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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 :

(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

(35) 
given the denominator, in order that the entire term represents an energy. In other words
this term represents the orbital kinetic energy

(36) 
It follows that the separation constant
quantises the orbital angular momentum squared.
This explains why we called the quantum number the orbital quantum number above.
It labels orbitals in such a way that the magnitude of the angular momentum of each orbital is

(37) 
You will notice that this is different to the Bohr model, where angular momentum was quantised as
. Bohr's quantisation condition was derived for flat 2dimensional 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 ``matterwave'' 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 classmates.
Next: The Magnetic Quantum Number
Up: Quantum numbers
Previous: The Principal Quantum Number
Simon Connell
20041004