2.3.11 · D1Quantum tunneling — concept, transmission coefficient
Ek tiny particle koi hard dot nahi balki ek spread-out wave hai jiska height har point pe batata hai ki particle wahan milne ki kitni probability hai. Kyunki wo wave smoothly bend karti hai aur kabhi zero pe snap nahi ho sakti, uska thoda sa hissa hamesha us wall ke andar aur aar-paar leak karta rehta hai jise particle cross nahi kar sakta — aur yahi leak hai quantum tunneling .
Is page pe assume kiya gaya hai ki aapne kuch nahi dekha . Hum har letter, squiggle, aur symbol parent topic mein use hone wale ko ek ek karke build karenge, har cheez pehle wali ke upar tiki hui. Ant tak, master formula plain English jaisi lagegi.
Kisi bhi symbol se pehle, physical situation picture karo. Ek particle left se aata hai, ek raised "wall" of energy se milta hai, aur hum poochte hain: kitna uss paar guzarta hai?
Teen flat/raised zones teen regions hain jinke baare mein parent baat karta hai. Neeche sab kuch un tools ke baare mein hai jo describe karte hain ki wave har region mein kya karti hai.
Definition Position variablex x straight line ke saath distance hai, metres mein measure kiya gaya. Left chhota (yaani negative bhi) x x
Picture: upar wali figure mein horizontal axis. Barrier x = 0 x = L
Topic ko iske zaroorat kyun hai: tunneling poori tarah ek sawaal hai ki wave barrier ke aar-paar chalte waqt kaise badlti hai — isliye hume ek naam chahiye "kitna aage" ke liye.
E E energy hai jo particle carry karta hai (uski motion energy, kyunki humne flat ground ko zero energy set kiya hai). Ise poori picture pe khichi ek horizontal water-level line ki tarah socho.
Woh picture jo matter karti hai: energy level E V 0
Agar E ≥ V 0
Agar E < V 0 neeche ): classically particle zaroor bounce back karega. Yeh "forbidden" case hai — aur bilkul wahi case hai jahan tunneling rehti hai.
V 0 − E jitni height se tum kam pad rahe ho , V 0 − E
Topic ko dono kyun chahiye: tunneling sirf tab exist karti hai jab E < V 0
ψ ( x ) ψ x matter-wave ki height hai. Jahan ψ ψ
Picture: ek jagah dot ki jagah, x ψ
∣ ψ ∣ 2 chance ∣ ψ ∣ 2 squared (isliye yeh hamesha positive hai). Yeh squaring baad mein matter karegi (yahi woh jagah hai jahan se mysterious "2" aata hai).
Topic ko iske zaroorat kyun hai: tum kisi particle ke baare mein "wall se leak karna" nahi bol sakte jab tak particle koi aisi cheez na ho jo spread ho sake — ek wave. ψ Wavefunction and Boundary Conditions mein develop ki gayi hai aur de Broglie Wavelength pe tiki hai (yeh notion ki matter ki wavelength hoti hai).
Yeh sab kuch ka dil hai, isliye hum ise ek figure dete hain.
Definition Oscillating wavee ik x hamesha upar-neeche ripple karti rehti hai — ek sine-jaisi wiggle. Hum ise e ik x travelling wave hai: ek moving particle. Wavy magenta curve dekho.
e − κ x wiggle nahi karti . Yeh smoothly neeche zero ki taraf slide karti hai bina kabhi cross kiye — ek exponential decay. Smooth violet curve dekho jo fade hoti hai par kabhi khatam nahi hoti. Yeh "leak" hai.
Intuition Do alag shapes kyun?Ek wave jo axis ki taraf wapas curve karti hai wiggle karti hai (oscillates). Ek wave jo axis se door curve karti hai decay ki taraf daud jaati hai. Tumhe kaun sa milta hai yeh sirf energy gap V 0 − E E < V 0
Topic ko iske zaroorat kyun hai: tunneling ka poora trick yeh hai ki wall ke andar wave decay karti hai par vanish nahi hoti , isliye far side tak ek sliver survive karta hai.
