Why the Schrodinger equation?

The first thing which the Newton second law failed to foretell were distances between electron possible orbitals in an atom. The electron’s energy spectrum appeared to be discontinuous so a novel equation had to give a set of discrete solutions. In mathematics, such a set appears as a sequence of eigenvalues when a functional operator is applied to its eigen- or proper functions. Ervin Schrodinger was good in math so he wrote an equation for operators whose set of eigenvalues represented a discrete spectrum of energy. In there, he kept the classical link between energy and momentum using the derivatives WRT time and coordinate respectively as their operators. For stationary states, the Schrodinger equation reduces into differentiation WRT coordinates only. In a space region where the potential energy of a system is absent, the equation has the simplest structure looking like the Newton equation describing oscillation of a pendulum or a spring. The proper solutions are harmonic functions of multiples of coordinates with details being defined by boundary conditions. Their eigenvalues constitute the discrete set of energy. Now, if the potential do depends on coordinates, the Schrodinger equation becomes more complicated yet can be solved analytically for a number of practically interesting potentials. For instance, for a quantum harmonic oscillator, the proper solutions are Chebyshev-Hermite polynomials. In addition to a discrete energy spectrum, the eigen solutions give a nonzero probability for an oscillator of a given energy to be outside classically possible space region.

Quantum oscillator energy levels

In addition to a discrete energy spectrum, the eigen solutions give a nonzero probability for an oscillator of a given energy to be outside classically possible space region. Furthermore, the lowest possible eigenvalue energy appears to be not zero. This means that unlike a classical oscillator, a quantum one could not be in rest, which is in agreement with Heisenberg uncertainty principle.

like(1)dislike(0)

Leave a Reply