--- title: "Tunneling: wave function in different regions" subject: "Electrical Engineering Material" chapter: 1 key: "semester_3/electrical_engineering_material:tunneling-wave-function-regions" source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/tunneling-wave-function-regions site: BCT Engineering --- # Tunneling: wave function in different regions Electrical Engineering Material · Chapter 1: Theory of Metals - Asked in: 2081 Chaitra (new course), 5 marks; 2080 Bhadra, 8 marks; 2076 Asoj, 8 marks; 2069 Chaitra, 8 marks; 2068 Chaitra, 8 marks (asked 5 times) - Video: [Quantum Tunneling |Tunnel effect | Quantum Physics | BTech | BSc | GATE | UPSC | Physics | IGNOU](https://www.youtube.com/watch?v=rdNfjN9H-14) — Gautam Varde - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/tunneling-wave-function-regions - Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-1 ## Question What is tunneling in quantum mechanics? With necessary mathematical expression, explain the nature of wave function in different regions in case of tunneling. ## Short answer Tunneling occurs when a particle passes through a potential barrier higher than its energy. In regions I and III (V=0), wavefunctions are oscillatory (e^(±jkx)); inside region II (barrier, V>E) they decay exponentially as e^(-αx). Transmission probability T ∝ e^(-2αa). ## Answer **Quantum tunneling** is the passing of a particle through a potential barrier whose height $V_0$ is greater than the particle's energy $E$. In classical mechanics this is impossible, but in quantum mechanics the wave function does not become zero inside the barrier. It decays, so a small part of it comes out on the other side and there is a finite chance of finding the particle there. #### Barrier and regions ```text V(x) ^ | +----------+ V = V0 | | | | - - - - - |- - - - - |- - - - - E < V0 | Region I | Region II| Region III | V = 0 | V = V0 | V = 0 --+------------+----------+-----------> x 0 a ``` The time-independent Schrödinger equation is $$ \frac{d^2\psi}{dx^2} + \frac{2m}{\hbar^2}(E - V)\psi = 0 $$ #### Region I ($x < 0$, $V = 0$) $$ \psi_1 = A e^{jkx} + B e^{-jkx}, \qquad k = \frac{\sqrt{2mE}}{\hbar} $$ $A e^{jkx}$ is the incident wave and $B e^{-jkx}$ the reflected wave. The wave function is **oscillatory**. #### Region II ($0 \le x \le a$, $V = V_0 > E$) $$ \frac{d^2\psi_2}{dx^2} - \alpha^2\psi_2 = 0, \qquad \alpha = \frac{\sqrt{2m(V_0 - E)}}{\hbar} $$ $$ \psi_2 = C e^{\alpha x} + D e^{-\alpha x} $$ The solution is **exponential, not oscillatory**. For a wide barrier the decaying term dominates, so $\psi$ falls off roughly as $e^{-\alpha x}$ inside the barrier. #### Region III ($x > a$, $V = 0$) $$ \psi_3 = F e^{jkx} $$ Only a transmitted wave travels to the right (there is nothing to reflect it back). It is oscillatory with the **same $k$** (same energy) but a **much smaller amplitude**. #### Boundary conditions $\psi$ and $d\psi/dx$ must be continuous at $x = 0$ and $x = a$. These four conditions fix $B, C, D, F$ in terms of $A$. #### Transmission probability $$ T = \frac{|F|^2}{|A|^2} \approx T_0\, e^{-2\alpha a}, \qquad T_0 = 16\frac{E}{V_0}\left(1 - \frac{E}{V_0}\right) $$ (valid when $\alpha a \gg 1$). $T$ falls very quickly as the barrier width $a$, the mass $m$ or $(V_0 - E)$ increases. ```text psi(x) Region I Region II Region III /\ /\ /\ |\ | / \ / \ / \| \ | / \/ \/ | \__ |/\/\/\/\/\/\ | ``--..__| (small amp.) 0 a ``` #### Examples Alpha decay of nuclei, the tunnel diode, the scanning tunneling microscope (STM), and conduction through thin oxide layers between metal contacts all rely on tunneling. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is quantum tunneling? A. A particle gains extra energy to climb over the barrier B. A particle passes through a barrier higher than its energy because its wave function decays but is not zero inside C. A particle is totally reflected by the barrier D. A particle splits into two parts Answer: B. The wave function does not vanish in the barrier, so a small part emerges and there is a finite chance of finding the particle beyond it. (Key point: Definition of tunneling) 2. What is the nature of the wave function in Region II (E < V0)? A. Oscillatory with the same k B. Zero everywhere C. Exponential (decaying), not oscillatory D. Constant Answer: C. With V0 > E the equation gives psi = C e^(alpha x) + D e^(-alpha x), exponential rather than oscillatory. (Key point: Wave function in barrier) 3. How does the wave function in Region III (x > a) compare with that in Region I? A. Same k but much smaller amplitude, only a transmitted wave B. Larger k and larger amplitude C. Exponentially decaying D. It contains both incident and reflected waves Answer: A. Beyond the barrier only a transmitted wave F e^(jkx) exists, with the same energy (same k) but small amplitude. (Key point: Region III wave) 4. How does the transmission probability T behave as the barrier width a increases? A. It increases linearly B. It is unchanged C. It increases exponentially D. It falls exponentially as e^(-2 alpha a) Answer: D. T is approximately T0 e^(-2 alpha a), so it falls very quickly with width, mass or (V0 - E). 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