Einstein later expressed to Erwin Schrödinger that, " it did not come out as well as I had originally wanted ; rather, the essential thing was, so to speak, smothered by the formalism.

Since all other atomic, or molecular systems, involve the motions of three or more " particles ", their Schrödinger equations cannot be solved exactly and so approximate solutions must be sought.

Attempted replacements for the Schrödinger equation, such as the Klein-Gordon equation or the Dirac equation, have many unsatisfactory qualities ; for instance, they possess energy eigenvalues that extend to –∞, so that there seems to be no easy definition of a ground state.

Now, if we wish to maintain the notion of a convected density, then we must generalize the Schrödinger expression of the density and current so that the space and time derivatives again enter symmetrically in relation to the scalar wave function.

However, one needs at least 4 × 4 matrices to set up a system with the properties required — so the wave function had four components, not two, as in the Pauli theory, or one, as in the bare Schrödinger theory.

This Hamiltonian is now a 2 × 2 matrix, so the Schrödinger equation based on it must use a two-component wave function.

* The Schrödinger equation includes the wavefunction, so its wave packet solution implies the position of a ( quantum ) particle is fuzzily spread out in wave fronts.

When c is very large and psi is slowly varying so that the lattice can be thought of as a line, this becomes the free Schrödinger equation:

Interestingly, Schrödinger did encounter an equation for which the wave function satisfied relativistic energy conservation before he published the non-relativistic one, but it led to unacceptable consequences for that time so he discarded it.

and so the Schrödinger equation becomes

In the case of a Tonks – Girardeau gas ( TG ), so many properties of this one-dimensional string of bosons would be sufficiently fermion-like that the situation is often referred to as the ' fermionization ' of bosons. Tonks – Girardeau gas coincide with quantum Nonlinear Schrödinger equation for infinite repulsion, which can be efficiently analyzed by Quantum inverse scattering method.

Even though the Schrödinger equation was developed two years later, Wentzel, Kramers, and Brillouin were apparently unaware of this earlier work, so Jeffreys is often neglected credit.

Although the Schrödinger equation was developed two years later, Wentzel, Kramers, and Brillouin were apparently unaware of this earlier work, so Jeffreys is often neglected credit for the WKB approximation.

For a free particle the potential is, so the Schrödinger equation for the free electron is

so that the Schrödinger equation for the spherical rotor ()

is still an operator that acts on the Hilbert space of wave functions, but it is not the same Hilbert space as in the nonrelativistic case, and the Hamiltonian no longer determines evolution of the system, so the Schrödinger equation no longer applies.

This Hamiltonian is now a 2 × 2 matrix, so the Schrödinger equation based on it must use a two-component wave function.

The pseudopotential is an attempt to replace the complicated effects of the motion of the core ( i. e. non-valence ) electrons of an atom and its nucleus with an effective potential, or pseudopotential, so that the Schrödinger equation contains a modified effective potential term instead of the Coulombic potential term for core electrons normally found in the Schrödinger equation.

The atomic Hamiltonian was unaffected by the approximation, so the total Hamiltonian in the Schrödinger picture under the rotating wave approximation is

This covers a wide range of electronic problems in condensed matter physics, so if we could solve the Schrödinger equation for a given system, we could predict its behavior, which has important applications in fields from computers to biology.

The dispersive character of solutions of the Schrödinger equation has played an important role in rejecting Schrödinger's original interpretation, and accepting the Born rule.

The Many-Worlds Interpretation deals with it by discarding the collapse-process, thus reformulating the relation between measurement apparatus and system in such a way that the linear laws of quantum mechanics are universally valid ; that is, the only process according to which a quantum system evolves is governed by the Schrödinger equation or some relativistic equivalent.

In 1926, Erwin Schrödinger used this idea to develop a mathematical model of the atom that described the electrons as three-dimensional waveforms rather than point particles.

With de Broglie's suggestion of the existence of electron matter waves in 1924, and for a short time before the full 1926 Schrödinger equation treatment of hydrogen like atom, a Bohr electron " wavelength " could be seen to be a function of its momentum, and thus a Bohr orbiting electron was seen to orbit in a circle at a multiple of its half-wavelength ( this historically incorrect Bohr model is still occasionally taught to students ).

The solution of the Schrödinger equation goes much further than the Bohr model, because it also yields the shape of the electron's wave function (" orbital ") for the various possible quantum-mechanical states, thus explaining the anisotropic character of atomic bonds.

This model is described by a quantum nonlinear Schrödinger equation.

As it turns out, analytic solutions of the Schrödinger equation are only available for a very small number of relatively simple model Hamiltonians, of which the quantum harmonic oscillator, the particle in a box, the hydrogen molecular ion, and the hydrogen atom are the most important representatives.

