# Quantum Mechanical Tunneling

## Tunnelling through step barier

$$V(x) = V_0$$ for $$0\lt x\lt a$$ 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.

$$\kappa = \sqrt{\frac{2mE}{\hbar^2}}$$, $$\Psi_{I,III} = A_{I,III}\exp(i\kappa x) + B_{I,III}\exp(-i\kappa x)$$.

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

### Initial Condition

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

### Energy below Barrier

In region II it will be a decaying exponential. $$k=\sqrt{2m(V_0-E)}/\hbar$$.

### Energy above Barrier

In region II it will be a oscillatory exponential. $$k=\sqrt{2m(E-V_0)}/\hbar$$.

### Probability Flux

Region I. $$j=\frac{\hbar}{m}\Im\left[\Psi^*\frac{d}{dx}\Psi\right] = \frac{\hbar}{m}\kappa\left(|A|^2-|B|^2\right)$$. $$j_{inc}=\frac{\hbar\kappa}{m}|A|^2$$ is the incoming wave velocity. $$j_{ref}=\frac{\hbar\kappa}{m}|B|^2$$ is the reflected wave velocity.

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

Reflection Coefficient. $$R=\frac{j_{ref}}{j_{inc}} = \frac{|B|^2}{|A|^2}$$.

Transmission Coefficient. $$T=\frac{j_{trans}}{j_{inc}} = \frac{|C|^2}{|A|^2}$$.

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

### Classical Expectations

If $$E\lt V_0$$ then we expect $$R=1$$ and $$T=0$$. If $$E\gt V_0$$ then we expect $$T=1$$ and $$R=0$$.

For $$E\lt V_0$$, $$T=\left[1+\frac{V_0^2\sinh^2 ka}{4E(V_0-E)}\right]^{-1}$$.

For $$E\gt V_0$$, $$T=\left[1+\frac{V_0^2\sin^2 ka}{4E(E-V_0)}\right]^{-1}$$.

### Parity Operator

#### Recall

Created: 2023-06-25 Sun 02:33

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