de Broglie Explanation of Bohr Quantisation

Step 1: State that electron has de Broglie wavelength λ = h/mv. Step 2: State stable orbit condition 2πr = nλ. Step 3: Substitute λ. Step 4: Rearrange carefully to get mvr = nh/2π. Step 5: Explain that n must be an integer for a standing wave.

A diagram may show a circular orbit with one, two, or three complete matter wavelengths fitted around it. Label orbit radius r, wavelength λ, and the condition 2πr = nλ.

A common wrong answer is saying quantisation happens because the electron's speed is zero in a stationary orbit. Stationary orbit means fixed allowed energy, not a stationary electron.

de Broglie's explanation says that an electron in a stable Bohr orbit behaves as a matter wave forming a standing wave around the nucleus, so the circumference of the orbit must contain an integral number of wavelengths: 2πr = nλ.

de Broglie's explanation says that an electron in a stable Bohr orbit behaves as a matter wave forming a standing wave around the nucleus, so the circumference of the orbit must contain an integral number of wavelengths: 2πr = nλ. Standing-wave condition: 2πr = nλ, where r is orbit radius in m, n is a positive integer, and λ is de Broglie wavelength in m. de Broglie relation: λ = h/mv, with h in J s, m in kg, and v in m s^-1. Combining them gives mvr = nh/2π. For the first Bohr orbit, n = 1, so the circumference contains one complete de Broglie wavelength. For n = 2, it contains two complete wavelengths.

If the electron wave does not fit exactly around the orbit, the wave would not join smoothly with itself and the orbit would not be stable. Stable orbits occur only when the circular path length equals a whole number of de Broglie wavelengths. Using λ = h/mv, the condition 2πr = nλ gives mvr = nh/2π, which is Bohr's angular momentum quantisation. Standing-wave condition: 2πr = nλ, where r is orbit radius in m, n is a positive integer, and λ is de Broglie wavelength in m. de Broglie relation: λ = h/mv, with h in J s, m in kg, and v in m s^-1. Combining them gives mvr = nh/2π. Show that de Broglie's condition gives Bohr's rule. Start with 2πr = nλ. Substitute λ = h/mv. Then 2πr = nh/mv. Multiplying both sides by mv gives 2πmvr = nh. Therefore, mvr = nh/2π. Final result: angular momentum is quantised. Usually asked as a derivation or short conceptual explanation: why only certain Bohr orbits are allowed, or how standing matter waves lead to angular momentum quantisation. A common wrong answer is saying quantisation happens because the electron's speed is zero in a stationary orbit. Stationary orbit means fixed allowed energy, not a stationary electron.