13.4.3 Quantum numbers
Completenessd/The three states inside the box
are a complete set of quantum
numbers for`= 1. Other states
with`= 1, such as the one on the
right, are not really new: they can
be expressed as superpositions
of the original three we chose.For a given, consider the set of states with all the possible values of the angular momentum’s component along some fixed axis. This set of states iscomplete, meaning that they encompass all the possible states with this.
For example, figure d shows wavefunctions with= 1 that are solutions of the Schr ̈odinger equation for a particle that is confined to the surface of a sphere. Although the formulae for these wave- functions are not particularly complicated,^8 they are not our main focus here, so to help with getting a feel for the idea of complete- ness, I have simply selected three points on the sphere at which to give numerical samples of the value of the wavefunction. These are the top (where the sphere is intersected by the positivezaxis), left (x), and front (y). (Although the wavefunctions are shown using the color conventions defined in figure u, p. 914, these numerical samples should make the example understandable if you’re looking at a black and white copy of the book.) Suppose we arbitrarily choose thezaxis as the one along which to quantize the component of the angular momentum. With this choice, we have three possible values forz: −1, 0, and 1. These
three states are shown in the three boxes surrounded by the black
rectangle. This set of three states is complete.
Consider, for example, the fourth state, shown on the right out-
side the box. This state is clearly identifiable as a copy of thez= 0 state, rotated by 90 degrees counterclockwise, so it is thex= 0
state. We might imagine that this would be an entirely new prize
to be added to our stamp collection. But it is actually not a state
that we didn’t possess before. We can obtain it as the sum of the
z=−1 andz= 1 states, divided by an appropriate normalization
factor. Although I’m avoiding making this example an exercise in
(^8) They are Ψ1,− 1 = sinθe−iφ, Ψ 10 =√2 cosθ, and Ψ 11 = sinθeiφ, whereθis
the angle measured down from thezaxis, andφis the angle running counter-
clockwise around thezaxis. These functions are called spherical harmonics.
Section 13.4 The atom 923