Quantum Mech (Barriers and Tunneling)

[zbhest123]

1. Two nanowires are separated by 1.3 nm as measured by STM. Inside the wires the potential energy is zero, but between the wires the potential energy is greater than the electron's energy by only 0.9 eV. Estimate the probability that the electron passes from one wire to the other.



2. k=sqrt(2m(V_0-E))/hbar
T=(1+V^2sinh^2(kL)/(4E(V_0-E)))^-1
T=16E/V_0(1-E/V_0)e^-2kL When kL>>1 (kL=6.31)
V_0-E=0.9 eV




3. The problem here is that I have 2 unknowns for both T equations, and I don't know if it would make sense to set them equal to each other and then solve for one of the unknowns. For either equation on its own the unknown V_0 and E do not cancel themselves out. Do I possibly need to use the equation K_min=(P)^2/(2m)=V_0-E? Or could you point me towards something I have overlooked? Would I be able to use any simple harmonic oscillator equations? Such as E_0=1/2*hbar*omega where omega^2=k/m? And then assume E_0=E? But I feel like this wouldn't make sense..

 

[NateD]

The probability of an electron passing from one wire to the other can be estimated using the equation for transmission probability, which is given by:T = (1 + (V^2 * sinh^2(kL))/(4E(V_0 - E)))^-1where V is the potential difference between the two wires, k is the wave vector of the electron, L is the separation between the two wires, and E is the energy of the electron. We can rearrange this equation to solve for k:k = sqrt(2m(V_0 - E))/hbar We can then substitute this into the transmission probability equation and solve for T:T = 16E/V_0(1 - E/V_0)e^-2kL Given the values for V_0 and E, we can calculate the transmission probability T, which will give us an estimate of the probability that an electron will pass from one wire to the other.