Suppose a beam of 5.0 eV protons strikes a potential energy barrier of height 5.5 eV and thickness 0.57 nm, at a rate equivalent to a current of 825 A.

(a) How long would you have to wait-on average-for one proton to be transmitted?

(b) How long would you have to wait if the beam consisted of electrons rather than protons?

This is a quantum mechanical "barrier penetration" problem.

See

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/barr.html

or any of many other websites about this phenomenon. There you will find the formula you need.

Convert the current to a particle arrival rate. Multiply that by the probability of penetration to get the penetration rate. The reciprocal of that is the time you would have to wait.

To calculate the average waiting time for one particle to be transmitted, we need to determine the transmission probability of the particle passing through the potential energy barrier. The transmission probability is given by the formula:

T = e^(-2kL)

Where:
T is the transmission probability,
k is the wavevector, and
L is the thickness of the barrier.

To find the wavevector, we can use the equation:

k = sqrt(2m(E - V))/ħ

Where:
m is the mass of the particle,
E is the energy of the particle,
V is the height of the barrier, and
ħ is the reduced Planck's constant.

For a proton:
m = 1.67 × 10^(-27) kg
E = 5.0 eV = 5.0 × 1.6 × 10^(-19) J
V = 5.5 eV = 5.5 × 1.6 × 10^(-19) J
ħ = 6.626 × 10^(-34) J·s

(a) For protons:
Let's calculate the wavevector:

k = sqrt(2 * 1.67 × 10^(-27) kg * (5.0 × 1.6 × 10^(-19) J - 5.5 × 1.6 × 10^(-19) J)) / (6.626 × 10^(-34) J·s)

Once we have the wavevector, we can plug it into the transmission probability formula:

T = e^(-2 * k * 0.57 × 10^(-9) m)

The average waiting time for one proton to be transmitted is the inverse of the transmission rate:

t = 1 / (T * 825 A)

(b) For electrons:
The only difference for electrons is their mass:

m = 9.11 × 10^(-31) kg

We can repeat the same steps as above to find the transmission probability and the average waiting time.

Please note that this calculation assumes the potential energy barrier is a square barrier and neglects other factors such as interactions and temperature effects.