Chapter 1: And Then Also — Delta E and Energy Difference Between Orbits

Chapter 1: And Then Also

• Transcript excerpt focuses on the phrase: "delta e" and identifies it as the energy difference between the orbits. The rest of the content is not present, so the notes below expand around this central concept and its standard implications in quantum/atomic physics.

Delta E: Energy difference between orbits

• Delta E (1) is defined as the energy difference between an initial and a final orbital state:

  • If an electron transitions from an initial state Ei to a final state Ef, then

  $\Delta E = E<em>f - E</em>i.$

• This energy difference determines whether the transition involves emission or absorption of a photon:

  • Absorption: the system gains energy (Delta E > 0) as the electron moves to a higher-energy (less tightly bound) orbit.
  • Emission: the system loses energy (Delta E < 0) as the electron moves to a lower-energy (more tightly bound) orbit, with the emitted photon carrying energy |Delta E|.

• The photon energy corresponding to the transition is related to Delta E by:

  $h\nu = |\Delta E|,$
  where h is Planck's constant and \nu is the photon frequency.

Conceptual context and significance

• Delta E captures energy conservation during a quantum transition between bound states.
• It governs spectral lines: each allowed transition corresponds to a photon with a specific energy, producing emission/absorption lines at characteristic frequencies or wavelengths.
• The notion of orbits here aligns with quantized energy levels (e.g., atomic orbitals) rather than classical continuous orbits.

Formulas: energy levels and transitions (standard context)

• For a hydrogen-like atom (n is the principal quantum number, Z is the atomic number), the energy of a bound level is:

  $E_n = -\frac{13.6\,\text{eV}\; Z^2}{n^2}.$

• The energy difference between an initial level ni and a final level nf is:

  $\Delta E = E<em>f - E</em>i = -13.6\text{ eV}\; Z^2\left(\frac{1}{n<em>f^2} - \frac{1}{n</em>i^2}\right).$

• The corresponding photon energy relation is:

  $h\nu = |\Delta E|.$

• If desired, the hydrogen Rydberg wavelength formula connects to transitions via:

  $\frac{1}{\lambda} = R<em>H\left(\frac{1}{n</em>f^2} - \frac{1}{n<em>i^2}\right),$ where (RH \approx 1.097\times 10^7\ \text{m}^{-1}) is the Rydberg constant for hydrogen.

Example calculations (illustrative)

• Example 1: Transition from ni = 3 to nf = 2 (hydrogen, Z = 1)

  • Calculate Delta E:

  $\Delta E = -13.6\text{ eV}\left(\frac{1}{2^2} - \frac{1}{3^2}\right) = -13.6\text{ eV}\left(\frac{1}{4} - \frac{1}{9}\right) = -13.6\text{ eV}\left(\frac{5}{36}\right) \approx -1.89\text{ eV}.$

  • Photon energy magnitude: (|\Delta E| \approx 1.89) eV.
  • Frequency:
    $\nu = \frac{|\Delta E|}{h} \approx \frac{1.89\text{ eV}}{4.135\times 10^{-15}\text{ eV s}} \approx 4.57\times 10^{14}\text{ Hz}.$
  • Wavelength:
    $\lambda = \frac{c}{\nu} \approx 6.56\times 10^{-7}\text{ m} = 656\text{ nm}.$ (Red H-alpha line in the Balmer series context.)

• Example 2: Absorption from ni = 2 to nf = 3

  • Delta E is positive; photon is absorbed with energy (h\nu = \Delta E).

Connections to broader principles

• Energy quantization: only discrete Delta E values are allowed, leading to discrete spectral lines.
• Conservation of energy: the energy difference must be carried away by a photon (or supplied by a photon) during transitions.
• Relationship to emission/absorption spectra across atoms and ions; foundational for spectroscopy, lasers, and astrophysical observations.

Practical implications and applications

• Understanding spectral lines helps identify elements in stars and interstellar matter.
• Design and analysis of light sources (LEDs, lasers) rely on specific Delta E transitions.
• Selection rules (not discussed here) govern which transitions are allowed, influencing which lines appear.

Notes on the provided transcript excerpt

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