Spectroscopic Notation - Byu Physics And Astronomy

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Physics 571 Lecture #27
1
Spectroscopic notation
Decades ago, atomic physicists came up with a method of labeling atomic energy levels. A level is
determined by its orbital, spin, and total angular momenta and also by its parity. We keep track
of these four items in the following symbol:
2S+1
Π
J
The total orbital angular momentum is labeled as L. It is an upper case letter, and in the usual
cryptic atomic physics manner, if L = 0, then L is really S. If L=1, then L is really P. If L = 2,
then L is really D. Values of L greater than 2 are labeled alphabetically as F, G, H, and so on for
L=3, 4, 5, etc.
The total spin of the system is labeled as S (not to be confused with the S used to label the
value of the orbital angular momentum — I know, this is like an Abbott and Costello routine). To
the upper left of the orbital angular momentum symbol we write the value of 2S+1. Sometimes
this is called the multiplicity, and in many situations this tells the total number of J values allowed
in a term (see below).
The total spin (S) and the total orbital angular momentum (L) add together to make the total
angular momentum, J. The allowed values of J are L+S
L-S, in integer steps. The addition
and allowed values of angular momentum are related to the fact that space is quantized.
The parity of the state is noted on the upper right-hand side of the symbol. If the parity is
even, then we don’t write anything. If it is odd, we write a little “o”. The parity depends on
whether the state is even or odd with respect to inversion of the coordinates in the usual way.
You know, in one dimension a function has even parity is ( ) = (
) and it has odd parity
if (
) =
( ). The sine function has odd parity. The cosine function has even parity. It
is possible to show that the stationary states in an atom (the solutions to the time independent
Schr¨ o dinger equation) have definite parity. They are either even or odd. When you are thinking
about parity in a multi-electron atom, you add up the
values of all the individual electrons (the
real sum, not the goofy quantum mechanical sum — you know, 1+1=2 and NOT 1+1=0,1,2). If
this number is even, then the state has even parity. If it is odd, the state has odd parity. Actually,
the more technically correct statement is that Π = ( 1)
. When Π = 1 the parity is even and
when Π =
1 the parity is odd.
Sometimes you will see something that looks like an energy level symbol without the J value
2
1
or parity specified, like
(read “doublet D”) or
(read “singlet P”). These are called terms.
A quantum mechanical state is specified by the quantum numbers , , ,
, and
for each
electron. In our spectroscopic notation, each energy level has 2 + 1 “magnetic” sub-levels. If we
specify a particular sub-level, then we have identified a state.
2
Hydrogen
Let’s consider a few examples. First let’s think about hydrogen. There is one electron. It has
an angular momentum
and a spin . Let’s put the electron in the ground state. The electron
configuration is 1 , meaning that
= 1 and = 0. If you were a chemist you would say the electron
is in the
orbital of the first shell. The total orbital angular momentum is just L = 0 because
1
1
= 0. The total spin angular momentum is S =
because
=
. The total angular momentum
2
2

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