Schrödinger equation
The Schrödinger equation is a fundamental equation in quantum mechanics . In the form of a partial differential equation, it describes the change in the quantum mechanical state of a non-relativistic system over time. The equation was first established by Erwin Schrödinger in 1926 as a wave equation and, when it was first applied, it was successfully used to explain the spectra of the hydrogen atom .
In the Schrödinger equation, the state of the system is represented by a wave function . The equation describes their change over time in that a Hamilton operator acts on the wave function. If the quantum system has a classical analogue (e.g. particles in three-dimensional space), the Hamilton operator can be obtained schematically from the classical Hamilton function . For some systems, Hamilton operators are also constructed directly according to quantum mechanical considerations (example: Hubbard model ).
In general, the wave function changes shape as a function of time. This can be used to describe physical processes such as B. the propagation, scattering and interference of particles. In the case of special wave functions, the Hamilton operator does not change the shape, but only the complex phase , so that the square of the magnitude of the wave function does not change over time. The corresponding states are stationary states , also referred to as eigen-states of the Hamilton operator. The Schrödinger equation enables the calculation of the energy levels defined by such states .
The Schrödinger equation forms the basis for almost all practical applications of quantum mechanics. Since 1926 it has been used to explain many properties of atoms and molecules (in which the electron wave functions are called orbitals ) as well as of solids ( ribbon model ).
History of the Schrödinger equation
The equation named after him was postulated by Schrödinger in 1926 . The starting point was the idea of matter waves going back to Louis de Broglie and the Hamilton-Jacobi theory of classical mechanics. The effect of classical mechanics is identified with the phase of a matter wave. As soon as typical distances are smaller than the wavelength, diffraction phenomena play a role, and classical mechanics must be replaced by wave mechanics.
The Schrödinger equation cannot be derived from classical physics, but is a postulate. Formally, however, the Schrödinger equation can be derived from the Hamilton function (expression for the energy) of the problem under consideration
can be derived by replacing the classical quantities energy, momentum and location with the corresponding quantum mechanical operators according to the correspondence principle :
Subsequent application to the unknown wave function gives the Schrödinger equation
- .
In the same way, the Hamilton function can be converted into a Hamilton operator.
Historically, Schrödinger started from Louis de Broglie's description of free particles and introduced analogies between atomic physics and electromagnetic waves in the form of De Broglie waves (matter waves):
- ,
where is a constant. This wave function is a solution of the Schrödinger equation just mentioned . In the statistical interpretation of quantum mechanics (founded by Max Born ), the square of the magnitude of the wave function indicates the probability density of the particle.
Another possibility to set up the Schrödinger equation uses the concept of the path integral introduced by Richard Feynman . This alternative derivation considers the probabilities for the various movements (paths) of the particle to be examined from one place to and thus leads back to the same Schrödinger equation. The classic effect also plays a central role here.
Schrödinger equation in generic form
The Schrödinger equation in its most general form is
In this case, referred to the imaginary unit , the reduced Planck's constant , the partial derivative with respect to time and the Hamiltonian (energy operator) of the system. The Hamilton operator works in a complex Hilbert space , the variable to be determined is a state vector in this space. This generic form of the Schrödinger equation also applies in relativistic quantum mechanics and in quantum field theory . In the latter case, the Hilbert space is a Fock space .
A quantum mechanical state is described by a vector in Hilbert space . Mostly the Dirac notation is used with Bra and Ket. 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, the Hilbert space is, for example, two-dimensional , for a harmonic oscillator its dimension is countably infinite . A free particle is described in an (improper) Hilbert space with uncountable infinite dimensions.
The time development described by the Schrödinger equation is a unitary transformation of the state vector in the Hilbert space. Since this is a linear transformation, the superposition principle applies . Another consequence is the possibility of quantum mechanical entanglement of non-interacting subsystems.
Time evolution operator
The time evolution of the states is described by applying a Hamilton operator to the states. "Integrated" you get the time development operator :
The time evolution operator for time-independent Hamilton operators has the form:
The norm of a state is equal to the L ^{2} norm , which is induced by the scalar product :
The conservation of probability (conservation of the norm of the state) is expressed by the unitarity of the time evolution operator , which in turn is based on the fact that it is self-adjoint . With and follows:
If one assumes the preservation of the probability density in the theory, then the time evolution operator must be unitary. The change in a time-dependent state is therefore determined by an anti-Hermitian operator, which means that even before knowledge of the Schrödinger equation, without loss of generality
can start. This reduces the postulating of the Schrödinger equation to the determination of the form of the Hermitian operator .
