Schrödinger equation, developed by the Austrian physicist Erwin Schrödinger in 1925, is a fundamental equation in quantum mechanics. It describes the evolution over time of a massive non-relativistic particle, and thus fulfills the same role as the fundamental relation of dynamics in classical mechanics.
At the beginning of the twentieth century, it became clear that light has a wave-particle duality, that is to say, it could manifest, depending on the circumstances, either as a particle, the photon, or as a electromagnetic wave. Louis de Broglie proposed to generalize this duality to all known particles, although this hypothesis had the paradoxical consequence of the production of interferences by electrons-like light – which was later verified by Davisson-Germer experiment. By analogy with the photon, Louis de Broglie thus associates with each free particle of energy E and momentum p a frequency ν and a wavelength λ:
E = hν
p = h/λ
In the two expressions above, the letter h denotes the Planck constant. The Schrödinger equation, established by the physicist Erwin Schrödinger in 1925, is a functional equation whose unknown is a function, the wave function, which generalizes the approach of Louis de Broglie above to the massive non relativistic particles subject to a force deriving from a potential energy, the total mechanical energy of which is conventionally:
E = p2/2m + V (r).
The success of the equation, deduced from this extension by the use of the correspondence principle, was immediate as regards the evaluation of the quantified energy levels of the electron in the hydrogen atom, because it made it possible to explain the hydrogen emission lines: series of Lyman, Balmer, Brackett, Paschen, etc.
The commonly accepted physical interpretation of the Schrödinger wave function was given only in 1926 by Max Born. Because of the probabilistic nature it introduced, Schrödinger’s wave mechanics initially aroused distrust in some renowned physicists like Albert Einstein, for whom “God does not play dice”.
The historical derivation
The conceptual scheme used by Schrödinger to derive his equation is based on a formal analogy between optics and mechanics:
- In physical optics, the equation of propagation in a transparent medium of real index n varying slowly on the scale of the wavelength leads – when looking for a monochromatic solution whose amplitude varies very slowly in front of the phase – to an approximate equation called the eikonale. This is the approximation of geometrical optics, to which Fermat’s variational principle is associated.
- In the Hamiltonian formulation of classical mechanics, there is a so-called Hamilton-Jacobi equation. For a non-relativistic massive particle subjected to a force deriving from a potential energy, the total mechanical energy is constant and the Hamilton-Jacobi equation. For a non-relativistic massive particle subjected to a force deriving from a potential energy, the total mechanical energy is constant and the Hamilton-Jacobi equation for the “Hamilton characteristic function” then formally resembles the equation of the eikonale (the associated variational principle being the principle of least action.)
This parallel had been noted as early as 1834 by Hamilton, but he had no reason to doubt the validity of classical mechanics. After the De Broglie hypothesis of 1923, Schrödinger said to himself: the equation of the eikonale being an approximation of the wave equation of the physical optics, let us look for the wave equation of the “mechanical waveform “(to be constructed) whose approximation is the Hamilton-Jacobi equation. What he did, first for a standing wave (E = cte), then for any wave.
Note: Schrödinger had in fact begun by treating the case of a relativistic particle – as indeed de Broglie before him. He then obtained the equation known today as Klein-Gordon, but his application to the case of the Coulomb potential giving energy levels incompatible with the experimental results of the hydrogen atom, he would have fallen back on the non-relativistic case, with the success that we know.
In quantum mechanics, the state at time t of a system is described by an element |Ψ(t)› of the Hilbert complex space – with the use of Paul Dirac’s bra-ket notation. The square of |Ψ(t)› represents the probability densities of the results of all possible measurements of a system.
The temporal evolution of |Ψ(t)› is described by the Schrödinger equation:
where: i is the imaginary unit: i2 = -1; ℏ is Dirac constant: ℏ = h/2π; h is the Planck constant = 6.62607004 × 10-34 m2kg/s; Ĥ = p^2/2m + V(r̂,t) is the Hamiltonian, depending on the time in general, the observable corresponding to the total energy of the system; r̂ is the observable position; p^ is the observable impulse.
Unlike Maxwell’s equations governing the evolution of electromagnetic waves, the Schrödinger equation is non-relativistic. This equation is a postulate. It was supposed correct after Davisson and Germer had experimentally confirmed the hypothesis of Louis de Broglie.