Bohr–de Broglie Standing-Wave Condition (True/False) In the Bohr–Sommerfeld / de Broglie picture for quantized circular orbits, the circumference of an allowed electron orbit equals an integer multiple of the electron’s wavelength (2πR = nλ). True or false?

Introduction / Context:
De Broglie associated a wavelength λ = h/p with matter particles. The Bohr–Sommerfeld quantization condition can be viewed as a standing-wave requirement: only those orbits are stable where an integer number of wavelengths fits around the circumference, ensuring constructive interference and a stationary state.

Given Data / Assumptions:

• Circular orbit of radius R.
• Wavelength λ = h/p for the electron.
• Old quantum theory framework (pre-Schroedinger), used as an intuitive model.

Concept / Approach:

The standing-wave condition is 2πR = nλ for integer n. This is equivalent to quantized angular momentum L = nħ because p = m v and L = m v R; substituting λ = h/(m v) gives m v R = nħ. Thus, both the de Broglie and Bohr views are consistent for circular orbits in hydrogen-like systems.

Step-by-Step Solution:

Verification / Alternative check:

The derived energy levels E_n from this quantization match the Rydberg series for hydrogen, validating the physical picture within its domain.

Why Other Options Are Wrong:

• “False” choices contradict the established equivalence in the old quantum model.
• Ground-state-only or spin–orbit caveats are not part of the standing-wave criterion.

Common Pitfalls:

Confusing the heuristic Bohr–de Broglie model with exact quantum mechanical orbitals; the standing-wave idea is an analogy but correctly leads to L = nħ for circular motion.

Final Answer:

True