Exact Solutions: Oscillator and Hydrogen Atom

Exact Solutions: Quantum Oscillator and Hydrogen Atom

The Importance of Exact Solutions

Most quantum problems are solved numerically or approximately. But two special cases have exact analytic solutions: the quantum harmonic oscillator and the hydrogen atom. They serve as "building blocks" for all of quantum physics—from molecular vibrations to atomic spectra, from photons in a field to nuclear states.

Quantum Harmonic Oscillator

Potential: $V(x) = m\omega^2x^2/2$. Hamiltonian: $\hat{H} = \hat{p}^2/(2m) + m\omega^2\hat{x}^2/2$.

The Algebraic Method—Ladder Operators. Introduce dimensionless operators:

$\hat{a} = \sqrt{m\omega/2\hbar}\ (\hat{x} + i\hat{p}/m\omega)$ — annihilation operator
$\hat{a}^\dagger = \sqrt{m\omega/2\hbar}\ (\hat{x} - i\hat{p}/m\omega)$ — creation operator

Their commutator: $[\hat{a}, \hat{a}^\dagger] = 1$. Hamiltonian through these: $\hat{H} = \hbar\omega(\hat{a}^\dagger\hat{a} + 1/2) = \hbar\omega(\hat{N} + 1/2)$, where $\hat{N} = \hat{a}^\dagger\hat{a}$—operator of quantum number.

Eigenvalues: From $[\hat{N}, \hat{a}^\dagger] = \hat{a}^\dagger$ it follows that $\hat{a}^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle$ ($\hat{a}^\dagger$ "raises" to the next level), and $\hat{a}|n\rangle = \sqrt{n}|n-1\rangle$ ($\hat{a}$ "lowers"). Ground state: $\hat{a}|0\rangle = 0$. All levels: $E_n = \hbar\omega(n + 1/2)$, $n = 0, 1, 2, ...$

Key conclusions: energy is quantized (only discrete values); zero energy $E_0 = \hbar\omega/2 \neq 0$—quantum "zero-point vibration," a consequence of the uncertainty principle; levels are equally spaced ($\Delta E = \hbar\omega = const$).

Wave Functions: $\psi_n(x) = (m\omega/\pi\hbar)^{1/4} \cdot (1/\sqrt{2^nn!}) \cdot H_n(\xi) \cdot e^{-\xi^2/2}$, where $\xi = \sqrt{m\omega/\hbar}x$, $H_n(\xi)$—Hermite polynomials.

$H_0 = 1$, $H_1 = 2\xi$, $H_2 = 4\xi^2 - 2$. Functions $\psi_n$ have $n$ zeros—nodes—the number of nodes increases with $n$.

Numerical Example: CO molecule, vibration frequency $\nu = 6.5 \times 10^{13}$ Hz. Quantum of vibrational energy: $\hbar\omega = h\nu \approx 6.626\times10^{-34} \times 6.5\times10^{13} \approx 0.269$ eV. At $T = 300$ K, thermal energy $kT \approx 0.026$ eV

lt;< 0.269$ eV $\rightarrow$ the molecule is in the ground state. Vibrational excitation of CO requires infrared photons $\lambda \approx 4.6\ \mu$m.

Hydrogen Atom

Problem: electron in the Coulomb potential of the proton $V(r) = -e^2/(4\pi\varepsilon_0 r)$.

Separation of Variables: $\psi(r, \theta, \phi) = R(r) \cdot Y_l^m(\theta, \phi)$. In spherical coordinates the Schrödinger equation separates into radial and angular parts.

Angular Part—Spherical Harmonics: $\hat{J}^2Y_l^m = \hbar^2 l(l+1) Y_l^m$, $\hat{J}_z Y_l^m = \hbar m\ Y_l^m$. Quantum numbers: $l = 0, 1, 2, ...$ (orbital); $m = -l, -l+1, ..., l$ (magnetic). Spherical harmonics $Y_l^m(\theta, \phi)$—complete orthonormal basis on the sphere. They describe the angular structure of orbitals (the shape of s, p, d orbitals).

