2.3.8
Classical physics mein ek particle ki ek definite position hoti hai . Quantum mechanics mein particle ko ek wave function ψ ( x , t ) kahan hai , balki yeh batata hai ki use har jagah pane ke kya chances hain . Jo cheez actually probability ke roop mein measure hoti hai woh hai ==∣ ψ ∣ 2 probability density .
Square kyun? Kyunki ψ ∣ ψ ∣ 2 = ψ ∗ ψ
Wave function ψ ( x , t ) ψ ( x , t ) = a ( x , t ) + i b ( x , t ) nahi hai. Sirf isse bani quantities (jaise ∣ ψ ∣ 2
Definition Born rule (probability density)Particle ko tiny interval [ x , x + d x ] t d P = ∣ ψ ( x , t ) ∣ 2 d x , ∣ ψ ∣ 2 = ψ ∗ ψ ∣ ψ ∣ 2 probability per unit length hai (units: 1D mein m − 1 m − 3 ψ m − 1/2
ψ ∗ ψ = a + ib ψ ∗ = a − ib ψ ∗ ψ = ( a − ib ) ( a + ib ) = a 2 + b 2 = ∣ ψ ∣ 2 ≥ 0.
∣ ψ ∣ 2 ψ ∣ ψ ∣ ψ 2
ψ Complex ψ ψ 2 i 2 = − 1
∣ ψ ∣ ∣ ψ ∣ 2 phir square hoti hain. Nature ne ∣ ψ ∣ 2
Normalize kyun karte hain? Particle kahi na kahi toh hoga. Toh poore space par total probability exactly 1 honi chahiye.
KAISE karte hain: Har position par d P ∫ − ∞ ∞ d P = ∫ − ∞ ∞ ∣ ψ ( x , t ) ∣ 2 d x = 1.
Finite region [ a , b ]
P ( a ≤ x ≤ b ) = ∫ a b ∣ ψ ( x , t ) ∣ 2 d x .
Intuition Ek physical ψ ko kaun si conditions satisfy karni chahiye∫ ∣ ψ ∣ 2 d x ψ
Single-valued (har point par ek hi probability),Finite everywhere (infinite probability nahi),Continuous (aur usually ψ ′ Square-integrable → ψ → 0 x → ± ∞
Worked example Example 1 — Gaussian ko normalize karoDiya gaya hai ψ ( x ) = A e − x 2 / ( 2 σ 2 ) A
Step 1: ∣ ψ ∣ 2 = A 2 e − x 2 / σ 2 Kyun? Yahan ψ ψ ∗ ψ = ψ 2
Step 2: Gaussian integral use karo: ∫ − ∞ ∞ e − x 2 / σ 2 d x = σ π Kyun? Standard result hai; ∫ e − α x 2 d x = π / α α = 1/ σ 2
Step 3: Total = 1 A 2 σ π = 1 ⇒ A = ( σ π ) − 1/2 Kyun? Normalization condition demand karti hai ki ∣ ψ ∣ 2
Worked example Example 2 — Particle in a box (ground state)ψ ( x ) = A sin L π x 0 ≤ x ≤ L 0 P ( 0 ≤ x ≤ L /2 )
Step 1 (normalize): ∫ 0 L A 2 sin 2 L π x d x = A 2 ⋅ 2 L = 1 2 L sin 2 2 1 L A = 2/ L
Step 2 (probability): sin 2 x = L /2 aadha area [ 0 , L /2 ] P = ∫ 0 L /2 L 2 sin 2 L π x d x = 2 1 . Yeh step kyun? Probability density box ke center ke around symmetric hai, isliye yeh 50/50 hai — integrate karne se pehle ek quick Forecast-then-Verify sanity check.
