Particle in a box — solving TISE, energy levels, wavefunctions

2.3.10 · Physics › Modern Physics

The setup

KYA chahiye: allowed wavefunctions $ψ (x)$ aur energies $E$ . KAISE: box ke andar time-independent Schrödinger equation (TISE) ko boundary conditions ke saath solve karo.

Figure — Particle in a box — solving TISE, energy levels, wavefunctions

Deriving from scratch

Rearrange karo: $\frac{d ^{2} ψ}{d x ^{2}} = - \frac{2 m E}{ℏ ^{2}} ψ = - k^{2} ψ, k^{2} \equiv \frac{2 m E}{ℏ ^{2}}$

$k$ define kyun kiya? Kyunki andar $E > 0$ hai, coefficient negative hai, jo oscillatory solutions deta hai. $k$ naam rakhne se algebra clean rehti hai — aur $k$ wave number nikalta hai.

Toh wave number quantised hai: $k_{n} = \frac{nπ}{L}$

Common mistakes

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho ek jump rope dono walls se bandhi hai. Jab tum use hilao, toh sirf kuch neat patterns aate hain: ek bada arch, phir do arches, phir teen — kabhi "dedh" arch nahi, kyunki ends hamesha still rehne chahiye. Har neat pattern ek certain shaking speed = ek certain energy maangta hai. Ek tiny particle ek box mein exactly usi rope jaisa behave karta hai: sirf special wave patterns fit hote hain, toh uske paas sirf special energies ho sakti hain. Aur woh kabhi poori tarah se wiggling band nahi kar sakta — sabse calm pattern (ek arch) mein bhi kuch energy hoti hai.

Recall checkpoint

Infinite square well kaunsa potential define karta hai?

V = 0

for

0 < x < L

aur

V = \infty

bahar (walls particle ko trap karti hain).

ψ

kaunsi boundary conditions satisfy karta hai?

ψ (0) = ψ (L) = 0

(continuity; particle wahan nahi ho sakta jahan

V = \infty

ho).

Cosine term kyun vanish ho jaata hai?

ψ (0) = 0

B = 0

force hota hai, sirf

A sin (k x)

bachta hai.

k

par quantisation condition kya hai?

sin (k L) = 0 \Rightarrow k_{n} = nπ / L, n = 1, 2, 3, \dots

Particle in a box ke energy levels?

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

Normalised wavefunctions?

ψ_{n} (x) = 2/ L sin (nπ x / L)

Nonzero ground-state energy kyun hoti hai?

Confinement + uncertainty: flat/zero wave forbidden hai, toh

E_{1} = h^{2} /8 m L^{2} > 0

(zero-point energy).

Level spacings kaisa behave karte hain?

Woh badhte hain:

E_{n + 1} - E_{n} = (2 n + 1) E_{1}

, equal nahi.

ψ_{n}

mein kitne interior nodes hote hain?

n - 1

nodes (aur

n

antinodes).

Probability density aur wavefunction mein kya fark hai?

Density

∣ ψ_{n} ∣^{2}

hai; probability ke liye ise region par integrate karo.

Bade box

L

ka energies par kya effect hota hai?

E_{n} \propto 1/ L^{2}

, toh levels girate hain aur paas aa jaate hain.

Connections

Schrödinger Equation (TISE) — woh master equation jo humne solve ki.
Wavefunction and Born Rule — $∣ ψ ∣^{2}$ ka matlab aur normalisation.
Heisenberg Uncertainty Principle — zero-point energy explain karta hai.
Quantum Harmonic Oscillator — ek aur bound state, lekin equally spaced levels.
Quantum Tunnelling and Finite Well — jab walls finite hoti hain toh kya badalta hai.
Standing Waves on a String — quantisation ka classical analogue.
Quantum Dots — real-world particle-in-a-box (size se colour tune hota hai).

Particle in a box — solving TISE, energy levels, wavefunctions

The setup

Deriving from scratch

Key consequences

Worked examples

Common mistakes

Recall checkpoint

Connections

Concept Map