Atoms: what we should know, part II

As we discussed today in class, for an electron in an atom there is a sequence of states, or energy levels, in which the electron can reside. (See part I.) These wave-like states have characteristics which are reminiscent of the modes of a string of length L stretched between two posts.

5) The lowest energy state in this sequence is called the ground state. Rather amazingly, the ground state has kinetic energy.

6) This is one of the most surprising results of quantum physics and is intimately related to the uncertainty principle.

7) The uncertainty principle says that the more an electron is confined, the higher its momentum will be. Momentum, speed and kinetic energy (K.E.) all go together, so confining an electron causes it to have kinetic energy, and the more confined it is the more kinetic energy it must have.

8) Just as the fundamental mode of a guitar string has a higher frequency when the string is shortened (f=v/2L), so will an electron have a higher kinetic energy when it is more confined (K.E.~h^2/mL^2). Here L is the length of the string, which is a measure of the confinement of the string wave; and L is the diameter of the electron cloud around the atom, which is a measure of the confinement of the electron. (Smaller L in each case means more confinement.)

9) Note that in each case there is an inverse relationship: smaller L means higher frequency (for the fundamental mode of the string, smaller L means higher kinetic energy for the electron in the atom.