Question

Do standing waves explain why electron orbitals are quantized?

Answer 1

Electrons, due to wave-particle duality, can behave both as waves and as particles. The wavelength of the electron is called the "De Broglie Wavelength", which is given by the equation lambda = h/p, where lambda is the wavelength, p is the momentum and h is Planck's constant.

Thus, as electrons are waves with specific wavelengths, there are certain "orbits" around the nucleus which are integral multiple of the wavelength, which causes resonance. These orbits are the s, p, d and f orbitals, along with their respective sub-orbitals. While this analogy is good for visualizing it, but it has many limitations. In reality, we have to solve the Schrodinger equation to find the resonance and then take the normalization and square it to find the probability of an electron being in a specific place.

Now consider a string with both ends clamped. Now assume that something causes the string to begin to vibrate. Since both ends are clamped, the only wave motions that can be sustained on the string are those which have nodes at both ends. This means that only waves whose wavelength is an integer multiple of the string length can exist on the string (otherwise it would not have nodes at both ends). Thus the allowed vibration modes are catalogued by an integer, there is no continuum of allowed vibration modes for the string, only those that fit the integer constraint on the wavelength.

Standing waves naturally give rise to integral quantisation (i.e. a set of allowed states enumerated by an integer).

In quantum mechanics a particle is described by a wavefunction which contains all the information about the particle and behaves very much like our intuitive concept of a wave. When determining energy levels in an atom we can focus on the outer-most electron, assuming each state this electron is allowed to occupy corresponds to an energy level for the atom. The only orbits the electron can be allowed to have are those that result in a meaningful wavefunction (otherwise we will lose physical information about the electron). If we think of the orbits as circles, the only orbits that give us meaningful wavefunctions are those with a circumference which accommodate the wavelength of the wavefunction an integer number of times. Though it is harder to see than the requirement of having nodes at both string ends, exactly the same principle is in operation here, we are fitting a wave into a set of "boundary conditions", this time our boundary condition is that the wavefunction joins up seamlessly after moving around the closed orbit. Again we find a situation where the allowed number of states is restricted to fit a condition involving an integer. We can think of the wavefunction being restricted to forming standing waves in each allowed orbit and this produces the quantisation of allowed states.

This is how standing waves explain the quantization of electronic orbitals.