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=¯h22m2ψ+Uψ.
ψ is the wave function of a particle; a function that describes the nature of your particle.

2=2x2+2y2+2z2 in Cartesian coordinate system.
2=1rr(r2ψr)+sin(θ)θ(sin(θ)ψθ)+2ψϕ2 in Polar Coordinate system.

Lets see the hydrogen atom, the easiest one.

E=E1n2. 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 ψ.

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

Solving this implies that ψ has non-zero value if the values of l has integer values and does not exceed n1 .

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

In other words, if n=3 you must have l=0,1and2. Energy level 3 has 3 orbitals.
s,pandd orbitals. Other values of l in this case will cause ψ to breakdown.

The electron will cease to exist.

Therefore, if n=3, you will have 3 solutions for ψ 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.