Equation of Schrödinger

The equation of Schrödinger , conceived by the Austrian Physicist Erwin Schrödinger in 1925, is a fundamental equation in Quantum physics not-relativist. It describes the evolution in the time of a massive particle not-relativist, and thus fulfills the same role as the fundamental relation of dynamics in traditional Mécanique.

Birth of the equation

Historical context

At the beginning of the 20th century, it had become clearly that the light presents a duality wave-corpuscle, i.e. it could appear, according to the circumstances, either like a particle, the Photon, or like a electromagnetic Onde. Louis de Broglie proposed to generalize this duality with all the known particles although this assumption had as a paradoxical consequence that the electron S were to be able to produce Interférence S like the light, which was checked later on by the Expérience. By analogy with the Photon, Louis de Broglie associated thus with each free particle of energy $E$ and Quantité of movement $p$ a frequency $\backslash \; nu$ and a wavelength $\backslash \; lambda$:

$\backslash \; left\; \backslash \; \{\backslash \; begin\; \{matrix\}\; naked\; E=h\; \backslash \; \backslash \; \backslash \; p=h/\backslash \; lambda\; \backslash \; end\; \{matrix\}\; \backslash \; right.$

The equation of Schrödinger, found by the physicist Erwin Schrödinger in 1925, is a equation of wave which generalizes the approach of of Broglie above to the massive particles not-relativists subjected to a force drifting of a potential energy, whose total mechanical energy is classically:

$E\; =\; \{p^2\; \backslash \; over\; 2m\}\; +\; V\; (R)$.

The success of the equation, deduced from this extension by the use of the principle of correspondence, was immediate as for the evaluation of the quantified levels of energy of the electron in the Atome of Hydrogène, because it made it possible to explain the emission lines of the Hydrogène: series of Lyman, Balmer, Bracket, Paschen, etc

The correct physical interpretation of the function of wave of Schrödinger was given only in 1926 by max Born. Because of the probabilistic character that it introduced, the wave mechanics of Schrödinger initially caused mistrust in some physicists of reputation like Albert Einstein, for which “ God does not play dice ”.

Historical derivation

The conceptual diagram used by Schrödinger to derive its equation rests on a formal analogy between optics and mechanics:

• In Optical undulatory, the equation of propagation in a transparent medium of real index N varying slowly on the scale wavelength led - when a monochromatic solution is sought whose amplitude varies very slowly in front of the phase - to an approximate equation known as of the eikonale. It is the approximation of the geometrical Optique, with which the variational principle with Fermat is associated.

• In the Hamiltonian formulation of traditional mechanics, there exists an equation known as of Hamilton-Jacobi. For a nonrelativistic massive particle subjected to a force deriving from a potential energy, total mechanical energy is constant and the equation of Hamilton-Jacobi for the " function characteristic of Hamilton" resemble then formally the equation of the eikonale (the variational principle associated being the Principe with less action.)

This parallel had been noted since 1834 by Hamilton, but this one did not have then by reason to doubt the validity of traditional mechanics. After the assumption of De Broglie of 1923, Schrödinger was said: the equation of the eikonale being an approximation of the equation of wave of undulatory optics, let us seek the equation of wave of the " mechanics ondulatoire" (to be built) whose approximation is the equation of Hamilton-Jacobi. What it did, initially for a standing wave ( E = cte), then for an unspecified wave.

Note: Schrödinger had in fact started by treating the case of a relativistic particle - as besides of Broglie before him. It then obtained the equation known today under the name of Klein-Gordon, but its application to the case of the Coulomb potential giving of the energy levels incompatible with the experimental results of the hydrogen atom, it would have been folded back on the case not-relativist, with the success which one knows.

