2.3.9
Classical mechanics mein ek particle ki definite position x ( t ) wavefunction Ψ ( x , t ) ∣Ψ ∣ 2 probability hai. Schrödinger equation simply woh rule hai jo bataata hai ki Ψ — yeh Newton ke F = ma
Intuition Equation ki zaroorat kyun haiNewton ka law present state ( x , p ) wahi power chahiye: abhi ka Ψ Ψ first-order-in-time differential equation chahiye. Time mein first order ka matlab: Ψ ( x , t 0 ) Ψ interpretation ∣Ψ ∣ 2
Definition Free particle ke liye plane waveDefinite momentum p E Ψ ( x , t ) = A e i ( k x − ω t ) , p = ℏ k , E = ℏ ω .
Yahan se kyun shuru karein? Kyunki yeh woh ek quantum state hai jiska energy aur momentum hum pehle se exactly jaante hain. Agar equation yahan kaam kare, toh hum generalize kar sakte hain.
Step 1 — time mein differentiate karke E Kyun? Hume ek aisa operator chahiye jo E ∂ t ∂ Ψ = − iω Ψ = − ℏ i E Ψ ⇒ E Ψ = i ℏ ∂ t ∂ Ψ .
Step 2 — space mein differentiate karke p Kyun? Kinetic energy banane ke liye hume p 2 ∂ x 2 ∂ 2 Ψ = ( ik ) 2 Ψ = − k 2 Ψ = − ℏ 2 p 2 Ψ ⇒ p 2 Ψ = − ℏ 2 ∂ x 2 ∂ 2 Ψ .
Step 3 — energy conservation impose karte hain. Kyun? Yahi physics ka input hai. Free particle ke liye
E = 2 m p 2 . Ψ i ℏ ∂ t ∂ Ψ = − 2 m ℏ 2 ∂ x 2 ∂ 2 Ψ .
Step 4 — potential add karte hain. Kyun? Real particle forces feel karta hai. Total energy E = 2 m p 2 + V ( x ) V ( x ) Ψ
Intuition Time aur space ko alag kyun karein?Agar V Ψ ( x , t ) = ψ ( x ) ϕ ( t )
TDSE mein substitute karo:
i ℏ ψ ( x ) d t d ϕ = ϕ ( t ) H ^ ψ ( x ) . ψ ϕ sirf time i ℏ ϕ 1 d t d ϕ = sirf space ψ 1 H ^ ψ = E .
Constant E Sirf t x x , t Us constant ki units energy ki hain aur woh total energy E
Time part: i ℏ ϕ ˙ = E ϕ ⇒ ϕ ( t ) = e − i E t /ℏ
Space part:
Definition Born interpretation & normalizationP ( x ) d x = ∣Ψ ( x , t ) ∣ 2 d x x x + d x ∫ − ∞ ∞ ∣Ψ ∣ 2 d x = 1.
Worked example 1 — Verify karo ki plane wave free TDSE solve karta haiΨ = A e i ( k x − ω t ) V = 0
LHS: i ℏ ∂ t Ψ = i ℏ ( − iω ) Ψ = ℏ ω Ψ Kyun? time-derivative − iω
RHS: − 2 m ℏ 2 ∂ x 2 Ψ = − 2 m ℏ 2 ( − k 2 ) Ψ = 2 m ℏ 2 k 2 Ψ Kyun? do space derivatives ( ik ) 2 = − k 2
Equate karo: ℏ ω = 2 m ℏ 2 k 2 E = 2 m p 2 Yeh kyun matter karta hai? Confirms karta hai ki equation sahi free-particle energy reproduce karta hai.
Worked example 2 — Infinite square well, ground stateV = 0 0 < x < L Infinite walls kyun? Woh force karte hain ki ψ = 0 x = 0 , L − 2 m ℏ 2 ψ ′′ = E ψ ⇒ ψ ′′ = − k 2 ψ k = 2 m E /ℏ ψ = A sin ( k x ) + B cos ( k x ) ψ ( 0 ) = 0 ⇒ B = 0 Kyun? cos 0 = 1 = 0 ψ ( L ) = 0 ⇒ sin ( k L ) = 0 ⇒ k L = nπ n = 0 ψ ≡ 0 E n = 2 m ℏ 2 k 2 = 2 m L 2 n 2 π 2 ℏ 2 , n = 1 , 2 , 3 … Yeh kyun matter karta hai? Quantization (discrete energies) automatically emerge karta hai boundary conditions se — assume nahi kiya jaata.
