Modern Physics

Explanation: de Broglie wavelength ( ) is the wavelength associated with a material particle, proposing that matter has wave-like properties. It is given by , where is Planck’s constant and is momentum.
Derivation: For photons, , where .
de Broglie extended this to particles: , where is mass, is velocity.
Threshold Wavelength: Threshold frequency , where .
.
Answer: de Broglie wavelength ; threshold wavelength = 2960 Å.

Angular momentum , where is the electron mass ( ), , .
Answer: (to 3 significant figures).

For a hydrogen atom, the electron revolves in a circular orbit. The centripetal force is provided by the electrostatic force between the proton and electron.
Step 1: Electrostatic force: where is the electron charge, is the radius, and is the permittivity of free space.
Centripetal force: where is the electron mass, is velocity.
Step 2: Bohr’s quantization (third postulate): where is the principal quantum number, is Planck’s constant.
Step 3: Substitute into (1): Answer: .

Bohr’s third postulate states that the angular momentum of an electron in a stable orbit is quantized. The angular momentum (where is electron mass, is velocity, is orbit radius) is an integer multiple of : where is the principal quantum number and is Planck’s constant.
Answer: .

Photoelectric emission occurs when the frequency of incident light exceeds the threshold frequency of the metal, where ( is the work function). Red light has the lowest frequency in the visible spectrum ( ). If the metal’s threshold frequency is higher than that of red light, emission will not occur. Higher frequency light (e.g., blue, violet, or ultraviolet) is often required.
Answer: No, red light is not always necessary; light with frequency above the metal’s threshold frequency is needed.