Modern formulation of the equation

In Mechanical quantum, the state at the moment T of a system is described by an element $\backslash \; left|\; \backslash \; Psi\; (T)\; \backslash \; right\; \backslash \; rangle$ of the complex space of Hilbert - is used the Notation bra-ket of Paul Dirac. $\backslash \; left|\; \backslash \; Psi\; (T)\; \backslash \; right\; \backslash \; rangle$ represents the probabilities of results of all possible measurements of a system.

Temporal evolution of $\backslash \; left|\; \backslash \; Psi\; (T)\; \backslash \; right\; \backslash \; rangle$ is described by the equation of Schrödinger:

| | | |} where

• $i\; \backslash ,$ is the imaginary unit;
• $\backslash \; hbar\; \backslash ,$ is the reduced Constante of Planck (h/2π) ;
• $\backslash \; hat\; \{H\}\; \backslash ,$ is the Hamiltonian , depend on time in general, the Observable corresponding to the total energy of the system;
• $\backslash \; hat\; \{\backslash \; vec\; \{\backslash \; mathbf\; \{R\}\}\}\; \backslash ,$ is the observable position;
• $\backslash \; hat\; \{\backslash \; vec\; \{\backslash \; mathbf\; \{p\}\}\}\; \backslash ,$ is observable the impulse.

It should be noted that, contrary to the Maxwell's equations managing the evolution of the electromagnetic waves, the equation of Schrödinger is not relativistic. Also let us note that this equation is not shown: it is a postulate. It was supposed to be correct after Davisson and Germer had confirmed in experiments the assumption of Louis de Broglie.

Resolution of the equation

The equation of Schrödinger being a vectorial equation one can rewrite it in an equivalent way in a bases particular space of the states. If one chooses for example the base $\backslash \; left|\backslash \; vec\; \{R\}\; \backslash \; right\; \backslash \; rangle$ corresponding to the Representation of position defined by

$\backslash \; hat\; \{\backslash \; vec\; \{\backslash \; mathbf\; \{R\}\}\}\; \backslash \; left|\backslash \; vec\; \{R\}\; \backslash \; right\; \backslash \; rangle=\; \backslash \; vec\; \{R\}\; \backslash \; left|\backslash \; vec\; \{R\}\; \backslash \; right\; \backslash \; rangle$

then the function of wave $\backslash \; Psi\; (T,\; \backslash \; vec\; \{R\})\; \backslash \; equiv\; \backslash \; left\; \backslash \; langle\; \backslash \; vec\; \{R\}\; \backslash \; right|\backslash \; left.\; \backslash \; Psi\; (T)\; \backslash \; right\; \backslash \; rangle\; \backslash ,$ satisfy the following equation

$I\; \backslash \; hbar\; \{\backslash \; partial\; \backslash \; Psi\; (T,\; \backslash \; vec\; \{R\})\; \backslash \; over\; \backslash \; partial\; T\}\; =\; \{\backslash \; hbar^2\; \backslash \; over\; 2m\}\; \backslash \; overrightarrow\; \{\backslash \; nabla\}\; ^2\; \backslash \; Psi\; (T,\; \backslash \; vec\; \{R\})\; +V\; (\backslash \; vec\; \{R\},\; T)\; \backslash \; Psi\; (T,\; \backslash \; vec\; \{R\})$

where $\backslash \; overrightarrow\; \{\backslash \; nabla\}\; ^2\; \backslash ,$ is the Laplacien.

In this form one sees that the equation of Schrödinger is a partial derivative equation utilizing linear operators , which makes it possible to write the generic solution like summons particular solutions. The equation is in the large too complicated majority of the cases to admit an analytical solution so that its resolution is approximate and/or numerical.

Seek clean states

The operators appearing in the equation of Schrödinger are linear operators; it follows that any linear combination of solutions is solution of the equation. That carries out to support the research solution which has a great theoretical and practical interest: namely the states who are clean of the Hamiltonian operator.

