The hyperfine interaction in a hydrogen atom between the magnetic dipole moment of the proton and the spin magnetic dipole moment of the electron splits the ground level into two levels separated by 5.9 * 10-6 eV. (a) Calculate the wavelength and frequency of the photon emitted when the atom makes a transition between these states, and compare your answer to the value given at the end of Section 41.5. In what part of the electromagnetic spectrum does this lie? Such photons are emitted by cold hydrogen clouds in interstellar space; by detecting these photons, astronomers can learn about the number and density of such clouds.

Verified step by step guidance

1. First, understand that the energy difference between the two hyperfine levels is given as 5.9 * 10^-6 eV. This energy difference corresponds to the energy of the photon emitted during the transition.
2. Convert the energy difference from electron volts to joules using the conversion factor: 1 eV = 1.602 x 10^-19 J. This will allow us to use the energy in the context of the Planck-Einstein relation.
3. Use the Planck-Einstein relation to find the frequency of the photon: E = h * f, where E is the energy in joules, h is Planck's constant (6.626 x 10^-34 J*s), and f is the frequency. Rearrange the formula to solve for frequency: f = E / h.
4. Once you have the frequency, use the speed of light equation to find the wavelength: c = λ * f, where c is the speed of light (3.00 x 10^8 m/s), λ is the wavelength, and f is the frequency. Rearrange to solve for wavelength: λ = c / f.
5. Finally, compare the calculated wavelength to the electromagnetic spectrum to determine the type of radiation. The wavelength should fall within the radio wave region, which is consistent with the 21 cm line often used in radio astronomy to study hydrogen clouds.

Key Concepts

Hyperfine Interaction

Hyperfine interaction refers to the interaction between the magnetic dipole moments of the nucleus and the electron in an atom. In hydrogen, this interaction causes a splitting of energy levels, resulting in two closely spaced levels in the ground state. This splitting is crucial for understanding transitions that emit photons, as seen in the question.

Photon Energy and Transition

Photon energy is related to the transition between energy levels in an atom. When an electron moves from a higher to a lower energy level, a photon is emitted with energy equal to the difference between these levels. This energy can be calculated using E = hf, where h is Planck's constant and f is the frequency, allowing us to determine the wavelength and frequency of the emitted photon.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, categorized by wavelength and frequency. Understanding where a photon lies within this spectrum helps identify its properties and applications. In this context, the emitted photon from hydrogen's hyperfine transition is typically in the radio wave region, which is significant for astronomical observations of hydrogen clouds.