Schroedinger equation

Schroedinger equation is the basic equation of the Quantum mechanics in the Schroedinger representation.

The quantum system system is determined with its Hamiltonian $$H$$, any state of this system is determined with the wave function $$\Psi$$, and time evolution is determined with the Schroedinger equation

$$i \hbar \dot \Psi = H \Psi$$

where $$\hbar$$ is Planck constant and dot differentiates with respect to time. Usually, time is denoted with letter $$t$$.

The Schroedinger equation should not be confused with the Schroeder equation, used to construct a regular iterates of a holomorphic function at its fixed point.

Quantisation

Quantum mechanics happens to be especially efficient in the atomic physics and the molecular physics with the specific rule of quantisation.

Quantisation is Heuristic hint to construction of Hamiltonian of a quantum system, based on the analogy with the classical system and the classical Hamiltonian. In such a way, the quantisation refers to the classical mechanics. This is only analogy and only hint. The attempts to postulate this analogy as a basic physical law leads to interpretation of quantum mechanics in terms of classical mechanics and causes contradictions.

The classical Hamiltonian expresses the energy of a system in terms of its classical coordinates $$x$$ and the corresponding momenta $$p$$. In the simple case, the classical Hamiltonian of a single particle of mass $$m$$ can be expressed with

$$\displaystyle H=\frac {p^2}{2m} + U(x)$$

where $$x$$ refers to coordinates, $$p$$ are corresponding momenta, and $$U$$ is potential, that does not depend on momenta.

Then, in order to get the quantum Hamitonian, the momentum is assumed to be a linear operator with certain combination property with the coordinate, namely,

$$p x - x p = - \mathrm i \hbar$$

In the simple case, the wave function can be realised as function of coordinates, that are treated as real numbers. The operator of coordinate appears as multiplication of the wave function to its argument, and operator of momentum appears to be proportional to operator of differentiation with respect to this coordinate,

$$p= - \mathrm i \hbar \partial_x$$

Such a choice provides the correct commutation relation for the coordinate and the momentum. The resulting form of the Schroedinger equation is denoted with term Coordinate representation.

Stationary Schroedinger

Stationary Schroedinger refers to the specific case of the Schroedinger equation, where the wave function $$\Psi$$ is eigenfunction of Hamiltonian H. Then, the time dependence of wave function is exponential.

The eigenvalue is usually interpreted as energy,

$$H \Psi(t) = E \Psi(t)$$

Such an assumption greatly simplifies the consideration, dropping out one variable, one parameter from the problem, because the wave function can be written as follows:

$$\Psi(t)= \exp(-\mathrm i \omega t) \psi$$

where $$\psi$$ does not depend on time; here $$\omega = E / \hbar$$.

In the simple case, psi is just function of coordinates, and the stationary schroedinger can be written as follows:

$$\displaystyle \frac{p^2}{2m} \psi(x) + (U(x)-E) \psi =0$$

or, after to choose the coordinate representation,

$$\displaystyle -\frac{\hbar^2}{2m} \psi ^{\prime\prime}(x) + (U(x)-E) \psi =0$$

where prime differentiates with respect to coordinate(s).