Quantum Mechanical Tunneling

Tunnelling through step barier

$V (x) = V_{0}$ for $0 < x < a$ and $V (x) = 0$ for all other $x$ .

For regions I, III, the solutions will be oscillatory exponentials. At the boundary the function and the derivative are continuous.

$κ = \sqrt{\frac{2 m E}{ℏ^{2}}}$ , $Ψ_{I, I I I} = A_{I, I I I} \exp (i κ x) + B_{I, I I I} \exp (- i κ x)$ .

$A$ is the incoming wave, from $-$ to $+$ . $B$ is going the opposite direction.

Initial Condition

Incoming from $- \infty$ and no reflective wall at $+ \infty$ . Then $B_{I I I} = 0$ .

Energy below Barrier

In region II it will be a decaying exponential. $k = \sqrt{2 m (V_{0} - E)} / ℏ$ .

Energy above Barrier

In region II it will be a oscillatory exponential. $k = \sqrt{2 m (E - V_{0})} / ℏ$ .

Probability Flux

Region I. $j = \frac{ℏ}{m} ℑ [Ψ^{*} \frac{d}{d x} Ψ] = \frac{ℏ}{m} κ (| A |^{2} - | B |^{2})$ . $j_{i n c} = \frac{ℏ κ}{m} | A |^{2}$ is the incoming wave velocity. $j_{r e f} = \frac{ℏ κ}{m} | B |^{2}$ is the reflected wave velocity.

Region III. $j = \frac{ℏ κ}{m} | C |^{2}$ is the transmitted wave velocity.

Reflection Coefficient. $R = \frac{j_{r e f}}{j_{i n c}} = \frac{| B |^{2}}{| A |^{2}}$ .

Transmission Coefficient. $T = \frac{j_{t r a n s}}{j_{i n c}} = \frac{| C |^{2}}{| A |^{2}}$ .

$R + T = 1$ for a non-absorbing wall.

Classical Expectations

If $E < V_{0}$ then we expect $R = 1$ and $T = 0$ . If $E > V_{0}$ then we expect $T = 1$ and $R = 0$ .

For $E < V_{0}$ , $T = {[1 + \frac{V_{0}^{2} \sinh^{2} k a}{4 E (V_{0} - E)}]}^{- 1}$ .

For $E > V_{0}$ , $T = {[1 + \frac{V_{0}^{2} \sin^{2} k a}{4 E (E - V_{0})}]}^{- 1}$ .

Parity Operator

Recall

Probability Conservation and Probability Flux