Ab hum e ik x e − κ x
e e ≈ 2.718 e ( something ) e − κ x fraction se fade karo". Yeh constant-fraction fading bilkul wohi smooth curve hai jo upar violet plot mein hai. Hum ise isliye use karte hain kyunki yeh single aisi shape hai jiska steepness uski apni height ke proportional rehta hai — exactly wahi jo Schrödinger's equation wall ke andar demand karti hai.
Definition Imaginary uniti i i 2 = − 1 e ik x rippling wave ban jaata hai (sines aur cosines likhne ka ek compact tarika). Rule of thumb: exponent mein i i
k k = ℏ 2 m E k fast wiggle (short wavelength) matlab bada k E
Definition Decay constantκ κ = ℏ 2 m ( V 0 − E ) κ rate hai jis se violet curve wall ke andar fade hoti hai . Bada κ
κ m ( V 0 − E ) κ bhaari ho (m oonchi ho (V 0 dheema/less energetic ho (E κ
Note karo ki k κ k E paas hai), κ V 0 − E chahiye thi par nahi hai ). Ek waves banata hai, doosra decay banata hai.
m κ
Definition Reduced Planck constantℏ ℏ ≈ 1.055 × 1 0 − 34 k κ
L x = 0 x = L
§1 ki figure mein teen arrows appear hote hain: ek incoming, ek bounce back karta hua, ek guzarta hua.
A B F A B F heights hain. Bada letter ⇒ us outcome ki zyada chance.
Definition Transmission coefficientT T = ∣ A ∣ 2 ∣ F ∣ 2 T fraction hai jo particles aar-paar nikalta hai — 0 aur 1 ke beech ek probability. Yeh hai "kitna guzara" divided by "kitna aaya tha", dono squared kyunki probability ∣ ψ ∣ 2 T = 0.024
e − 2 κ L Amplitude F e − κ L T squared use karta hai, aur e − κ L ( e − κ L ) 2 = e − 2 κ L
sinh sinh ( y ) = 2 1 ( e y − e − y ) do decay/growth exponentials ka combination hai, aur yeh exact formula mein isliye appear hota hai kyunki barrier ke andar wave actually ek fading part e − κ x e + κ x y sinh ( y ) ≈ 2 1 e y e − 2 κ L
ψ ψ ′ mile bina sudden jump ke (ψ aur bina sudden kink ke (ψ ′ ψ ′ A , B , F T Wavefunction and Boundary Conditions mein hai.
Topic ko iske zaroorat kyun hai: ek wave jisme break ya kink ho usse infinite energy chahiye hogi — nature ise forbid karta hai. Dono faces pe smoothness force karna exactly wahi hai jo far side pe amplitude leak karta hai.
Position x and the barrier picture
Energy E vs barrier height V0
Wavefunction psi = matter wave
Sign of gap decides shape
Transmission T = F squared over A squared
Smooth-join of psi and slope
Har upstream cheez transmission coefficient T mein pour hoti hai — woh ek number jo parent topic chase kar raha hai.
Reveal karne se pehle answer karne ki koshish karo. Agar kar sako, tum parent derivation ke liye ready ho.
x x x = 0 x = L
Kaun sa case tunnelling allow karta hai, E > V 0 E < V 0 E < V 0
ψ ∣ ψ ∣ 2 ψ ∣ ψ ∣ 2
Wave ko kya alag batata hai: wiggle vs. decay? Exponent mein i e ik x i e − κ x
k k = 2 m E /ℏ ; badi energy
E ⇒ tighter ripples.
κ κ = 2 m ( V 0 − E ) /ℏ ; bada mass
m , badi height
V 0 , chhoti energy
E .
e − 2 κ L Amplitude e − κ L T
T Transmitted fraction, T = ∣ F ∣ 2 /∣ A ∣ 2
Har wall face pe kya do cheezein match karni chahiye? Wave ψ ψ ′
Exact formula mein sinh Barrier ke andar wave e − κ x e + κ x sinh
Recall Har idea aage kahan develop hoti hai
Full wave-matching algebra → Wavefunction and Boundary Conditions
Matter waves exist kyun karte hain → de Broglie Wavelength
Woh equation jo shapes force karti hai → Schrödinger Equation
Step pe bouncing vs. crossing → Potential Barrier and Reflection
Real-world payoffs → Alpha Decay , Scanning Tunneling Microscope