Quantum Tunneling

Using probability flux $j$

j_{A} = \frac{ℏ k}{m} A A^{*}

Negative $ℏ k$ term for B. Applications of Quantum Tunneling

Scanning tunneling microscopy
Field emission scanning electron microscopy
Alpha decay
Dielectric breakdown
Ivar Giaever (Nobel Prize winner for quantum tunneling) Also recall photoelectric effect: Note figure on right utilizing quantum tunneling

In class assignment 1: From boundary conditions at x=0 and L, determine the four algebraic equations that relate A,B,C,D,E,F and G

A e^{ik x} + B e^{- ik x} = G e^{κ x} + F e^{- κ x}

At x=0

A + B = F + G

ik (A - B) = κ (- F + G)

At x=L

G e^{κ x} + F e^{- κ x} = C e^{ik x}

- κ F e^{- κ L} + κ G e^{κ L} = ik C e^{ik x}

Now considering a finite barrier as a scattering problem with a wave coming from the left.

@ x < 0 : A e^{ik x} + B e^{- ik x}

@0 < x < L : F e^{κ x} + G e^{- κ x}

@ x > L : C e^{ik x}

In class Assignment 2:

(\frac{4 kκ}{k ^{2} + κ ^{2}})^{2} = 16 \frac{E}{V _{0}} (1 - \frac{E}{V _{0}})

Consider a barrier with an arbitrary shape: With transmission coeff.

T \approx C e^{- 2 \sum κ_{i} d x} \approx C e^{- 2 \int_{a}^{b} d x 2 m (V_{0} - E) /ℏ}

There are many situations where we need to consider a barrier that is not rectangular: Recall: Electric field, $E_{0}$ w/t $V = q E_{0} x$ Now,

E = W + E - e E_{0} x_{0}

0 = W - e E_{0} x_{0}

x_{0} = \frac{w}{e E _{0}}

∴ T \approx C e^{- 2 \int_{0}^{x_{0}} d x 2 m /ℏ (W - e E_{0} x)}

This is known as the Fowler-Nordheim Relationship

Paul's Physics Notes