This was an assumption in the earlier Bohr model of the atom, but it is a prediction of the Schrödinger equation.

The nearly free electron model rewrites the Schrödinger equation for the case of a periodic potential.

If one inserts a more realistic barrier model into the simplest form of the Schrödinger equation, then an awkward mathematical problem arises over the resulting differential equation: it is known to be mathematically impossible in principle to solve this equation exactly in terms of the usual functions of mathematical physics, or in any simple way.

Although the cubical model of the atom was soon abandoned in favor of the quantum mechanical model based on the Schrödinger equation, and is therefore now principally of historical interest, it represented an important step towards the understanding of the chemical bond.

In the Schrödinger model, the Heisenberg group acts on the space of square integrable functions.

The two researchers simultaneously solved the Schrödinger equation for a model nuclear potential and derived a relationship between the half-life of the particle and the energy of emission that depended directly on the mathematical probability of tunnelling.

One simple model for this is a wave equation known as the nonlinear Schrödinger equation ( NLS ), in which a normal and perfectly accountable ( by the standard linear model ) wave begins to ' soak ' energy from the waves immediately fore and aft, reducing them to minor ripples compared to other waves.

In the original Anderson tight-binding model, the evolution of the wave function ψ on the d-dimensional lattice Z < sup > d </ sup > is given by the Schrödinger equation

Atomic orbitals can be the hydrogen-like " orbitals " which are exact solutions to the Schrödinger equation for a hydrogen-like " atom " ( i. e., an atom with one electron ).

# the hydrogen-like atomic orbitals are derived from the exact solution of the Schrödinger Equation for one electron and a nucleus.

In the quantum picture of Heisenberg, Schrödinger and others, the Bohr atom number n for each orbital became known as an n-sphere in a three dimensional atom and was pictured as the mean energy of the probability cloud of the electron's wave packet which surrounded the atom.

In the Schrödinger equation for this system of one negative and one positive particle, the atomic orbitals are the eigenstates of the Hamiltonian operator for the energy.

Dirac realised that his relativistic version of the Schrödinger wave equation for electrons predicted the possibility of antielectrons.

For example, in the Schrödinger picture, there is a linear operator U with the property that if an electron is in state right now, then in one minute it will be in the state, the same U for every possible.

* Solutions to the radial Schrödinger equation ( in spherical and cylindrical coordinates ) for a free particle

This molecular orbital theory represented a covalent bond as an orbital formed by combining the quantum mechanical Schrödinger atomic orbitals which had been hypothesized for electrons in single atoms.

In principle, it is possible to solve the Schrödinger equation in either its time-dependent or time-independent form, as appropriate for the problem in hand ; in practice, this is not possible except for very small systems.

The total energy is determined by approximate solutions of the time-dependent Schrödinger equation, usually with no relativistic terms included, and by making use of the Born – Oppenheimer approximation, which allows for the separation of electronic and nuclear motions, thereby simplifying the Schrödinger equation.

Once the electronic and nuclear variables are separated ( within the Born – Oppenheimer representation ), in the time-dependent approach, the wave packet corresponding to the nuclear degrees of freedom is propagated via the time evolution operator ( physics ) associated to the time-dependent Schrödinger equation ( for the full molecular Hamiltonian ).

Complex-crater morphology on rocky planets appears to follow a regular sequence with increasing size: small complex craters with a central topographic peak are called central peak craters, for example Tycho ; intermediate-sized craters, in which the central peak is replaced by a ring of peaks, are called peak-ring craters, for example Schrödinger ; and the largest craters contain multiple concentric topographic rings, and are called multi-ringed basins, for example Orientale.

Provided the theory is linear with respect to the wavefunction, the exact form of the quantum dynamics modelled, be it the non-relativistic Schrödinger equation, relativistic quantum field theory or some form of quantum gravity or string theory, does not alter the validity of MWI since MWI is a metatheory applicable to all linear quantum theories, and there is no experimental evidence for any non-linearity of the wavefunction in physics.

Bohr's results for the frequencies and underlying energy values were confirmed by the full quantum-mechanical analysis which uses the Schrödinger equation, as was shown in 1925 – 1926.

The solution to the Schrödinger equation for hydrogen is analytical.

The solution of the Schrödinger equation ( wave equations ) for the hydrogen atom uses the fact that the Coulomb potential produced by the nucleus is isotropic ( it is radially symmetric in space and only depends on the distance to the nucleus ).

There are several important effects that are neglected by the Schrödinger equation and which are responsible for certain small but measurable deviations of the real spectral lines from the predicted ones: