Hydrogen energy levels Eₙ = −13.6 - n² eV

WHAT is being claimed

Why negative? We choose $E=0$ to mean "electron infinitely far away, at rest" (free). Any bound state has less energy than that, so it's negative. More negative = more tightly bound.

HOW to derive it from scratch (Bohr model)

We need only three physical ideas. Watch how $1/n^2$ pops out.

Formula Step 4 — Total energy = kinetic + potential

$E = \underbrace{\tfrac12 m v^2}_{\text{KE}} + \underbrace{\left(-\frac{1}{4\pi\varepsilon_0}\frac{e^2}{r}\right)}_{\text{PE}}$ From $(\star)$ , $\tfrac12 m v^2 = \dfrac{1}{2}\dfrac{1}{4\pi\varepsilon_0}\dfrac{e^2}{r}$ . So $E = \frac{1}{2}\frac{1}{4\pi\varepsilon_0}\frac{e^2}{r} - \frac{1}{4\pi\varepsilon_0}\frac{e^2}{r} = -\frac{1}{2}\frac{1}{4\pi\varepsilon_0}\frac{e^2}{r}$ Why this step? Notice $E = -\text{KE}$ : the total energy is half the potential energy and is negative automatically. Now substitute $r = a_0 n^2$ : $\boxed{\,E_n = -\frac{m e^4}{8\varepsilon_0^2 h^2}\cdot\frac{1}{n^2} = -\frac{13.6\ \text{eV}}{n^2}\,}$

The constant $\dfrac{m e^4}{8\varepsilon_0^2 h^2} = 13.6\ \text{eV}$ (the Rydberg energy) is built purely from $m,e,\varepsilon_0,h$ — fundamental constants. Plug numbers → $13.6$ eV. That is the whole secret.

Figure — Hydrogen energy levels Eₙ = −13.6 - n² eV

Transitions & photons (where spectra come from)

Worked examples

Common mistakes (Steel-man + fix)

Recall Feynman: explain to a 12-year-old

Imagine a staircase inside the atom. The electron can stand on step 1, step 2, step 3… but never floating between steps. Step 1 is the lowest, deepest basement; the higher steps are squished close together near the ground floor (which means "escaped"). To climb up the electron must swallow exactly the right amount of light; to fall down it spits out light of an exact color. The number $13.6$ just tells you how deep the basement is. Divide it by the step-number squared to find how deep each step sits.

Flashcards

What is the hydrogen energy level formula?

E_n = -13.6/n^2

eV, with

n=1,2,3,\dots

Why are hydrogen energy levels negative?

Reference

E=0

is the free electron; bound states have less energy, so they're negative (energy needed to free the electron).

What is the ionization energy of hydrogen from the ground state?

13.6

eV (from

-13.6

eV up to

0

What are the two Bohr postulates used to derive

E_n

(1) Coulomb force = centripetal force; (2) angular momentum quantized:

mvr=n\hbar

Why does total energy equal

-\tfrac12\,

PE and also

-

KE?

From the force balance

mv^2 = ke^2/r

, KE

=\tfrac12 ke^2/r

and PE

=-ke^2/r

, so

E=-\tfrac12 ke^2/r = -

KE.

How does orbit radius scale with

n

r_n = a_0 n^2

, with

a_0=0.529

Å.

Energy of

n=2

-3.40

eV.

Photon energy formula for a transition

n_i\to n_f

\Delta E = 13.6(1/n_f^2 - 1/n_i^2)

eV.

Wavelength of the

3\to2

(H-alpha) line?

\approx 656

nm (red), since

\Delta E=1.89

eV and

\lambda=hc/\Delta E

Handy constant for eV→nm conversion?

hc = 1240

eV·nm.

How does the formula change for a one-electron ion of charge

Z

E_n = -13.6\,Z^2/n^2

eV.

Why do energy levels crowd together as

n

increases?

Because

1/n^2

shrinks rapidly, so gaps narrow approaching

E=0

Connections

Bohr model of the atom — source of the quantization postulate.
Hydrogen spectral series — Lyman, Balmer, Paschen from these levels.
Photon energy E = hf = hc/λ — converts level gaps to light.
Rydberg constant and formula — $13.6$ eV $= R_H hc$ .
Ionization energy — directly the depth of $n=1$ .
Coulomb's law and Centripetal force — the two forces balanced in Step 1.
Quantum numbers — $n$ is the principal quantum number.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, hydrogen atom mein ek proton aur ek electron hota hai. Electron sirf kuch fixed "shelves" ya energy levels pe hi reh sakta hai — beech mein kahin nahi. Inn levels ki energy ka formula hai $E_n = -13.6/n^2$ eV. Yahaan $n$ shelf ka number hai (1, 2, 3...). Sabse important baat: energy negative hai, kyunki electron proton ke saath bandha (bound) hua hai. Negative ka matlab — usko free karne ke liye energy deni padegi. Ground state ( $n=1$ ) sabse deep hai ( $-13.6$ eV), aur jaise $n$ badhta hai, levels $0$ ke paas crowd ho jaate hain.

Yeh formula aaya kahan se? Bohr ne do simple ideas use kiye: (1) Coulomb force hi centripetal force ka kaam karta hai (proton electron ko andar kheechta hai, isi se circle banta hai), aur (2) angular momentum quantize hota hai, $mvr = n\hbar$ — yani har orbit allowed nahi, sirf whole-number wale. In dono ko mila ke radius $r_n = a_0 n^2$ nikalta hai, aur phir total energy (KE + PE) lagao to $E_n = -13.6/n^2$ nikal aata hai. Yeh $13.6$ koi magic number nahi — bas $m, e, \varepsilon_0, h$ constants se bana hai.

Yeh matter kyun karta hai? Jab electron upar wale level se neeche girta hai, to difference $\Delta E = E_{n_i}-E_{n_f}$ ek photon (light) ke roop mein bahar aata hai. Isi liye hydrogen ki spectrum mein fixed colored lines dikhti hain — jaise $3\to2$ wali red light $656$ nm. Exam tip: hamesha difference lo, single level ki energy se photon mat nikaalo. Aur yaad rakho: $-13.6$ ko $n^2$ se bhag do, multiply mat karo!

Hydrogen energy levels Eₙ = −13.6 - n² eV

WHAT is being claimed

HOW to derive it from scratch (Bohr model)

Transitions & photons (where spectra come from)

Worked examples

Common mistakes (Steel-man + fix)

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Modern Physics

Connections