1. HUP, Pauli, and Quantized Energy
What if there is more than one particle?
The area of the plot has to equal h.
So, p = h/2L
E = h2/8mL2.

2. HUP, Pauli, and Quantized Energy
What if there is more than one particle?
For n = 1, E = h2/8mL2, as before.
Δx = L still, but Pauli’s Exclusion Principle (rule) states that no two particles can overlap in an x-p plot.
Thus, ΔxΔp = h, so (L)(2p2) = 2h.
So, p = 2h/2L.
Because E = p2/2m, E = 4h2/8mL2.
Thus, E = n2h2/8mL2, as before.

3. HUP, Pauli, and Quantized Energy
But, when we add more electrons to an atom, we see that two electrons can be in each energy level. How does this not violate the Pauli Exclusion Principle?

4. HUP, Pauli, and Quantized Energy
“Spin”
Provides an extra dimension for quantum particles to “spread out” (thus two can occupy the same energy level).
The electron produces a magnetic field (intrinsic).
Spin, like everything, is quantized. In this case, it has two possible values (ms = ±½).
Misnomer – comes from classical mechanics; electrons do not “spin”.

5. “Electrons in a Box”
The model can be supported by considering electrons in a long molecule. Consider curcumin, which is in turmeric powder.

6. “Electrons in a Box” E = n2h2/8mL2
ΔE = (nf2 – ni2)[h2/8mL2] ≈ 4.6 x 10–19 J
nf = 6; ni = 5
L ≈ 12 Å = 1.2 x 10-9 m
m = x 10–31 kg
h = x 10–34 Js
ΔE = hc/λ
λ ≈ 4.3 x 10–7 m or about nm

7. “Electrons in a Box”
Expected to absorb around nm (literature value is ~425 nm).

8. Pictures or Math?
Heisenberg’s work gave us the same answers as Bohr but did not rely on a physical picture.
Heisenberg only cared about what we could observe (which does not include orbits).
Not interested in a given “state” of an atom of electron.
Only interested in associations between pairs of states – we can measure the change (spectral lines).
Matrix mechanics: HUP from this; purely mathematical
How are transitions made? Don’t know (and it is irrelevant for Heisenberg)
Chess Example

9. Watching a Chess Game e4 e5
Ng3
Nc6
Bb5
Nf6
Nc3
Bc5
O-O d5

10. Schrödinger Wave mechanics instead of matrix mechanics.
Quantum particle in a 1-D box, acting as a wave.
Boundary conditions: bound in the box, total probability in the box = 1, continuous wave function.
ĤΨ = EΨ

11. Schrödinger Wave mechanics instead of matrix mechanics.
Still follows Heisenberg and Pauli (these are rules).
Only certain wavelengths (thus energies) allowed. They must “fit” the box (Ψ, the wave function or function of electron coordinates, must equal 0 at each wall).