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I am reading how de Broglie justified the 2nd postulate of Niels Bohr (i.e. angular momentum of an electron to be integral multiple of $\frac{h}{2\pi}$). I get his explanation of electron acting like standing wave and circumference should hold integral multiples of wavelengths. I get till here.

Where I am struggling is the normal equation for a standing wave with both ends fixed is wavelength $= \frac{2L}{n}$ where $L$ is length of the string and $n = 1,2,\cdots$. So here integral multiples of wavelength is twice the length. But in case of atom integral multiples of wavelength is equal to length (circumference of orbit). There seems to be discrepancy of 2 times in both the equation. Can someone please explain what is going on here?

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The factor of $2$ difference is because a standing wave can only be formed if a wave's phase goes through an integer number of full cycles before closing up on itself. For the Bohr orbits, these cycles go around the circle once, but for standing waves on a string with both ends fixed, these cycles traverse the length of the string twice. They go to the right, bounce off the end, then come back.

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