Probability of tunneling through square barrier

Tunneling through square barrier

An electron accelerated through a potential difference of $E=1.5 eV$ approaches a potential barrier having a height of $U = 3 eV$ and width of $d =2 nm$. Write down the continuity equations for the wave function in the regions 1, 2 and 3 and find the probability of tunneling (transmission probability).

Swow My Homework-Square Barrier 

The continuity conditions at interface 1 are

$psi_1(0)=psi_2(0)$   and $frac{dpsi_1(0)}{d x}=frac {dpsi_2(0)} {d x}$
At interface 2 the continuity conditions are

$psi_2(d)=psi_3(d)$   and $frac{dpsi_2(d)}{d x}=frac {dpsi_3(d)} {d x}$

The tunneling probability is just (from continuity conditions):
$P=|psi_3(d)/psi_1(0)|^2=|(psi_2(d))/psi_2(0))|^2=exp{(-2*k_2*d)}$

Since the wave function is decreasing exponential in the barrier. K2 is the wave number in the barrier region 2.

$k_2 = 2pi/lambda_2 =(2pi p_2)/h=sqrt{(2m(U-E))}/hbar$

U is the barrier height, E is the total energy.

$P = exp{{-2[sqrt{2m(U-E)}/hbar]*d}}$

For U =3 eV, E =1.5 eV and d =2 nm for an electron we have $P=1.31*10^{-11}$