Question #cb6b8

1 Answer
Feb 7, 2016

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.