Question

One of the difficulties of the Bohr model was that it assumes the existence of stationary orbits. Why was this problematic from a classical physics point of view? How does the solution of the Schrödinger equation remedy this situation?

Answer 1

According to Bohr's theory, the electrons can only orbit stably without radiating in certain orbits (called "stationary orbits") at a certain discrete set of distances from the nucleus.

These orbits are associated with definite energies and are also called energy shells or energy levels. In these orbits, the electron's acceleration doesn’t result in radiation and energy loss as required by classical electromagnetics.

This assumption of Bohr's atomic model violates the rule of classical physics because there was no accounting for the fact that the electron would spiral into the nucleus.

The model also violates the uncertainty principle in that it considers electrons to have known orbits and locations, two things which cannot be measured simultaneously.

In 1926 the Schrödinger equation, essentially a mathematical wave equation, established quantum mechanics in widely applicable form. In order to understand how a wave equation is used, it is helpful to think of an analogy with the vibrations of a bell, violin string, or drumhead. These vibrations are governed by a wave equation, since the motion can propagate as a wave from one side of the object to the other. Certain vibrations in these objects are simple modes that are easily excited and have definite frequencies. For example, the motion of the lowest vibrational mode in a drumhead is in phase all over the drumhead with a pattern that is uniform around it; the highest amplitude of the vibratory motion occurs in the middle of the drumhead. In more-complicated, higher-frequency modes, the motion on different parts of the vibrating drumhead are out of phase, with inward motion on one part at the same time that there is outward motion on another.

Schrödinger postulated that the electrons in an atom should be treated like the waves on the drumhead. The different energy levels of atoms are identified with the simple vibrational modes of the wave equation. The equation is solved to find these modes, and then the energy of an electron is obtained from the frequency of the mode and from Einstein’s quantum formula, E = hν. Schrödinger’s wave equation gives the same energies as Bohr’s original formula but with a much more-precise description of an electron in an atom. The lowest energy level of the hydrogen atom, called the ground state, is analogous to the motion in the lowest vibrational mode of the drumhead. In the atom the electron wave is uniform in all directions from the nucleus, is peaked at the centre of the atom, and has the same phase everywhere. Higher energy levels in the atom have waves that are peaked at greater distances from the nucleus. Like the vibrations in the drumhead, the waves have peaks and nodes that may form a complex shape. The different shapes of the wave pattern are related to the quantum numbers of the energy levels, including the quantum numbers for angular momentum and its orientation.