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I was reading about how de Broglie's electron-wave theory explained the quantisation of energy in a Bohr atom. What I understand is as follows:

If the wave is to be arranged in the form of a circle so that it attaches to itself, the waves can only occur if there is a whole number of waves in the circle.

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The standing waves on the left-hand side of the figure are allowed to form in the brown box, because they fit perfectly inside the box. In contrast, the waves on the right-hand side of the figure cannot form in the brown box because they do not fit perfectly inside the box.

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The left-hand standing wave fits in the electron cloud while the right-hand one does not.

(Source)

Bohr said that electrons could exist at specific "allowed" energy levels, but that they couldn't exist between those energy levels. And, apparently, these whole number wavelengths are supposed to explain that.

But, can't the wavelength of an electron increase or decrease to "fit" the wave perfectly in any radius? Or, are there some other conditions/constraints involved?

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    $\begingroup$ There corresponds a unique energy to every $\ce n^{th}$ shell. And unique energy means unique frequency which in turn means fixed wavelength. $\endgroup$ May 24, 2023 at 3:41
  • $\begingroup$ You can search for other related posts. Look for "Bohr's postulates". See for instance chemistry.stackexchange.com/questions/121488/… $\endgroup$
    – Buck Thorn
    May 24, 2023 at 5:30
  • $\begingroup$ A complete derivation: alpha.chem.umb.edu/chemistry/ch115/carter/files/103more/… $\endgroup$
    – Buck Thorn
    May 24, 2023 at 5:31
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    $\begingroup$ If anyone saw the source, it has 7 sub-sections of the Schörodinger's equation. It doesn't present a single equation, because it is for higschool. Hard to grasp anything of this without math... I'd rather state that Bohr said that the angular momentum had "allowed" values, and that energy also does is a consequence of it (although he wanted the latter to explain the spectral series...) $\endgroup$ May 24, 2023 at 11:05

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They do adjust to 'fit' the box/ring as you put it, those on the right are the wavefunctions for a bigger box than shown in the picture. You can have any size box/ring you want but the restriction is that the wavefunctions have to fit exactly. (This is called a boundary condition). If this were not the case, as in the pic at the bottom right, there will be a place where the wavefunction can have two values at the same position which is not physically possible. As the size of the box/ring changes so do the energy levels, you should have seen the equations that demonstrate this.

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