Hermiticity is a requirement made on all operators of quantum mechanics who represent measurement results according to the correspondence principle. Since measurement results must always be real, only Hermitian operators can be used as assigned operators . Such operators are also called observables .
Non-relativistic quantum mechanics of point particles
The equation established by Schrödinger is the prototype and special case of the general scheme. It describes the quantum mechanics of non-relativistic point particles, the Hilbert space is the space of complex-valued functions in the configuration space.
A single particle with a scalar potential
The complex-valued wave function of a point particle in a potential is a solution of the Schrödinger equation
- ,
where are the mass of the particle, its location, the Laplace operator and time.
The Schrödinger equation is a linear partial differential equation of the second order. Due to the linearity, the superposition principle applies: If and are solutions, then there is also a solution, where and are arbitrary complex constants.
With the Hamilton operator
the Schrödinger equation in its general form
write.
A charged particle in an electromagnetic field
Note: Electrodynamic sizes are here in the CGS system of units specified
If the particles, such as has in the case of an electron or proton, an electric charge, so an external electromagnetic field of the one-particle Hamiltonian generalizes in the presence in the spatial representation to
- ,
where here denote the electrical charge of the particle ( for electrons), the speed of light in a vacuum, the vector potential and the scalar potential. The resulting Schrödinger equation takes the place of the classical equation with Lorentz force . The potentials are linked to the electric field or the magnetic field by the following relationships :
The Hamilton operator of a many-particle system is the sum of the one-particle Hamilton operators and the interaction energies (for example, the Coulomb interactions between the particles)
Multiple particles
Multiple particles are represented by a single wave function . This wave function has the positions of all particles as well as time as parameters.
Meaning of the Schrödinger equation and explanations
General explanations
With the Schrödinger equation, the ad hoc construction of Bohr's atomic model was overcome (as before with the more complicated Heisenberg matrix mechanics ). The discrete energy levels of the hydrogen atom, which are assigned to stationary classical orbits of an electron in the Coulomb potential of the atomic nucleus in Bohr's model , result from the Schrödinger equation as eigenvalues of the Schrödinger equation for an electron in the potential of the atomic nucleus.
While the orbit of a particle in classical mechanics is determined by Newton's equation of motion , in quantum mechanics the Schrödinger equation provides a probability distribution for the location of the particle instead . One also speaks illustratively of the fact that the particle is delocalized over space. However, as a more comprehensive theory, quantum mechanics must contain classical mechanics. One form of this correspondence is established by the Ehrenfest theorem . The theorem says u. a. that the mean value of the particle coordinate satisfies the classical equation of motion. The correspondence becomes relevant and evident with localized coherent wave packets. Such wave packets can be used with higher quantum numbers, so z. B. construct at higher excited states of the hydrogen atom.
In the Schrödinger equation, the wave function and the operators appear in the so-called Schrödinger picture , in which an equation of motion for the states is considered. In the Heisenberg picture , equations of motion for the operators themselves are considered instead. These equations of motion are called Heisenberg's equation of motion . The two formulations are mathematically equivalent.
The Schrödinger equation is deterministic, which means that its solutions are unique when given initial conditions. On the other hand, the solutions of the Schrödinger equation according to the Copenhagen interpretation are statistical quantities from which only statements can be made 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, rather it is due to nature itself.
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 can be found anywhere in the entire space is 1. For the standardized solutions obtained in this way then corresponds to the probability density of the particle at the location at the point in time . However, not every solution of a Schrödinger equation can be normalized. If it exists, this standardized solution is uniquely determined except for a phase factor of the form for a real one , which is however physically meaningless.
Since the Schrödinger equation is invariant under the phase transformation ( U (1) symmetry), it follows from the Noether theorem that normalization is maintained; the probability is therefore a conserved quantity.
Expected values of measured quantities
The physical properties of the particle result from the wave function. For example, the classical value for the location of the particle is given by the mean location of the particle at the time , that is
while the classical value for the momentum of the particle is replaced by the following mean value:
- .
