Quantum chemistry — particle in a box revisited; H-atom solutions

5.1.1 · Chemistry › Physical Chemistry (Advanced)

1. The master equation

WHY yeh form? Total energy = kinetic + potential. QM mein, momentum operator ban jaata hai $\overset{p}{^} = - i ℏ \nabla$ , toh kinetic energy $\frac{p ^{2}}{2 m} \to - \frac{ℏ ^{2}}{2 m} \nabla^{2}$ . Yeh demand karna ki $ψ$ $\hat{H}$ act karne ke baad same shape mein rahe (sirf $E$ se rescale ho) — iska matlab hi "stationary state" hai.

2. Particle in a 1D box — scratch se derive kiya hua

Setup (WHAT): Ek particle $x = 0$ aur $x = L$ ke beech trapped hai jahan andar $V = 0$ hai aur bahar $V = \infty$ hai. Infinite walls ⇒ bahar $ψ = 0$ , toh continuity se $ψ (0) = ψ (L) = 0$ — yeh boundary conditions hain.

HOW hum solve karte hain. Box ke andar $V = 0$ : $- \frac{ℏ ^{2}}{2 m} \frac{d ^{2} ψ}{d x ^{2}} = E ψ \Rightarrow \frac{d ^{2} ψ}{d x ^{2}} = - k^{2} ψ, k^{2} = \frac{2 m E}{ℏ ^{2}}$

Yeh step kyun? Yeh classic "second derivative = negative constant × itself" hai ⇒ sine/cosine solutions.

General solution: $ψ (x) = A sin k x + B cos k x$ .

$ψ (0) = 0$ apply karo: $B = 0$ . (cosine 0 par zero nahi hota, toh woh mar jaana chahiye.)
$ψ (L) = 0$ apply karo: $A sin k L = 0$ . Kyunki $A \neq = 0$ (warna $ψ \equiv 0$ ), humein $k L = nπ$ chahiye.

$k = \frac{nπ}{L}, n = 1, 2, 3, \dots$

Quantization kyun appear hoti hai: box ki length mein half-wavelengths ki poori number fit honi chahiye. Yahi poora reason hai ki energy discrete hoti hai.

$k^{2} = 2 m E / ℏ^{2}$ mein substitute karo:

Normalization $A$ ko fix karta hai. $\int_{0}^{L} ∣ ψ ∣^{2} d x = 1$ require karo: $A^{2} \int_{0}^{L} sin^{2} \frac{nπ x}{L} d x = A^{2} \cdot \frac{L}{2} = 1 \Rightarrow A = \frac{2}{L}$

Figure — Quantum chemistry — particle in a box revisited; H-atom solutions

3. 1D box se hydrogen atom tak

WHY H-atom harder hai: yeh 3D hai, aur potential central Coulomb attraction hai $V (r) = - \frac{e ^{2}}{4 π ε _{0} r}$ jo sirf $r$ par depend karta hai. Central potentials humein spherical coordinates mein variables separate karne dete hain.

HOW quantum numbers arise hote hain (same boundary-condition logic):

Quantum number	Kis condition se	Allowed values
$n$ (principal)	radial BC: $R \to 0$ as $r \to \infty$	$1, 2, 3, \dots$
$ℓ$ (azimuthal)	$θ$ -equation poles par finite	$0, 1, \dots, n - 1$
$m_{ℓ}$ (magnetic)	$ϕ$ -equation single-valued: $Φ (ϕ) = Φ (ϕ + 2 π)$	$- ℓ, \dots, + ℓ$

WHY $Φ$ single-valued ⇒ integer $m$ : $Φ (ϕ) = e^{im ϕ}$ ko $2 π$ ke baad repeat karna chahiye ⇒ $e^{im \cdot 2 π} = 1$ ⇒ $m \in Z$ . Bilkul same "poori number fit karo" idea jaise box mein.

4. Common mistakes (Steel-manned)

5. Active recall

Recall Quick self-test (answers cover karo)

Kaunse boundary conditions 1D box ko quantize karte hain? → $ψ (0) = ψ (L) = 0$ .
$m_{ℓ}$ integer kyun nikalta hai? → $ϕ$ ki single-valued requirement.
Ek orbital mein radial nodes ki number? → $n - ℓ - 1$ .

TISE mein kinetic operator ki jagah

p^{2} /2 m

ko kya replace karta hai?

- \frac{ℏ ^{2}}{2 m} \nabla^{2}

Box mein energy quantized kyun hoti hai?

Boundary conditions force karti hain ki half-wavelengths ki poori number fit ho, jisse

k = nπ / L

milta hai.

1D box ke energy levels?

E_{n} = \frac{n ^{2} h ^{2}}{8 m L ^{2}}

n = 1, 2, 3, \dots

Box mein

n = 0

kyun nahi ho sakta?

Yeh

ψ \equiv 0

deta hai, yaani koi particle nahi — valid state nahi hai.

ψ_{n}

mein kitne interior nodes hote hain?

n - 1

1D box wavefunction ke liye normalization constant?

2/ L

H-atom ke liye variables separate karne par wavefunction kya milta hai?

ψ = R_{n ℓ} (r) Y_{ℓ}^{m} (θ, ϕ)

ℓ

aur

m_{ℓ}

ke allowed values?

ℓ = 0, \dots, n - 1

;

m_{ℓ} = - ℓ, \dots, + ℓ

Hydrogen energy levels?

E_{n} = - 13.6/ n^{2}

eV (sirf

n

par depend karta hai).

H-atom energies negative kyun hain?

Electron bound hai; energy free (

r \to \infty

) zero reference se neeche hai.

Electron milne ke liye radial distribution function?

P (r) = 4 π r^{2} ∣ ψ (r) ∣^{2}

1 s

electron ka most probable radius?

r = a_{0} \approx 52.9

pm (Bohr radius).

Radial nodes vs angular nodes?

radial

= n - ℓ - 1

; angular

= ℓ

; total

= n - 1

Recall Feynman: ek 12-saal ke bachche ko explain karo

Ek guitar string imagine karo jo dono ends par fix hai. Woh sirf special patterns mein vibrate kar sakti hai — ek bump, do bumps, teen bumps — kabhi aadha bump nahi, kyunki ends pinned hain. Ek trapped electron usi string ki tarah hai: sirf certain "vibration patterns" (energies) allowed hain. Hydrogen atom mein, electron do walls ke beech ki jagah ek positive nucleus ke around trapped hai; rule "wave perfectly fit honi chahiye aur door tak blow up nahi honi chahiye" phir se sirf special patterns allow karta hai — aur woh patterns hi orbitals hain (1s, 2p, etc.). Same idea, fancier cage.

Connections

Schrödinger Equation — woh master equation jo dono problems solve karte hain.
Quantum Numbers — $n, ℓ, m_{ℓ}$ H-atom states ko kaise label karte hain.
Atomic Orbitals and Shapes — spherical harmonics s, p, d shapes dete hain.
Zero-point Energy — kyun sabse neechi box state $E = 0$ nahi hai.
Quantum Dots — PIB nanocrystals par apply hota hai.
Bohr Model vs Quantum Mechanics — $a_{0}$ most probable radius ki tarah reinterpret hota hai.
Operators and Eigenvalues — $\hat{H} ψ = E ψ$ ke peeche math machinery.

1. The master equation

2. Particle in a 1D box — scratch se derive kiya hua

3. 1D box se hydrogen atom tak

4. Common mistakes (Steel-manned)

5. Active recall

Connections

Concept Map