de broglie wavelength

Question

An electron in the first Bohr orbit of hydrogen has a de Broglie wavelength λ. What is the wavelength of the electron in the second orbit?

RankGuru · Accepted Answer

1. **Bohr's Postulate**: For an orbit to be stable, its circumference must equal an integer multiple of the de Broglie wavelength:
   - $2\pi r_n = n \lambda_n$.
2. **Radius Relation**: Radius of $n^{th}$ orbit is $r_n = a_0 n^2$.
3. **Substitution**:
   - $2\pi (a_0 n^2) = n \lambda_n$
   - $\lambda_n = (2\pi a_0) n$
4. **Proportionality**: $\lambda_n \propto n$.
5. **Comparison**:
   - For $n=1$, $\lambda_1 = \lambda$.
   - For $n=2$, $\lambda_2 = 2 	imes \lambda_1 = 2\lambda$.

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