Quantum tunneling — concept, transmission coefficient

2.3.11 · Physics › Modern Physics

Tunneling KYA hai?

YEH KYUN hota hai: Ek particle ko ek wavefunction $ψ (x)$ ke zariye describe kiya jaata hai. Schrödinger's equation $ψ$ aur uski slope $ψ^{'}$ ko har jagah continuous rehne ke liye force karti hai. Ek barrier ke andar jahan $E < V_{0}$ ho, $ψ$ oscillate nahi kar sakti — lekin woh abruptly vanish bhi nahi kar sakti. Baaki bachat hai sirf exponential decay. Decay ≠ zero, toh kuch amplitude doosri taraf pahunch jaati hai.

Transmission coefficient KAISE derive karte hain

Setup: $E$ energy ka ek particle height $V_{0}$ aur width $L$ ke rectangular barrier se takraata hai ( $0 < x < L$ ), jahan $E < V_{0}$ .

Step 1 — Har region mein Schrödinger's equation likho

Time-independent Schrödinger equation: $- \frac{ℏ ^{2}}{2 m} \frac{d ^{2} ψ}{d x ^{2}} + V (x) ψ = E ψ$

Kyun? Yeh energy eigenvalue equation hai; isko solve karne se matter wave ki spatial shape milti hai.

Region I ( $x < 0$ , $V = 0$ ): $ψ^{''} = - \frac{2 m E}{ℏ ^{2}} ψ \equiv - k^{2} ψ, k = \frac{2 m E}{ℏ}$ Solution: $ψ_{I} = A e^{ik x} + B e^{- ik x}$ (incoming + reflected waves).

Region II ( $0 < x < L$ , $V = V_{0}$ ): $ψ^{''} = + \frac{2 m ( V _{0} - E )}{ℏ ^{2}} ψ \equiv + κ^{2} ψ, κ = \frac{2 m ( V _{0} - E )}{ℏ}$ +sign kyun? Kyunki $V_{0} > E$ hai, bracket positive hai, toh equation real exponentials deti hai, oscillations nahi: $ψ_{I I} = C e^{- κ x} + D e^{+ κ x}$

Region III ( $x > L$ , $V = 0$ ): $ψ_{I I I} = F e^{ik x}$ Sirf $e^{+ ik x}$ kyun? $+ \infty$ se kuch wapas nahi aata; sirf ek transmitted wave rightward move karti hai.

Step 2 — Transmission define karo

Transmission coefficient = (transmitted probability flux)/(incident flux). Kyunki $k$ dono taraf same hai: $T = \frac{∣ F ∣ ^{2}}{∣ A ∣ ^{2}}$

Step 3 — Boundary conditions lagao

$x = 0$ aur $x = L$ par $ψ$ aur $ψ^{'}$ ki continuity se char equations milti hain. Solve karne par (algebra) exact result milta hai:

Step 4 — Thick/high-barrier limit jo useful hai

Jab $κ L ≫ 1$ , $sinh (κ L) \approx \frac{1}{2} e^{κ L}$ , toh $sinh^{2} \approx \frac{1}{4} e^{2 κ L}$ , jo "1" ko dominate karta hai: $T \approx \frac{16 E ( V _{0} - E )}{V _{0}^{2}} e^{- 2 κ L}$

Figure — Quantum tunneling — concept, transmission coefficient

Worked Examples

Common Mistakes (Steel-manned)

Recall Feynman: 12-saal ke bachche ko samjhao

Socho tum ek tennis ball wall pe maar rahe ho. Normally woh bounce karke wapas aati hai. Lekin ab ball ko ek fluffy cloud ki tarah imagine karo, na ki hard dot. Jab cloud wall se takraata hai, uska zyaadatar hissa bounce karta hai, lekin thoda sa faint fog wall ke through seep karta hai aur doosri taraf faintly dikhta hai. Agar wall patli ho, thoda fog paar nikal jaata hai — aur kabhi kabhi ball wahan paai jaati hai! Patli walls aur halki balls = zyada leaking. Wahi seepage tunneling hai.

Recall Active-recall checklist

$ψ$ barrier ke andar exponential (oscillatory nahi) kyun hai?
$κ$ kya hai aur isko kya control karta hai?
$e^{- 2 κ L}$ mein "2" ka factor kahan se aata hai?
Bahar nikalne wala particle zyada energy ke saath aata hai? Kam? Ya same?

Flashcards

Quantum tunneling kya hai?

Ek particle ka

V_{0} > E

wale barrier se guzarna, classically impossible, allowed isliye kyunki

ψ

forbidden region ke andar continuous aur nonzero (exponentially decaying) hoti hai.

ψ

barrier ke andar exponential kyun hoti hai?

Kyunki

V_{0} > E

Schrödinger's equation ko

ψ^{''} = + κ^{2} ψ

banata hai, jiske real solutions

e^{\pm κ x}

hain, oscillations nahi.

Decay constant

κ

define karo.

κ = 2 m (V_{0} - E) /ℏ

, woh rate jis par

ψ

barrier ke andar decay karti hai.

Approximate transmission coefficient?

T \approx G e^{- 2 κ L}

jahan

G = 16 E (V_{0} - E) / V_{0}^{2}

Exact transmission coefficient?

T = [1 + \frac{V _{0}^{2} s i n h ^{2} ( κ L )}{4 E ( V _{0} - E )}]^{- 1}

e^{- 2 κ L}

mein "2" ka factor kyun?

Amplitude

e^{- κ L}

ki tarah decay karta hai; probability

\propto ∣ ψ ∣^{2}

ise square karke

e^{- 2 κ L}

banata hai.

Kya tunneling particle energy gain karta hai?

Nahi — woh same energy

E < V_{0}

ke saath bahar aata hai; energy conserve hoti hai.

T

mass par kaise depend karta hai?

κ \propto m

ke through; bhaare particles exponentially kam tunnel karte hain, isliye macroscopic objects effectively kabhi tunnel nahi karte.

Tunneling ka real application?

Scanning Tunneling Microscope (STM), alpha decay, tunnel diodes, flash memory.

Connections

Schrödinger Equation — decaying solution ka source
Wavefunction and Boundary Conditions — continuity use ki gayi BCs deta hai
Alpha Decay — nuclei se $α$ particles ka Gamow's tunneling
Scanning Tunneling Microscope — exponential $T (L)$ → atomic resolution
Potential Barrier and Reflection — $R = 1 - T$
de Broglie Wavelength — $k = 2 m E /ℏ$ wave number ko energy se link karta hai