Worked example Example 3 — Complex ψ, phase cancel ho jaata haiψ ( x ) = A e ik x e − ∣ x ∣/ L ∣ ψ ∣ 2
Step 1: ∣ ψ ∣ 2 = ψ ∗ ψ = A ∗ e − ik x e − ∣ x ∣/ L ⋅ A e ik x e − ∣ x ∣/ L Kyun? Conjugate lene par phase mein i → − i
Step 2: e − ik x e ik x = e 0 = 1 ∣ ψ ∣ 2 = ∣ A ∣ 2 e − 2∣ x ∣/ L Yeh kyun important hai? Oscillating phase factor e ik x Jo do wave functions sirf global phase mein alag hon, woh identical predictions dete hain.
Common mistake "Probability = ek point par∣ ψ ∣ 2 Kyun sahi lagta hai: Hum kehte hain ∣ ψ ∣ 2 Galti yeh hai: ∣ ψ ∣ 2 density hai (per unit length). Exact ek point par probability ∣ ψ ∣ 2 d x → 0 Fix: Actual probability pane ke liye hamesha ek interval par integrate karo.
Common mistake "Main ψ ko N se divide karke normalize karta hoon."Kyun sahi lagta hai: Total area N Galti yeh hai: Normalization ∣ ψ ∣ 2 ψ quadratic hai. Fix: ψ N ∣ ψ ∣ 2 N
Common mistake "ψ khud measurable wave hai."Kyun sahi lagta hai: Water/EM waves directly observed amplitudes hoti hain. Galti yeh hai: ψ ∣ ψ ∣ 2 ψ Fix: ψ probability amplitude maano, physical displacement nahi.
Common mistake "ψ ko ek phasee i θ Kyun sahi lagta hai: Isse function badal jaata hai. Fix: ∣ e i θ ψ ∣ 2 = ∣ ψ ∣ 2
Recall Quick self-test (answers chhupao)
∣ ψ ∣ 2 density .Square kyun, sirf ψ ψ ∣ ψ ∣ 2 ≥ 0
Normalization condition? → ∫ − ∞ ∞ ∣ ψ ∣ 2 d x = 1
1D mein ψ m − 1/2
Global phase e i θ ∣ ψ ∣ 2
Recall Feynman: ek 12-saal ke bachche ko samjhaoSocho ek foggy ghost hai jo ek jagah nahi reh sakta. Fog wahan thick hai jahan ghost ke milne ka zyada chance hai aur wahan thin hai jahan kam chance hai. Wave function ψ square karo. Aur kyunki ghost pakka kahi na kahi hai, agar poore kamre ki saari fog ko add karo toh exactly 1 poora ghost milna chahiye.
"Square it to be fair, sum it to be there."
ψ square karo → fair (non-negative) probabilities milti hain. Sum (integrate) karke 1 banao → particle there hai (kahi na kahi).
Wave function ψ kya describe karta hai? Ek probability amplitude — ek complex function jiska squared magnitude particle pane ki probability density deta hai.
Probability ke liye |ψ|² kyun use karte hain, ψ kyun nahi? ψ negative ya complex ho sakta hai; |ψ|²=ψ*ψ hamesha real aur ≥0 hota hai, aur interference experiments se match karta hai.
Normalization condition batao. ∫_{-∞}^{∞} |ψ|² dx = 1 (particle kahi na kahi toh hoga).
Agar ∫|ψ|²dx = N ho toh ψ ko normalize kaise karte hain? ψ ki jagah ψ/√N rakho, taaki |ψ|² 1/N se scale ho.
[a,b] mein particle milne ki probability kya hai? P = ∫_a^b |ψ(x,t)|² dx.
Ek dimension mein ψ ke units? m^(−1/2), taaki |ψ|² ke units m^(−1) hon (probability per length).
Kya global phase e^{iθ} physics ko affect karta hai? Nahi — |e^{iθ}ψ|² = |ψ|² hai, isliye yeh physically invisible hai.
Ek physical ψ ko kaun si chaar conditions satisfy karni chahiye? Single-valued, finite, continuous, aur square-integrable (infinity par →0).
Box of width L mein ψ=A sin(πx/L) ke liye A kya hai? A = √(2/L).
integrated over all space
Complex conjugate psi star
Probability density mod psi squared
Born rule dP equals mod psi squared dx
Normalization integral equals 1
Probability in region a,b
Physical conditions single-valued finite continuous