39-5 The Hydrogen Atom

39-5 The Hydrogen Atom#

Prompts

  • The hydrogen atom is an electron trap with Coulomb potential \(U = -e^2/(4\pi\varepsilon_0 r)\). Write the quantized energies \(E_n\) in eV. What is the ground-state energy? Why are the energies negative?

  • For a jump from \(n_{\text{high}}\) to \(n_{\text{low}}\), write \(\Delta E\) and the photon wavelength \(\lambda\) using the Rydberg constant \(R\). What is \(1/\lambda\) in terms of \(n_{\text{low}}\) and \(n_{\text{high}}\)?

  • Identify the Lyman, Balmer, and Paschen series: which level is the “home base” for each? Which series has visible lines?

  • What is the series limit? What is ionization? If an electron in the ground state absorbs a photon with \(\lambda\) shorter than the Lyman limit, what happens?

  • The Bohr model (1913) gave the correct energy formula but is wrong in most other aspects. What did Bohr assume? What does Schrödinger’s equation give that is correct?

Lecture Notes#

Overview#

  • The hydrogen atom is a natural electron trap: the electron is bound by the Coulomb attraction to the proton. The potential is \(U = -e^2/(4\pi\varepsilon_0 r)\)—a 3D well with walls that vary with distance.

  • Quantized energies \(E_n = -13.6\,\text{eV}/n^2\) (\(n = 1, 2, 3, \ldots\)). Ground state \(n=1\) has \(E_1 = -13.6\) eV. Levels converge to \(E = 0\); \(E > 0\) is the nonquantized (ionized) region.

  • Spectral series: Lyman (to \(n=1\)), Balmer (to \(n=2\), visible), Paschen (to \(n=3\)). Wavelengths given by \(1/\lambda = R(1/n_{\text{low}}^2 - 1/n_{\text{high}}^2)\).

  • The Bohr model (1913) correctly predicted \(E_n\) but is wrong in most other aspects; Schrödinger’s equation gives the correct quantum description.

Hydrogen as an electron trap#

A hydrogen atom consists of an electron (charge \(-e\)) and a proton (charge \(+e\)). The electron is trapped by the Coulomb attraction. The potential energy is

(398)#\[ U(r) = -\frac{e^2}{4\pi\varepsilon_0 r} \]

This is a three-dimensional, finite well—no sharp walls; the “depth” varies with distance \(r\). Solving Schrödinger’s equation yields quantized energies and wave functions.

Quantized energies#

The allowed energies are

(399)#\[ E_n = -\frac{13.6\,\text{eV}}{n^2}, \quad n = 1, 2, 3, \ldots \]

  • Ground state (\(n=1\)): \(E_1 = -13.6\) eV.

  • Excited states (\(n \geq 2\)): \(E_2 = -3.4\) eV, \(E_3 = -1.51\) eV, etc.

  • The levels get closer as \(n\) increases and converge to \(E = 0\).

  • \(E > 0\): The electron is free (not bound); the atom is ionized. This is the nonquantized region.

The negative energies indicate bound states; zero is the threshold for ionization.

Spectral series and wavelengths#

When the atom jumps between levels, it emits or absorbs a photon with \(\Delta E = hf = hc/\lambda\). For a jump from \(n_{\text{high}}\) to \(n_{\text{low}}\) (emission) or \(n_{\text{low}}\) to \(n_{\text{high}}\) (absorption):

(400)#\[ \frac{1}{\lambda} = R\left(\frac{1}{n_{\text{low}}^2} - \frac{1}{n_{\text{high}}^2}\right) \]

where \(R \approx 1.097 \times 10^7\) m\(^{-1}\) is the Rydberg constant.

Spectral series are grouped by the “home-base” level (where downward jumps end):

Series

Home base

Wavelength range

Lyman

\(n = 1\)

UV

Balmer

\(n = 2\)

Visible (656, 486, 434, 410 nm) + UV

Paschen

\(n = 3\)

IR

The series limit is the shortest wavelength in a series—the jump from \(n \to \infty\) to the home-base level. Light with \(\lambda\) shorter than the Lyman limit can ionize the atom (electron escapes with \(E > 0\)).

Balmer lines

The four visible hydrogen lines (656 nm red, 486 nm cyan, 434 nm blue, 410 nm violet) are Balmer jumps: \(n = 3\to 2\), \(4\to 2\), \(5\to 2\), \(6\to 2\). Bohr’s model correctly predicted these wavelengths.

Bohr model vs Schrödinger#

Bohr (1913): Assumed the electron orbits the nucleus in a circle with quantized angular momentum \(L = n\hbar\). He derived the correct \(E_n\) and explained the hydrogen spectrum. But the model is wrong in most aspects: electron is not a classical particle in orbit; \(L = 0\) is incorrectly disallowed; the model fails for multi-electron atoms.

Schrödinger: Solving the Schrödinger equation for the Coulomb potential gives the correct wave functions, energies, and angular momentum values. The energy formula \(E_n = -13.6\,\text{eV}/n^2\) matches Bohr’s result, but the underlying physics is quantum mechanical—probability clouds, not orbits.

Summary#

  • Hydrogen: Coulomb trap \(U = -e^2/(4\pi\varepsilon_0 r)\); quantized \(E_n = -13.6\,\text{eV}/n^2\).

  • Spectral formula \(1/\lambda = R(1/n_{\text{low}}^2 - 1/n_{\text{high}}^2)\); \(\Delta E = hf = hc/\lambda\).

  • Series: Lyman (\(n=1\)), Balmer (\(n=2\), visible), Paschen (\(n=3\)).

  • Ionization: \(E > 0\); photon with \(\lambda\) shorter than series limit can ionize.

  • Bohr gave correct \(E_n\); Schrödinger gives correct full quantum description.