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Recall from last lecture:

From Bohr's theory for hydrogen we have:

• E_n = -(m_e k_e² e⁴ / 2 hbar²)(1/n²) = -13.6eV / n²
• r_n = n²a₀
• a₀ = hbar² / m_e k_e e² = 0.0529nm
• R_H = m_e k_e² e⁴ / 4p c hbar³

28.4 Modification of the Bohr Theory

Bohr's theory gives us a good model for the hydrogen atom, a model that is still used with a few modifications.

The Bohr theory can be applied to other "hydrogen-like" atoms (atoms with only one electron) by replacing the factor e² by Ze² wherever it appears, where Z is the number of protons in the nucleus. For singly ionized helium, Z=2; for doubly ionized lithium, Z=3; and so on.

The next refinement was to extend Bohr's theory to include elliptical orbits. This led to the concept of the orbital quantum number, l. (The level n is known as the principal quantum number.) For a given energy level, n, the orbital quantum number can take on values from 0 to n-1. The values of l are equated to the subshells: s is l=0, p is l=1, d is l=2, and f is l=3.

The next refinement came when people noticed that what appeared to be a single spectral line would split into several lines in the presence of a magnetic field (see Figure 28.9). This led to the concept of orbital magnetic quantum number, m_l. The results showed that m_l could take on integer values from -l to +l.

Lastly, it was noted that at very high resolutions, single spectral lines split into two, even in the absence of a magnetic field. This splitting is known as fine structure, and was connected to the concept of electron spin. The spin magnetic quantum number, m_s was introduced to account for this.

These four quantum numbers are what we still use to describe the structure of atoms: n, l, m_l, and m_s. We will discuss these more in the next sections.

28.5 De Broglie Waves and the Hydrogen Atom

Bohr's assumption that allowed orbits are those where the electron's orbital angular momentum is an integral multiple of hbar results in correct predictions for the energy levels of hydrogen. But this assumption comes out of the blue. Why it works was a mystery until de Broglie realized a connection between this assumption and his matter waves.

It turns out that Bohr's condition is equivalent to stating that the allowed orbits are those with an integral number of oscillations of the electron's de Broglie waves. If an orbit has an integral number of oscillations, then the circumfrence of the orbit is an integer multiple of the de Broglie wavelength, l:

28.6 Quantum Mechanics and the Hydrogen Atom

The use of matter waves was expanded on by Schrödinger when he devised his wave equation for quantum mechanics. One of the first problems analysed with the wave equation was the hydrogen atom. This problem can be solved exactly, and some of the results are:

• The energies of the allowed states agree exactly with the Bohr results.
• The state of an atom is described by three quantum numbers,
• the principal quantum number, n
• the orbital quantum number, l
• the orbital magnetic quantum number, m_l
• The quantum numbers come directly out of the mathematics of the wave equation. It also determines the allowed ranges of the quantum numbers:
  • n ranges from 1 to infinity in integer steps
  • l ranges from 0 to n-1 in integer steps
  • m_l ranges from -l to +l in integer steps

States that do not adhere to these rules are not allowed. The wave equations re-produces the results of the Bohr theory plus modifications, without ad-hoc assumptions. Further, it can be applied to a host of other problems. The significance is similar to the advent of Newton's laws, which not only re-produced the results of Kepler regarding the orbits of the planets, but can then be used to solve many other problems.

28.7 The Spin Magnetic Quantum Number

There is a fourth quantum number that is needed to complete the atomic model. This fourth quantum number is m_s, the spin magnetic quantum number. We usually picture this quantum number as an electron spinning in one of two possible directions (clockwise or counter-clockwise). This number was originally introduced to explain the splitting of certain spectal lines, like the orange doublet of sodium.

Sodium is a group I element, like hydrogen, lithium, and potassium. We now understand that it consists of a complete n=1 shell, a complete n=2 shell, plus one electron in the n=3 shell (1s²2s²2p⁶3s¹). This structure makes it "hydrogen-like". The ground-state energy depends on whether the outermost electron is "spinning" in the same direction as or opposite to the nucleus of the atom. Thus, the ground-state energy "splits" into two closely spaced levels, and the spectral line splits, one line corresponding to transitions to one of the levels and the other line to transitions to the other level.

It is interesting to note that the spin of the electron and other particles is still put into the theory by hand. A fundamental explanation of particle spin is still being sought.