Potential barrier

1. What a Potential Barrier Is

Consider 1D potential step/barrier:

V(x)=0 for x<0 (region I), V=V_0>0 for 0<x<a (region II, barrier width a), V=0 for x>a (region III)

Particle energy $E$ incident from left. Classically: if $E>V_0$ , transmit over; if $E<V_0$ , reflect, cannot enter barrier. Quantum: wave penetrates, even $E<V_0$ has non-zero transmission (tunneling, next topic); $E>V_0$ has partial reflection even above barrier due to wave mismatch.

2. TISE in Three Regions

Let

k_1 =√(2mE)/ħ

in I/III (free),

k_2 =√(2m(E-V_0))/ħ

if $E>V_0$ (propagating inside), or

κ =√(2m(V_0-E))/ħ

imaginary $k_2=iκ$ if $E<V_0$ (evanescent, decaying).

General solutions:

φ_I = A e^{ik_1 x} + B e^{-ik_1 x}

(incident + reflected)

φ_{II}=C e^{ik_2 x}+D e^{-ik_2 x}

if $E>V_0$ , or

C e^{κx}+D e^{-κx}

decaying/growing if $E<V_0$ (physically keep decaying).

III:

φ_{III}=F e^{ik_1 x}

(transmitted only, no incoming from +∞)

Apply continuity $φ$ and $dφ/dx$ at $x=0$ and $x=a$ (where $V$ jumps finite) ⇒ solve for $B/A$ (reflection amplitude $r$ ) and $F/A$ ( $t$ ) ⇒ reflection

R=|r|²

, transmission

T=|t|²

with $R+T=1$ .

3. Results

$E>V_0$ :

T =1/[1+ (V_0² sin²(k_2 a))/(4E(E-V_0))]

— oscillates with $a$ ; $T=1$ at resonance when $k_2 a=nπ$ (barrier transparent like Fabry-Perot). Even with enough energy, some reflection — unlike classical.

$E<V_0$ : replace $k_2→iκ$ ⇒

T =1/[1+ (V_0² sinh²(κ a))/(4E(V_0-E))]

⇒

T≈ (16E(V_0-E)/V_0²) e^{-2κa}

for thick barrier $κa≫1$ — exponential decay, not zero (next topic tunneling).

4. Worked Checks

(a) High barrier V0=2E, a=1 nm, electron E=1 eV: κ=√(2m(V0-E))/ħ=√(2·9.1e-31·1.6e-19)/1.05e-34≈5.1e9 m⁻¹ ⇒ κa≈5.1 ⇒ T≈ e^{-10.2}·factor≈3e-5 small but not zero ✓

(b) At resonance k2 a=π ⇒ T=1 even though V0≠0 ⇒ resonant tunneling.

(c) Classical would predict T=0 for E<V0, but quantum gives ~e^{-2κa}.

5. Exam Points

Sketch $V(x)$ step/barrier — 3 marks, frequent.
Write TISE solutions in three regions with $k_1,k_2,κ$ — 5 marks, very frequent.
State continuity of $φ, dφ/dx$ at boundaries — 3 marks.
Give $T$ formula for $E>V_0$ and mention resonance — 5 marks, frequent.
Mention $T≠0$ for $E<V_0$ preview — 3 marks.

Potential barrier