de Broglie's hypothesis gives a reason for Bohr's angular-momentum rule. If an electron has wavelength \(\lambda=h/p\), a stable circular orbit must close on itself as a standing wave. That condition is \(2\pi r=n\lambda\).
Substituting \(\lambda=h/(mv)\) immediately gives \(mvr=n\hbar\). The logic is important: quantization is no longer just an extra rule imposed on a particle orbit; it looks like a wave self-consistency condition. This is the conceptual step toward wave mechanics.
de Broglie's hypothesis gives Bohr's quantization a wave interpretation. Instead of saying that angular momentum is mysteriously restricted, we ask whether the electron wave closes consistently after one full orbit.
If the wave does not fit an integer number of wavelengths around the circumference, it destructively interferes with itself and cannot represent a stable stationary orbit.
For an electron in a circular orbit, \(p=mv\). The closure condition is then
The circumference must contain an integer number of wavelengths.
Substitute \(\lambda=h/(mv)\) into the closure condition:
Multiplying by \(mv/(2\pi)\) gives
The angular-momentum rule is no longer an isolated postulate; it becomes a standing-wave condition.
The idea shifts the problem from "which orbit is allowed?" to "which wave states satisfy the boundary conditions?" This is the conceptual entrance to Schrodinger's equation in Chapter 2.
The practical move in de Broglie's hypothesis is to attach a wavelength to matter through momentum:
For an electron on a circular Bohr orbit, a stable wave must close on itself. The circumference must contain an integer number of wavelengths:
Substituting \(\lambda=h/p\) and \(p=m_ev\) gives \(m_evr=n\hbar\). Thus de Broglie's idea turns Bohr's angular momentum rule into a standing-wave condition rather than an isolated assumption.
This page is designed to support short guided exercises on: The matter-wave proposal and its role as a bridge from old quantum physics to wave mechanics.
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Use these anchors to design compact exercises. The exercise should be answerable from this page plus standard algebra, while longer derivations, full worked examples and broader context should point back to the original book.
- Focus: The matter-wave proposal and its role as a bridge from old quantum physics to wave mechanics.
- Conceptual check: state what the main result says physically before using it algebraically.
- Equation: \[\lambda=\frac{h}{p}\]
- Equation: \[2\pi r=n\lambda\]
- Equation: \[2\pi r=n\frac{h}{mv}.\]
- Boundary: use this page for setup and first-step reasoning; cite the book for longer derivations, complete experimental history or solved-problem detail.
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