Worked example 3 — Well ke ground state ko normalize karnaψ 1 = A sin ( π x / L ) ∫ 0 L A 2 sin 2 ( π x / L ) d x = 1 ∫ 0 L sin 2 ( π x / L ) d x = L /2 Kyun? sin 2 1/2 A 2 ⋅ L /2 = 1 ⇒ A = 2/ L
Common mistake "TDSE wave equation hai, toh light ki tarah∂ t 2 Kyun sahi lagta hai: electromagnetic aur string waves ∂ t 2 u = c 2 ∂ x 2 u Fix: Schrödinger's time mein first order hai. Woh single time derivative (i Ψ
Common mistake "Wavefunction khud probability hai."Kyun sahi lagta hai: hum sirf Ψ Fix: Ψ ∣Ψ ∣ 2 = Ψ ∗ Ψ
Common mistake "Stationary state matlab kuch nahi badalata, tohΨ Kyun sahi lagta hai: "stationary" frozen jaisa lagta hai. Fix: Ψ = ψ e − i E t /ℏ observable ∣Ψ ∣ 2
Common mistake "TISE hamesha apply hota hai."Kyun sahi lagta hai: yahi woh equation hai jisse zyada compute karte hain. Fix: TISE sirf tab exist karta hai jab V time-independent ho; yeh variables separate karne se aata hai. Time-varying potentials ke liye poora TDSE chahiye.
Recall Self-test (answers chupaao)
∣Ψ ∣ 2 Time mein first order kyun? → taaki Ψ ( t 0 )
TDSE se TISE kaise milta hai? → separation of variables, V
Box mein energies quantized kyun hoti hain? → boundary conditions ψ ( 0 ) = ψ ( L ) = 0
Recall Feynman: ek 12-saal ke bacche ko explain karoSocho ek foggy cloud hai jo dikhata hai ki ek tiny ball kahan ho sakti hai . Fog jitna thikka, ball ke wahan milne ki utni zyada possibility. Schrödinger's equation woh weather rule hai jo bataata hai ki fog time ke saath kaise drift aur change karta hai. Agar ball ko ek box mein trap karo, toh fog sirf kuch khaas neat wavy patterns mein hi ban sakta hai — aur har pattern ki apni fixed energy hoti hai. Isliye ek trapped quantum particle sirf special "menu" energies hi rakh sakta hai, kuch bhi nahi.
Mnemonic Dono forms yaad karo"i-hat-bar pushes Time; Energy eats Shape."
TDSE: i ℏ ∂ t Ψ = H ^ Ψ H ^ ψ = E ψ "TIme-Independent = Time stripped off as e − i E t /ℏ
de Broglie Hypothesis — p = ℏ k Wavefunction and Born Interpretation — ∣Ψ ∣ 2 Particle in a Box — TISE ki pehli application.Quantum Harmonic Oscillator — TISE with V = 2 1 m ω 2 x 2 Hamiltonian Operator — H ^ ψ = E ψ H ^ Energy Quantization — boundary conditions se emerge karta hai.Heisenberg Uncertainty Principle — Ψ
Quantum particle ki state kaunsa field describe karta hai? Complex wavefunction Ψ ( x , t )
Time-dependent Schrödinger equation likho. i ℏ ∂ t Ψ = − 2 m ℏ 2 ∂ x 2 Ψ + V Ψ
Time-independent Schrödinger equation likho. − 2 m ℏ 2 ψ ′′ + V ψ = E ψ H ^ ψ = E ψ
TDSE time mein first order kyun hai? Taaki abhi ka Ψ
Plane wave se energy kaun sa operator deta hai? i ℏ ∂ t e − iω t E
p 2 − ℏ 2 ∂ x 2 e ik x p 2
TDSE se TISE kaise milta hai? Separation of variables Ψ = ψ ( x ) ϕ ( t ) V E
Stationary state ka time factor kya hai? ϕ ( t ) = e − i E t /ℏ
Stationary states "stationary" kyun hain? ∣Ψ ∣ 2 = ∣ ψ ∣ 2 Ψ
Width L E n = 2 m L 2 n 2 π 2 ℏ 2 n = 1 , 2 , 3 , …
[ 0 , L ] ψ = A sin ( π x / L ) ∣Ψ ∣ 2 d x Particle ko [ x , x + d x ]
Normalization condition kya hai? ∫ − ∞ ∞ ∣Ψ ∣ 2 d x = 1
H ^ Hamiltonian; total-energy operator.
Probability density |Psi|^2
Hamiltonian operator H-hat