Every classical measurand is replaced by an averaging of the associated operator over the space in which the particle is located:
- .
The expression is called the expectation of . The expected value of the energy is the same .
Method of solving the Schrödinger equation
Stationary solutions
For a system with a Hamilton operator without explicit time dependence, the approach is
obvious. The time dependency of the state vector is expressed here by a factor with a constant frequency . For the time-independent factor of the state vector, the Schrödinger equation becomes the eigenvalue equation
- .
According to Planck's formula , such a system has the energy
- .
Discrete eigenvalues correspond to discrete energy levels of the system (“quantization as eigenvalue problem”).
Note: A common spatial representation of the "time-free" (stationary) Schrödinger equation is:
Solution procedures in general
The solutions of the Schrödinger equation (or Pauli equation) contain in principle the entire solid state physics and chemistry (one restriction: for internal electrons of heavy atoms, relativistic corrections are no longer small). Solutions in closed form are only available for some 1-electron systems (hydrogen atom, potential barrier, harmonic oscillator, Morse potential, ...). From the helium atom or hydrogen molecule on, one is dependent on numerical techniques.
With computer support and suitable methods (perturbation calculation, variation approaches, ...), systems with up to about 10 electrons can be treated numerically without approximation. H. the methods converge towards the exact solution with increasing computational effort. An example of such a method is Configuration Interaction .
With this method, which is in principle exact, a wave function in the -dimensional configuration space has to be determined in the -particle case . If you use (support point or variation) values for each dimension, values must be calculated. As a result, this exponentially growing requirement for memory and computing power makes exact calculations impossible for most systems (an ethane molecule, for example, contains two carbon atoms and 18 electrons). Walter Kohn called this exponential growth in resources the "exponential barrier".
Larger systems are therefore examined using approximation methods. Well-known methods are the Hartree-Fock approximation , extensions and the split operator method in theoretical chemistry .
The density functional theory , which goes back to Walter Kohn, plays a special role , as it specifically circumvents the exponential barrier. With this, ab initio calculations can be used to calculate lattice constants and binding energies even for complex atoms and compounds with errors in the percentage range.
Solution examples
One-dimensional free particle
In the one-dimensional case of a free particle, the Laplace operator is reduced to a double derivative and the potential disappears.
In the case of a Gaussian amplitude distribution, a solution of the one-dimensional Schrödinger equation with vanishing potential is:
Here is half the width of the wave packet and the wavelength at the time . The following pictures show the spatial and temporal course of the wave function for different initial conditions. In the case of a pure Gaussian distribution, the wave function broadens on both sides. If the initial Gaussian distribution is multiplied by the complex oscillation , the result is a moving particle with dispersion.
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 in the state , or to an atomic nucleus 2 and is in the state . If no transitions are possible, the stationary Schrödinger equation applies. If transitions from to are possible, the Hamilton operator must produce an admixture of state when applied to state , and analogously for transitions from to . One parameter determines the transition rate. The system is then modeled as follows:
By addition and subtraction of these equations we see that new stationary states in the form of superpositions of and are:
because for this one finds 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 . The corresponding superposition state is the binding state of the molecule.
Schrödinger equation in mathematics
For the Schrödinger equation in a Hilbert space it can be shown mathematically that the Hamilton operator is self adjoint . Then it follows from Stone's theorem the existence of a unitary group and thus the unambiguous solvability of the initial value problem . From a mathematical point of view, it is important to differentiate self adjointness from the weaker property of symmetry . The latter can usually be shown by a partial integration ; for self-adjointness a detailed investigation of the domain of the adjoint operator is necessary. Both terms coincide for bounded operators, but Schrödinger operators are usually unbounded and, according to Hellinger-Toeplitz's theorem, can not be defined over the whole Hilbert space. Then it is necessary to examine the spectrum of in order to understand the dynamics.
Analytical procedures and investigation of the solution properties
Schrödinger equation without potential
The Schrödinger equation without potential (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 H ^{s} norms
The maintenance of standards
can be shown by Fourier transformation. In the case it expresses the conservation of the probabilities.
Dispersion
It applies
- .
This property expresses the dissolution of the wave packets.
