In addition to quantized energy (specified by principle quantum number n), the solutions subject to physical boundary conditions also have quantized orbital angular momentum L. The magnitude of the vector L is required to obey
where l is the orbital quantum number.
Recall that the Bohr model of the hydrogen atom also had quantized angular moment L = nħ, but the lowest energy state n = 1 would have L = ħ. In contrast, the Schrödinger equation shows that the lowest state has L = 0. This lowest energy-state wave function is a perfectly symmetric sphere. For higher energy states, the vector L has in addition only certain allowed directions, such that the z-component is quantized as
States with different quantum numbers l and n are often referred to with letters as follows:
Hydrogen atom states with the same value of n but different values of l and ml are degenerate (have the same energy).
Figure at right shows radial probability distribution for states with l = 0, 1, 2, 3 and different values of n = 1, 2, 3, 4.
Table 41.1 (below) summarizes the quantum states of the hydrogen atom. For each value of the quantum number n, there are n possible values of the quantum number l. For each value of l, there are 2l + 1 values of the quantum number ml.
Example 41.2: How many distinct states of the hydrogen atom (n, l, ml) are there for the n = 3 state? What are their energies?
The n = 3 state has possible l values 0, 1, or 2. Each l value has ml possible values of (0), (-1, 0, 1), or (-2, -1, 0, 1, 2). The total number of states is then 1 + 3 + 5 = 9. We will see later that there is another quantum number s, for electron spin (±½), so there are actually 18 possible states for n = 3.
Each of these states have the same n, so they all have the same energy.
States of the hydrogen atom with l = 0 (zero orbital angular momentum) have spherically symmetric wave functions that depend on r but not on or . These are called s states. Figure 41.9 (below) shows the electron probability distributions for three of these states.
Follow Example 41.4.
The hydrogen atom: Probability distributions II
States of the hydrogen atom with nonzero orbital angular momentum, such as p states (l = 1) and d states (l = 2), have wave functions that are not spherically symmetric. Figure 41.10 (below) shows the electron probability distributions for several of these states, as well as for two spherically symmetric s states.