Unit 23: Quantization of Energy

Bohr's Theory of the Hydrogen Atom

Niels Bohr proposed a model for the hydrogen atom that introduced quantization of angular momentum and energy levels, successfully explaining the observed spectral lines.

Key Postulates

1. Electrons revolve around the nucleus in fixed circular orbits without radiating energy.
2. Only those orbits are allowed for which the angular momentum is an integral multiple of h/2π: L = nħ = n h / (2π), where n = 1,2,3,… is the principal quantum number.
3. Electrons can jump between orbits by absorbing or emitting a photon whose energy equals the difference between the initial and final energy levels: E_photon = E_i - E_f = h f.

Energy Levels

The energy of an electron in the nth orbit is given by:

E_n = - \frac{13.6 \text{ eV}}{n^2}

The negative sign indicates that the electron is bound to the nucleus; zero energy corresponds to a free electron at infinity.

Spectral Transitions

When an electron drops from a higher orbit n_i to a lower orbit n_f, the emitted photon has energy:

ΔE = E_{n_i} - E_{n_f} = 13.6 \text{ eV} \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)

This relation reproduces the observed wavelengths of the hydrogen spectrum.

Spectral Series of Hydrogen

The spectral lines emitted by hydrogen are grouped into series according to the final orbit n_f of the transition.

For example, the first line of the Balmer series (Hα) corresponds to the transition n=3 → 2 and has a wavelength of about 656 nm (red light).

Excitation and Ionization Potentials

These concepts quantify the energy required to change the state of an electron in an atom.

Excitation Potential

The excitation potential is the minimum voltage that must be applied to an electron (or the energy that must be supplied) to raise it from the ground state to a specific excited state.

For hydrogen, the excitation potential to reach the first excited state (n=2) is:

V_{exc} = \frac{E_2 - E_1}{e} = \frac{(-3.4 \text{ eV}) - (-13.6 \text{ eV})}{e} = 10.2 \text{ V}

Ionization Potential

The ionization potential is the energy needed to remove the electron completely from the atom (i.e., to take it from n=1 to n=∞). For hydrogen:

V_{ion} = \frac{E_{∞} - E_1}{e} = \frac{0 - (-13.6 \text{ eV})}{e} = 13.6 \text{ V}

Thus, a 13.6 V accelerating potential can ionize a hydrogen atom.

Energy Level Diagrams and Spectra

An energy level diagram graphically represents the allowed energies of an atom.

Emission Spectrum

When electrons transition from higher to lower energy levels, they emit photons. The collection of these photons forms the emission spectrum, seen as bright lines on a dark background.

Absorption Spectrum

When a continuous spectrum passes through a cool gas, electrons absorb photons of specific energies to jump to higher levels, producing dark lines at those wavelengths—the absorption spectrum.

Both spectra are complementary; the wavelengths of the lines match those predicted by Bohr’s formula.

De Broglie Theory and Wave‑Particle Duality

Louis de Broglie hypothesized that particles exhibit wave‑like properties, introducing the concept of matter waves.

De Broglie Wavelength

Any moving particle of momentum p = mv is associated with a wavelength:

λ = \frac{h}{p} = \frac{h}{mv}

where h is Planck’s constant (6.626×10^{-34} J·s).

Experimental Confirmation

Electron diffraction experiments (Davisson–Germer, 1927) showed that electrons produce interference patterns when scattered from a nickel crystal, confirming their wave nature with a wavelength matching the de Broglie prediction.

Implications for Bohr’s Model

Quantization of angular momentum in Bohr’s model can be understood as a standing‑wave condition: the circumference of the orbit must contain an integer number of wavelengths, 2πr_n = n λ, leading to the same quantization rule.

Heisenberg Uncertainty Principle

Werner Heisenberg formulated a fundamental limit to the precision with which certain pairs of physical properties, such as position and momentum, can be known simultaneously.

Mathematical Form

For any particle:

Δx ⋅ Δp ≥ \frac{h}{4π}

where Δx is the uncertainty in position and Δp the uncertainty in momentum.

Physical Interpretation

The principle does not arise from experimental imperfection; it is intrinsic to quantum mechanics. Attempting to localize a particle more precisely increases the spread in its momentum, and vice versa.

Example

If an electron is confined to a region of size Δx = 1×10^{-10} m (approximately the size of an atom), the minimum uncertainty in its momentum is:

Δp ≥ \frac{h}{4π Δx} ≈ \frac{6.626×10^{-34}}{4π × 1×10^{-10}} ≈ 5.3×10^{-25} kg·m/s.

X‑rays: Nature, Production, and Applications

X‑rays are a form of electromagnetic radiation with very short wavelengths and high energies.

Nature

Wavelength range: approximately 0.1 nm to 10 nm (or photon energies from ~0.12 keV to 124 keV). They travel at the speed of light and exhibit typical EM wave properties (reflection, refraction, diffraction, interference).

Production Mechanisms

• Bremsstrahlung (braking radiation): When fast‑moving electrons are decelerated by the electric field of atomic nuclei in a metal target, they lose kinetic energy, which is emitted as a continuous spectrum of X‑rays.
• Characteristic X‑rays**: When an incident electron ejects an inner‑shell electron (e.g., from the K shell), an electron from a higher shell drops down to fill the vacancy, emitting a photon of discrete energy characteristic of the target material (e.g., Kα, Kβ lines).