Schrödinger equation with potential
The Schrödinger equation with a potential
can be treated with methods of perturbation theory . For example follows from the set of Kato-Rellich: Applies in three (or fewer) dimensions , wherein is limited and disappears at infinity and is square-integrable, then to self-adjoint and the main spectrum is . Under the essential spectrum there can be a maximum of countably many eigenvalues that can only accumulate at zero. These prerequisites in particular cover the Coulomb potential and thus the hydrogen atom ,
- ,
which can be solved explicitly by separation in spherical coordinates . If one considers atoms with more than one electron or molecule , the self adjointness was only later proven by Tosio Kato . The structure of the essential spectrum is described in this case by the HVZ theorem (after W. Hunziker, C. van Winter and GM Zhislin). Such models can usually only be solved numerically.
The one-dimensional Schrödinger equation is a special case of a Sturm-Liouville equation .
Extensions
Pauli and Dirac equation
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 .
The Pauli equation is not Lorentz invariant , but “only” Galileo invariant (not relativistic). The correct relativistic generalization of the Schrödinger and the general Pauli equation, the fermions lorentzinvariante Dirac equation is that in contrast to the Schrödinger equation is a partial differential equation first order.
Nonlinear extensions of the Schrödinger equation
A number of problems in physics lead to one generalization, the nonlinear Schrödinger equation
- ,
with a nonlinear self-interaction term . The explicit dependence of the solution function on time and place was left out. Especially in the case of the cubic nonlinear Schrödinger equation , and one dimension is an integrable wave equation with Solitonenlösungen . It appears, for example, when describing light waves in glass fibers and water waves. In dimension one has in the cubic case the Gross-Pitaevskii equation , which describes the Bose-Einstein condensate .
If one assumes a gravitational self-interaction of the particles, one obtains the non-linear Schrödinger-Newton equation .
literature
Schrödinger's original work
- Erwin Schrödinger: Quantization as an eigenvalue problem. In: Annals of Physics . Vol. 79, 1926, pp. 361, 489; Vol. 80, 1926, p. 437; and Vol. 81, 1926, p. 109. (original work)
- Erwin Schrödinger: The wave mechanics. Battenberg, Stuttgart 1963, DNB 454485557 . (Documents of the natural sciences. Physics department; Vol. 3) (Schrödinger's work on wave mechanics) - The work on wave mechanics is also reprinted in Günther Ludwig (Hrsg.): Wellenmechanik. Akademie-Verlag, Berlin 1970, DNB 458581941 .
- Erwin Schrödinger: The basic idea of wave mechanics . In: What is a law of nature? Contributions to the scientific worldview . 5th edition. Oldenbourg Wissenschaftsverlag, Munich 1997, ISBN 3-486-56293-2 , p. 86–101 ( limited preview in Google Book search).
Quantum Mechanics Textbooks
The Schrödinger equation is dealt with in all common textbooks in quantum mechanics, for example:
- Torsten Fließbach : Quantum Mechanics , Spectrum Academic Publishing House and other textbooks on quantum mechanics
- Peter W. Atkins: Molecular Quantum Mechanics and Other Textbooks of Theoretical and Physical Chemistry
- M. Alonso, EL Finn: Quantum Physics and Statistical Physics. Oldenbourg, Munich / Vienna 1998, ISBN 3-486-24836-7 .
Mathematics:
- Michael Reed, Barry Simon : Methods of Modern Mathematical Physics. 4 volumes, Academic Press 1978, 1980
- Hans Cycon, Richard G. Froese, Werner Kirsch , Barry Simon: Schrödinger Operators. Springer 1987
- Volker Bach : Schrödinger Operators. In: J.-P. Francoise, Gregory L. Naber, ST Tsou (Eds.): Encyclopedia of Mathematical Physics. Vol. 4, Academic Press, 2006, ISBN 0-12-512660-3 .
- Gerald Teschl : Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators. American Mathematical Society, 2009. (Free online version)
Remarks
- ↑ In his Nobel Lecture (1933) Schrödinger describes in a clear way (without mathematics) the mode of operation of Hamilton's principle in classical mechanics and quantum or wave mechanics.
- ^ Walter Kohn: Nobel Prize Lecture (1998)
- ^ RP Feynman, RB Leighton, ML Sands: lectures on physics. Vol. 3: Quantum Mechanics. Oldenbourg-Verlag, Munich.