Question

Question #cb6b8

Answer 1

Via Quantum Mechanics with lots and lots of math.

Explanation:

In Quantum Mechanics, there is the Schrodinger Equation . The equation is capable determining many properties of a particle in a system based on time and position of the particle.

In a steady state Schrodinger Equation (we are not interested in the time variable);
The equation is `E=-barh^2/(2m)grad^2psi+Upsi`.
`psi` is the wave function of a particle; a function that describes the nature of your particle.

`grad^2=del^2/(delx^2)+del^2/(dely^2)+del^2/(delz^2)` in Cartesian coordinate system.
`grad^2=1/rdel/(delr)(r^2delpsi/(delr))+Sin(theta)del/(deltheta)(Sin(theta)delpsi/(deltheta))+del^2psi/(delphi^2)` in Polar Coordinate system.

Lets see the hydrogen atom, the easiest one.

`E=E_1/n^2`. `n` is the principle quantum number which corresponds to the orbital shell (also energy state).

What we want to do is to solve the Schrodinger equation to find `psi`.

Solving the equation is very tedious and requires arbitrary constants `l` and `m_l`. Refer to Separation of Variables method .

Solving this implies that `psi` has non-zero value if the values of `l` has integer values and does not exceed `n-1` .

!! `l` will determine your orbitals. `n` determines the values of `l`.!!

In other words, if `n=3` you must have `l=0,1 and 2`. Energy level 3 has 3 orbitals.
`s, p and d` orbitals. Other values of `l` in this case will cause `psi` to breakdown.

The electron will cease to exist.

Therefore, if `n=3`, you will have 3 solutions for `psi` based on `l=0,1, 2`.

Sorry if I can't express this any simpler. The history of the atomic model from Neils Bohr onwards is very theoretical.