Barrier Tunneling and Kinetic Energy

[nickm4]

1. Homework Statement

b). Find the kinetic energy K (sub t), the proton will have on the other side of the barrier if it tunnels through the barrier.

c) Find the kinetic energy K (sub r), it will have if it reflects from the barrier.

Variables:

Transmission Coefficient (T)

T= e^-2bL
T was found to be T= e^-11.617 or (9.011*10^-6)
e= 2.718...
L= length of the barrier which is given as 10fm or (10.0*10^-15m)

b= sqrt(((8pie^2)(m)(U(sub b)-E))/(h^2))

m= mass of proton(1.673*10^-27kg)
Ub= height of the potential barrier(given= 10MeV)
E= energy of the proton (given= 3MeV)
h= plank's constant (6.62*10^-34)

2.

Homework Equations

T= e^-2bL
b= sqrt(((8pie^2)(m)(U(sub b)-E))/(h^2))

The Attempt at a Solution

I solved the first part of the question to find the transmission coefficient, T. But I'm not sure how Kinetic energy is related. Other than through b.

This question is taken from " Fundementals of Physics" Halliday/Resnick 7th ED. Question: 38-63

Thanks Tons.

 

[Nate810]

Kinetic energy is the energy that an object has due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its current velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body in decelerating from its current speed to a state of rest.For a proton, the kinetic energy K (sub t) it will have on the other side of the barrier if it tunnels through the barrier can be calculated using the following equation:K (sub t) = (1/2)mv^2where m is the mass of the proton and v is the velocity of the proton.Similarly, the kinetic energy K (sub r) it will have if it reflects from the barrier can be calculated using the same equation.

 

[Moetasim]

I would approach this problem by first understanding the concept of barrier tunneling and how it relates to kinetic energy. Barrier tunneling refers to the phenomenon where a particle, in this case a proton, can pass through a potential barrier even if its energy is lower than the barrier's height. This is possible due to the wave-like nature of particles at the quantum level.

To find the kinetic energy of the proton on the other side of the barrier, we can use the equation K(sub t) = E - U(sub b), where E is the initial energy of the proton and U(sub b) is the height of the potential barrier. In this case, E = 3MeV and U(sub b) = 10MeV. Therefore, K(sub t) = 3MeV - 10MeV = -7MeV. This means that the proton will have a negative kinetic energy on the other side of the barrier, which is not physically possible. This is because the proton has tunneled through the barrier and its energy has decreased.

On the other hand, if the proton reflects from the barrier, its kinetic energy can be found by using the equation K(sub r) = E - U(sub b), where E is the initial energy of the proton and U(sub b) is the height of the potential barrier. In this case, since the proton is reflecting, its energy remains the same, so K(sub r) = 3MeV. This means that the proton will have the same kinetic energy as its initial energy if it reflects from the barrier.

The transmission coefficient, T, is related to the probability of the proton tunneling through the barrier. The higher the transmission coefficient, the higher the probability of tunneling. We can use the equation T = e^-2bL to calculate the transmission coefficient, where b is the barrier width and L is the length of the barrier. In this case, b can be found using the equation b = sqrt(((8pie^2)(m)(U(sub b)-E))/(h^2)), where m is the mass of the proton, U(sub b) is the height of the potential barrier, and h is Planck's constant. Once we have the value of T, we can use it to calculate the probability of the proton tunneling through the barrier. This can be done by multiplying T by 100 to get a percentage probability.

In summary, barrier tunneling and kinetic