of the hydrogen atom. The transitions appear at different wavelengths of light,
however.Example 11.22
Predict the wavelength of light emitted by an excited Li^2 ion (Z3) as an
electron goes from the n4 state to the n2 state. Use the mass of the
electron in place of the reduced mass (this imparts a very minor 0.008% er-
ror in the calculation).Solution
We can use an expression for Esimilar to the one in equation 11.64, with
addition of the Z^2 term:E
Z
8
(^2) e
(^20)
4
h
(^2) n
1
22 n1
(^21)
For n 2 2 and n 1 4:
E
2
1
2 4
1
2
E3.677
10 ^18 J
Using Eh and c
as conversions, we can determine the wavelength
of the photon having this energy:
54.0 nm
This wavelength is in the vacuum ultraviolet region of the spectrum.11.11 The Hydrogen Atom Wavefunctions
Let us take a closer look at the wavefunctions themselves to finish this chapter.
Each wavefunction of a hydrogen atom is called an orbital.As mentioned, the
energy of an electron in an orbital (that is, an electron having its motion de-
scribed by a particular wavefunction) is dependent only on the principal quan-
tum number nand a collection of physical constants. Each group of wave-
functions having the same value of the quantized energy defines a shell.Each
shell has a degeneracy ofn^2. Each group of same-wavefunctions (for every
there are 21 wavefunctions, having different values ofm) constitutes a
subshell.In hydrogen and hydrogen-like atoms, all of the subshells within each
shell have the same energy. This is illustrated in Figure 11.17. In labeling shells
and subshells in hydrogen-like atoms (and other atoms, as we will see), we make
use of the quantum numbers nand . The numerical value of the principal
quantum number is used in the labeling, and for a letter designation is used: Letter designation
0 s
1 p
2 d
3 f
4 g
..
..
..32 (1.602
10 ^19 C)^4 (9.104
10 ^31 kg)
8[8.854
10 ^12 C^2 /(J m)]^2 (6.626
10 ^34 J s)^2358 CHAPTER 11 Quantum Mechanics: Model Systems and the Hydrogen Atom