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The equation was postulated by Schrödinger based on a postulate of Louis de Broglie that all matter has an associated matter wave. The equation predicted bound states of the atom in agreement with experimental observations. [4]: II:268 The Schrödinger equation is not the only way to study quantum mechanical systems and make predictions.
and substituting this into the above Schrödinger equation gives a non-relativistic QM equation for the wavefunction: the procedure is a straightforward substitution of a simple expression. By contrast this is not as easy in RQM; the energy–momentum equation is quadratic in energy and momentum leading to difficulties. Naively setting:
The Schrödinger equation determines how wave functions evolve over time, and a wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation. This explains the name "wave function", and gives rise to wave–particle duality.
The failure of classical mechanics applied to molecular, atomic, and nuclear systems and smaller induced the need for a new mechanics: quantum mechanics.The mathematical formulation was led by De Broglie, Bohr, Schrödinger, Pauli, and Heisenberg, and others, around the mid-1920s, and at that time was analogous to that of classical mechanics.
The quantum wave equation can be solved using functions of position, (), or using functions of momentum, () and consequently the superposition of momentum functions are also solutions: = + The position and momentum solutions are related by a linear transformation, a Fourier transformation. This transformation is itself a quantum superposition ...
Demanding that the classical equations of motion are preserved is not a strong enough condition to determine the matrix elements. The Planck constant does not appear in the classical equations, so that the matrices could be constructed for many different values of ħ and still satisfy the equations of motion, but with different energy levels.
and this is the Schrödinger equation. Note that the normalization of the path integral needs to be fixed in exactly the same way as in the free particle case. An arbitrary continuous potential does not affect the normalization, although singular potentials require careful treatment.
The Schrödinger equation applies to the new Hamiltonian. Solutions to the untransformed and transformed equations are also related by U {\displaystyle U} . Specifically, if the wave function ψ ( t ) {\displaystyle \psi (t)} satisfies the original equation, then U ψ ( t ) {\displaystyle U\psi (t)} will satisfy the new equation.