Foundations — Limitations of Bohr — fails for multi-electron atoms, fine structure

Before you can see where Bohr breaks, you must be fluent in every symbol his formula uses. We build them one at a time — each one anchored to a picture, each one earned before the next.

1. The cast of characters (the nucleus and the electron)

Look at the figure: the yellow dot is the nucleus, the blue ball is the electron, and the green circle is the orbit — the fixed path Bohr insists the electron must stay on.

2. Charge symbols: $e$, $Z$, and the "$+$/$-$"

3. The geometry symbols: $r$, $v$, and the circle

4. The force symbols: $k$, Coulomb pull, and centripetal force

The red arrow is the electric pull inward; the blue arrow is the electron's straight-line "want." They balance to give a circle. Why the topic needs this: this balance is line one of Bohr's whole derivation — it is where $r$ and $v$ first get tied together.

5. Mass symbols: $m$, $m_e$, $M$, and reduced mass $\mu$

6. The quantum leap: $n$, $\hslash$, and angular momentum $L$

The figure shows the allowed orbits as separated rings labelled $n=1,2,3$ — no rings between them are allowed. Why the topic needs it: this is the address label. Bohr's energy depends on $n$ alone, and that single-address limitation is the root of every failure. The richer address set ($n,l,m,s$) lives in Quantum Numbers n, l, m, s.

7. Energy symbols: $E_n$ and the minus sign

8. Light symbols: $\nu$, $h$, and $\Delta E=h\nu$

9. The correction symbols: $\alpha$, spin $s$, total angular momentum $j$

10. Two forbidding symbols: $r_{12}$ and $\Delta x\Delta p$

Prerequisite map

Read it top-down: charges and geometry build the force balance; angular momentum plus $\hslash$ build the quantum rule; together they give $E_n$; the fact that $E_n$ depends on $n$ and $Z$ alone — plus the extra symbols $r_{12}$, $s$, $j$, $\alpha$, uncertainty — is exactly what feeds into the Limitations topic.

Equipment checklist

Test yourself — cover the right side and answer aloud.