Energy Levels: $E_n = -\frac{me^4}{2\hbar^2(4\pi\varepsilon_0)^2} \cdot \frac{1}{n^2} = -13.6\ \text{eV}/n^2$, $n = 1, 2, 3, ...$

The principal quantum number $n$ determines the energy. For $l = 0$: s-orbital (spherically symmetric). For $l = 1$: p-orbital (dumbbell-shaped). For $l = 2$: d-orbital.

Degeneracy: for each $n$: $l = 0, 1, ..., n-1$; $m = -l, ..., l$. Total number of states: $\sum_{l=0}^{n-1}(2l+1) = n^2$. Taking into account spin ($s = \pm 1/2$): $2n^2$ states. This degeneracy explains the structure of the periodic table!

Atomic Size: Bohr radius $a_0 = \hbar^2(4\pi\varepsilon_0)/(me^2) \approx 0.529$ Å $\approx 0.53 \times 10^{-10}$ m. Maximum $|\psi_{100}|^2$ at $r = a_0$.

Practical Applications

Atomic Spectroscopy: A photon with wavelength $\lambda$ is absorbed in the $n \to n'$ transition: $1/\lambda = R_H(1/n'^2 - 1/n^2)$, $R_H = 1.097 \times 10^7$ m$^{-1}$—Rydberg constant. Lyman series ($n' = 1$): ultraviolet. Balmer series ($n' = 2$): visible light. It is the Balmer series that is observed in stellar spectra.

Quantum Numbers and Chemistry: Covalent bond—overlap of orbitals. Molecular shape is determined by orbital symmetry. The periodic table is the direct consequence of orbital filling: $1s^2\ 2s^2\ 2p^6\ 3s^2\ ...$

Hydrogen level degeneracy is lifted when fine structure (spin-orbit interaction, $\Delta E \sim \alpha^2$) and hyperfine structure (interaction with the nucleus spin, $\Delta E \sim \alpha^4$) are taken into account. Hyperfine splitting of hydrogen’s ground level (21 cm, 1420 MHz) is used in radio astronomy and atomic clocks with precision of about $10^{-18}$.

Exact Solutions in Atomic Spectroscopy and Laser Physics

Analytical solutions of quantum mechanics have direct technological applications. The discrete spectrum of the hydrogen atom became a historical test of the theory and to this day serves as the foundation of atomic spectroscopy. Lyman, Balmer, Paschen, and Brackett series are used for analyzing the composition of stars, plasma, and gas discharges in industrial applications. The harmonic oscillator forms the basis for describing molecular vibrations: infrared spectroscopy allows identification of molecular groups by the frequencies of bond stretching, which are close to the quantum level formula. Tunable lasers use resonator spatial modes close to Gaussian beams—a superposition of harmonic oscillator functions. The theory of excited atomic states, based on exact solutions for hydrogen-like ions, is applied in developing quantum frequency standards—atomic clocks with accuracy less than $10^{-18}$ seconds. Quantum cascade lasers for the mid-infrared range operate on transitions between subbands of a quantum well—a direct application of the "well" problem with infinite walls—and are used in atmospheric gas spectroscopy and the detection of explosives.

Assignment: (a) For the harmonic oscillator, calculate $\langle x \rangle$, $\langle x^2 \rangle$, $\langle p \rangle$, $\langle p^2 \rangle$ in the state $|n\rangle$ (in terms of $\hat{a}$ and $\hat{a}^\dagger$). Check: $\Delta x\Delta p = \hbar(n + 1/2) \geq \hbar/2$. (b) Hydrogen atom: calculate $\langle r \rangle$ and $\langle r^2 \rangle$ in the state $\psi_{100}$. How are they related to $a_0$? (c) Transition $3 \to 2$ in hydrogen: calculate the wavelength. To which series does it belong? What color in the visible spectrum?