# ELi5: Why electrons have quantised energy levels inside an atom?

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Why can’t electron just reside between two energy shells? What would happen if we grab an electron and forcefully keep it in between two shells?

In: Physics

The best answer has to do with the Schrodinger wave function which I’m not qualified to talk about. (More about that below.)

The more attainable version has to do with the deBroglie wavelength of the electron. If an electron is a standing wave, then it has a wavelength and it can only occupy a space that’s either 1, or 2, or 3… wavelengths long.

“What if we grab an electron and forcefully keep it between two shells” is analogous to “what if we take a spring (or an old-school phone cord) and try to set up a standing wave that’s 0.8, or 0.9, or 1.1, or 1.2 times the length of the spring/cord?”. The answer is you can’t get a standing wave that way; you’ll get a chaotic mess of interference because your wave doesn’t match the cord length. In other words, the electron will refuse to settle down there, and because it’s a probability function you can’t force it; it can literally teleport (tunnel) out of your grasp to get to a spot where it’s stable.

Example: hydrogen electron in ground state; mass is 9.11×10^-31 kg, energy is 13.6eV (2.176×10^-18 J). Velocity is sqrt(2E/m) = 2.19×10^6 m/s. deBroglie wavelength is h/mv = 3.32×10^-10 m. If you wrap that wavelength around a circle, the circle has a radius of 5.29×10^-11 meters and…oh, wow, that’s exactly the Bohr radius. 🙂

If you want to watch, Angela Collier on YouTube has a video that I think is called “how big is a hydrogen atom” where she hacks through the in-depth explanation. Even she shortcuts some of the math because it’s a tedious pain in the ass, but she shows more detail than I’ve ever seen anywhere else.

So to ELI5, think about a playground swing that you’re sitting on. You swing your legs in a particular way at the right time on the swing and you’ll go higher and higher. You’re swinging your legs at a particular frequency to match what the swing ‘wants’.

If you swing your legs at random times at random points, the swing won’t go higher and higher, you’ll end up going nowhere, just shaking around at the bottom.

This is because the swing has a rate that it naturally wants energy added, and when you match that rate, when you match that “natural frequency”, the swing can absorb that energy and take you higher and higher.

The physics of why an electron behaves the way it does is very different, but a similar principal applies. The electron is swinging around the nucleus of the atom at a particular frequency, and if you want to give it more energy, you need to match what it naturally ‘wants’, otherwise it’s like swinging your legs madly around on a stationary swing, not much is going to happen.

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Think of it like an electron in a hole.

It has a wavelength, because of wave particle duality, but the wave function needs to be at 0 at the edges of the hole.

The n=1 state is when the wavelength is twice the size of the hole (1 peak, half a wave)

n=2 is when the wavelength is the same length at the hole (2 peaks, a whole wave)

n=3 is when the wavelength is 2/3 the size of the hole (3 peaks, 1.5 waves)

And so on. The wavefunction couldn’t resonate in the hole if it doesn’t fit in the hole. The hole is having an effect on the wave function of the electron.

When you put multiple atoms text to each other, the exact energy levels for each orbital change slightly and it all gets blurred, so each energy state is more like a band of several possible energies than one specific one, but for the most part, the electron in a hole analogy works.