy levels of the electrons in atoms result here from the solution of a partial differential equation, namely the Schrödinger equation already mentioned -Potential at the hydrogen atom. The next section will explain this relationship in more detail.

Solution of the Schrödinger equation

The solutions to the Schrödinger equation are usually called Wave function but do not always resemble a classical wave function. Therefore, the solutions will be presented in more detail below.

  • In the case of a Hamilton operatorH = H (r, t) which depends explicitly on the time, an initial value problem has to be solved. Is the crowd m of the particle, the externally applied potential V.(r, t), the initial condition at the time t0 as well as the boundary conditions of ψ For t > t0 known, this results for the space under consideration and for all times t > t0 the wave function ψ (r, t) as a solution to the Schrödinger equation.
  • In the case of a time-independent Hamilton operator H = H (r) and thus also time-independent potentials V = V (r) and fixed edges provides the solution to the Schrödinger equation ψ = ψ (r, t) on the other hand represents a so-called stationary state or a superposition of stationary states. These states result as a solution to the following boundary value problem. By separating the variables according to the approach

with the real parameter E., which here stands for the energy of the steady state, one arrives immediately at the following time-independent Schrödinger equation:

.

Together with the boundary conditions ψ (r) the time-independent Schrödinger equation forms a so-called eigenvalue problem in which the energy eigenvalues E. and the associated eigenfunctionsψ (r) are to be determined. A simple example of such an eigenvalue problem is an electron in a box with no potential and impenetrable edges.

As in the case of the time coordinate just shown, a separation approach can often also be found for the location coordinates after a suitable coordinate system has been selected. The Schrödinger equation is then reduced from a partial to an ordinary differential equation of the second order. The vector space of the solutions of this ordinary differential equation of the second order is spanned by two linearly independent fundamental solutions. This applies in particular to problems with spherical symmetry (symmetrical under rotations about a point) and cylindrical symmetry (symmetrical under rotations about an axis).

The Schrödinger equation can be solved exactly for some simple potentials, e.g. B .:

In the case of more complicated potentials, an exact solution is usually no longer possible and a form of perturbation theory is applied. This means, if possible, the Hamilton operator of the Schrödinger equation is divided into one part , of which the solution is known, and a small remainder and then develops the solution as a fault series from the known solution.

The WKB approximation is a simple way of finding approximate solutions to the stationary Schrödinger equation in areas in which the potential changes only slightly over the mean wavelength of the particle.

In systems with several particles in one potential or in general with several (mutually interacting) particles, the Schrödinger equation must be simplified (e.g. Born-Oppenheimer approximation in molecular physics) or approximately solved. Despite the simplification, is an analytical solution still impossible, e.g. For example, for most atoms and all molecules, iterative approximation methods or other numerical methods must be used. In the case of many particles (e.g. in the field of theoretical chemistry), the Hartree-Fock method and its extensions are often used for this.

Lagrangian of the Schrödinger equation

The Lagrangian of the Schrödinger equation is

With . The Schrödinger equation then follows in analogy to the classical case from Lagrange's equations of motion, only here a differentiation is made according to the fields and the derivatives of the fields instead of location and speed.

Schrödinger equation for charged particles in an electromagnetic field

If the particle has an electrical charge, as in the case of an electron or proton, the Hamilton operator in the position representation becomes generalized in the presence of an external electromagnetic field

,

being here q the electrical charge of the particle (q = -e for electrons), c the speed of light in a vacuum, the so-called vector potential and Φ the scalar potential. The resulting Schrödinger equation takes the place of the classical equation with Lorentz force. The potentials are by the following relationships with the electric field or the magnetic field connected:

.

Usually the movement of charged particles in external (given) electromagnetic fields () considered. Conversely, of course, affects the wave function ψ also the external electromagnetic field in the following way. Out ψ and q an electrical current density and the charge density of the particle can be calculated again and these fields generate their own electromagnetic field, which reacts on all external charges and currents that cause the external field. If this were taken into account in the Schrödinger equation, this would be an example of a non-linear Schrödinger equation, which is much more difficult to handle except in the case of small disturbances.

Schrödinger equation in mathematics

Outside of physics, the Schrödinger equation enjoys a great deal of interest in mathematics. A large number of mathematicians are currently engaged in the investigation of existence and uniqueness questions, qualitative investigations of the properties of special solutions of the equations (e.g. investigating solitons of the nonlinear Schrödinger equation) and the numerical solution of the equations.

Form of the equation

Nature is not necessarily as important to a mathematician as it is to a physicist. Therefore the constants are left out, the notation is partially abstracted. The Schrödinger equation as a Cauchy problem has the following form:

.

The explicit dependence of the solution function u on time and place has been omitted. As the occurrence of the solution u in the potential term f (u) already indicates, the nonlinear Schrödinger equation is also a preferred object of investigation here.

Matching rooms

For the treatment of the Schrödinger equation, the Sobolev spaces are chosen uniformly. We denote here for

for integer s and
For ,

in which S.'the dual space of the Schwartz functions, the Fourier transform as well F. − 1 denotes the inverse transformation. It has z. B. for the use of language half a derivative enforced. You can also define negative exponents: A function in H − 2 is a function (or rather distribution) that can be generated by deriving a L.2Function arises.

Properties of solutions

Preservation of the HsStandards

Easily seen by Fourier transform of the semigroup.

This expresses the conservation of the probabilities.

Dispersion

It applies

.

This characteristic expresses that Dissolve of the wave packets. Mathematically, the Schrödinger equation is related to the heat conduction equation.

outlook

The interaction of the particle's spin or its own angular momentum with an external magnetic field is not taken into account in the above form of the Schrödinger equation. If this interaction should not be neglected, the Pauli equation should be used for an electron in the presence of an external magnetic field.

Unfortunately, the Pauli equation still has some fundamental flaws. For example, it is not Lorentz invariant but 'only' Galilei invariant. The correct relativistic generalization of the Schrödinger equation and the more general Pauli equation represents the Lorentzine variant Dirac equation for electrons, which, in contrast to the Schrödinger equation, is a partial differential equation of the first order.

  • Torsten Fließbach: Quantum mechanics, Spectrum Academic Publishing House and other textbooks on quantum mechanics
  • Erwin Schrödinger: Quantization as an eigenvalue problem. In: Annals of Physics. Volume 79, 1926, pp. 361, 489, 734, and Volume 81, 1926, p. 109 (original works)
  • Peter W.Atkins: Molecular Quantum Mechanics and other textbooks on theoretical and physical chemistry
  • M. Alonso / E.L. Finn: Quantum Physics and Statistical Physics, Oldenbourgh
  • John Gribbin: In Search of Schrödinger's Cat - Quantum Physics and Reality.

Categories: Quantum Physics | Theoretical chemistry