These states are thus solutions of the equation to the states and eigenvalues,

$H|\backslash \; varphi\_\; \{N\}\; \backslash \; rangle\; =E\_\; \{N\}|\backslash \; varphi\_\; \{N\}\; \backslash \; rangle$

who bears sometimes the name of equation of Schrödinger independent of time . The clean state $|\backslash \; varphi\_\; \{N\}\; \backslash \; rangle$ is associated with the eigenvalue $E\_\; \{N\}$, scalar reality, energy of the particle of which $|\backslash \; varphi\_\; \{N\}\; \backslash \; rangle$ is the state.

The values of energy can be discrete as the dependant solutions of a well of potential (e.g. levels of the hydrogen atom); it results a Quantification from it from the energy levels. They can also correspond to a continuous spectrum as the free solutions of a well of potential (e.g. an electron having enough energy to move away ad infinitum from the core from the atom from hydrogen).

It often happens that several states $|\backslash \; varphi\_\; \{N\}\; \backslash \; rangle$ corresponds to the same value of energy: one speaks then about degenerated energy levels.

Generally, determination of each clean state of the Hamiltonian, $|\backslash \; varphi\_\; \{N\}\; >$, and of associated energy, provides the stationary state corresponding, solution of the equation of Schrödinger:

$|\backslash \; psi\_\; \{N\}\; (T)\; \backslash \; rangle\; \backslash ,\; =\; \backslash ,\; |\backslash \; varphi\_\; \{N\}\; \backslash \; rangle\; \backslash ,\; \backslash \; exp\; (\backslash \; frac\; \{-\; iE\_\; \{N\}\; T\}\; \{\backslash \; hbar\}).$

A solution of the equation of Schrödinger can then be written very generally like a linear combination such states:

$|\backslash \; psi\; (T)\; \backslash \; rangle\; \backslash ,\; =\; \backslash ,\; \backslash \; sum\_\; \{N\}\; \backslash \; sum\_\; \{J\}\; c\_\; \{N,\; J\}|\backslash \; varphi\_\; \{N,\; J\}\; \backslash \; rangle\; \backslash \; exp\; (\backslash \; frac\; \{-\; iE\_\; \{N\}\; T\}\; \{\backslash \; hbar\}).$

According to the Postulates of quantum mechanics,

• the scalar complexes $c\_\; \{N,\; I\}$ is the amplitude of the state $|\backslash \; psi\; (T)\; >$ on the state $|\backslash \; varphi\_\; \{N,\; I\}\; >$;
• reality $\backslash \; Sigma\_\; \{I\}|c\_\; \{N,\; I\}|^2$ is the probability (in the case of a discrete spectrum) of finding energy $E\_\; \{N\}$ during a measurement of energy on the system.

Scarcity of an exact analytical resolution

The research of the clean states of the Hamiltonian is in general complex. Even the case analytically soluble of the hydrogen atom is it rigorously in simple form only if one neglects the coupling with the electromagnetic field which will allow the passage of the excited states, solutions of the equation of Schrödinger of the atom, towards the fundamental one.

Certain simple models, although not completely in conformity with reality, can be solved analytically and prove very useful:

• free particle (null potential);
• oscillating harmonic (potential quadratic);
• particle moving on a ring;
• particle in a rectangular well of potential;
• particle in an annular guide of wave;
• particle in a potential with spherical symmetry;
• particle in a unidimensional network (potential periodical).

In the other cases, it is necessary to call upon the various techniques of approximation:

• the Théorie of the disturbances provides analytical expressions in the form of asymptotic developments around an exactly soluble not-disturbed problem.
• the numerical Analyze makes it possible to explore inaccessible situations by the perturbation theory.

Generalization of the equation

Generalization with the relativistic field led to the equation of Klein-Gordon, then with the equation of Dirac; the latter naturally establishes the existence of the Spin and the antiparticles. However, there does not exist any entirely coherent interpretation of these equations of relativistic waves within the framework of a theory describing only one particle; the relevant framework for theoretical the quantum relativist is the Quantum theory of the fields.

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