Question

How does de Broglie's hypothesis account for the fact that the energies of the electron in a hydrogen atom are quantized?

Short answer

De Broglie's hypothesis, by introducing the idea of wave-particle duality, allows for the behavior of electrons in a hydrogen atom to be modeled as standing waves in certain permissible orbits. This leads to the concept of energy quantization, as the electron can only have the specific energy levels associated with these orbits, and changes in energy level require specific amounts of energy to be absorbed or emitted.

Step-by-step solution

Understand de Broglie's Hypothesis

De Broglie's hypothesis suggests that every particle has a dual nature. It can exhibit either wave-like properties or particle-like properties based on the situation. This idea of wave-particle duality was groundbreaking in quantum physics, and is the underlying theory behind quantum mechanics. In the context of electron behavior, this is why electrons exhibit wave-like properties when revolving around the nucleus.

Establish a Standing Wave in the Electron's Orbit

Now, consider the electron moving around the nucleus in a circular orbit. According to de Broglie, it behaves like a standing wave in this orbital path. A standing wave condition is one where the wave appears stationary, i.e., it does not seem to move in any particular direction. For a standing wave condition to exist, the circumference of the orbit needs to be an integral multiple of the wavelength (lambda) of the electron. So, 2 \pi r = n \lambda, the de Broglie relation, where 'r' is the radius of the orbit and 'n' is a natural number known as the quantum number.

Determine Energy Quantization from the Wave Condition

The wave condition reveals that only certain specific orbits are permissible, which are those where the circumference is an integral multiple of the wavelength. This leads to the concept of energy quantization - the electron can only occupy these certain orbits and therefore can only have certain specific energy levels. This results in the energies of the electron being quantized, because jumping from one orbit to another involves absorbing or emitting specific 'quanta' or amounts of energy, that correspond to the energy difference between those orbits.