Enter quantum mechanics.
In quantum mechanics, you throw out the idea of orbits entirely. Electrons are no longer particles or waves that have a specified position or velocity, but are described by the square root of a probability distribution of position called wave functions (yes, I know how arbitrary that sounds, but trust me, there's a good reason for this). They are thus incapable of travelling around the atom in an actual path, since they don't have an actual position.
Instead, electrons can occupy states of energy, which completely determines the sort of probability distribution they have. These energies are completely determined by the total potential and kinetic energies of the system - mathematically, you can get it by solving the Schrödinger equation, which is a complicated differential equation. If you're interested in the math, you can read a simple example here.
To get the model of the atom, you:
- Solve the Schrodinger equation in spherical coordinates with V being the electric potential between the atom and the electron. You can also include more complicated interactions like the magnetic force between the proton and the electron if you want.
- That's it.
The beauty of this is that these solutions to the Schrodinger equation tells you (mostly) everything you need to know about how the electrons in the atom behave.
Because electrons have no definite position, you might wonder what electrons might look like when they're in a definite energy state. Below you can see images of these states: the coloured regions are where the probability of finding the electron (for experts: the integral of the wavefunction modulus squared) is greater than 90%. In deference to classical mechanics and our old ways of thinking about electrons, we call these energy states orbitals.
The famous 1s, 2s, 3s, 3p ... orbitals of a hydrogen atom.
In the pictures above, you'll notice a series of numbers labeling each orbital. These are called quantum numbers, and they define the exact nature of every energy state. Let's go through them one by one:
- The first, most important quantum number is n, the aptly-named principal quantum number. This quantum number determines the energy of the state - however, as you'll quickly see, there's a lot more to a state than just its energy.
- The second quantum number is L, the azimuthal quantum number. It can only take on values between 0 to n. In other words, if n = 2, the only acceptable values of L that correspond to a real state are L=0,1 .
What is the significance of this number? Well, largely, it governs the orbital angular momentum of the electron and in turn determines the shape of the orbital (in other words, the regions you see in the images above).
- The third quantum number is m, the magnetic quantum number. It's very closely related to L : where L fixes the total angular momentum, \m fixes the angular momentum along the z -axis (or actually just any one particular axis). Obviously, m can take on only values between − L and L - what's not so obvious is that it can only take on integer values.
- This number specifies which sub-orbital an electron occupies. You read that right - electrons can occupy orbitals within orbitals. In actual practice, of course, this just means the shape is altered further.
- The fourth and final quantum number is the spin quantum number, which is somewhat unique in that it does not affect the orbital at all.
- To be even more precise, spin is not a number produced from orbital theory - it is something intrinsic to every particle. For electrons, it can only take on the values 1/2 or -1/2 ; for more exotic objects, such as photons or collections of particles, it can take on a much larger range of values.
- Why does it matter? We'll see in a minute.