The Orbital Quantum Number $l$

This quantum number arose in the solution of the polar part of the wave equation. To see its significance, we consider the terms in the radial wave equation :-

$$\frac{1}{R}\frac{d}{dr}\left( {r^2}\frac{dR}{dr}\right) + \... ...i \epsilon_0 r} + E - \frac{\hbar^2 l(l+1)}{2mr^2} \right) = 0$$ (34)

Clearly, the last term in brackets contains quantities which are various types of energies. The first quantity is the potential energy, the second is the total energy, so the third must be another type of energy. In fact, the numerator of this term must have the identification

$$\hbar^2 l(l+1) = (mvr)^2 = L^2 = \mbox{angular momentum squared}$$ (35)

given the denominator, in order that the entire term represents an energy. In other words this term represents the orbital kinetic energy

$$\frac{ l(l+1)\hbar^2}{2mr^2} = \mbox{orbital kinetic energy} = E_{k(\mbox{orbital})}$$ (36)

It follows that the separation constant $l(l+1))\hbar^2$ quantises the orbital angular momentum squared. This explains why we called the quantum number $l$ the orbital quantum number above. It labels orbitals in such a way that the magnitude of the angular momentum of each orbital is

$$L =\sqrt{l(l+1)}\hbar$$ (37)

You will notice that this is different to the Bohr model, where angular momentum was quantised as $L = n\hbar$. Bohr's quantisation condition was derived for flat 2-dimensional circular orbits for a ``particle''. Our new quantisation condition for orbital angular momentum is valid in the more general case of a delocalised quantum mechanical ``matter-wave'' in 3 dimensions. Curiously, both the Bohr model and the Quantum Mechanical Model lead to the same expression for the quantised energy levels.

Exercise 8
Ruminate on the relationships between the Bohr Model and the Quantum Mechanical Model and discuss it with your class-mates.