Barrier Tunneling: Proton Transmission Coefficient

[Kazza_765]

I can't work out what I am doing wrong in this question. Anyways, here it is:
A 3.0 MeV proton is incident on a potential energy barrier of thickness 10fm and height 10 MeV. What is the transmission coefficient T.
And here is what I have tried:
$$T=e^(-2bL)$$
$$b=\sqrt{\frac{8\pi^2m(U_b-E)}{h^2}}$$
$$b=\sqrt{\frac{8\pi^2(1.673*10^-27)\frac{10*10^6 - 3*10^6}{1.602*10^-19}}{(6.626*10^-34)^2}}$$
$$b=3.626*10^33$$
This gives me a transmission coeffecient of approximately 0, and the answer given in the book is 9.02*10^-6. Your help on this is much appreciated.

 

[Kazza_765]

Never mind, as soon as I posted it and saw my equation in front of me I worked it out. Silly mistake.

 

[quicknote]

Hello,

Thank you for your question. First, let's clarify what we mean by barrier tunneling. This is the phenomenon in which a particle, in this case a proton, can pass through a potential energy barrier that it does not have enough energy to overcome. This happens due to the wave-like nature of particles, which allows them to "tunnel" through the barrier.

Now, let's look at your calculation for the transmission coefficient T. The formula you have used, T = e^(-2bL), is correct. However, there is a mistake in your calculation of b. The correct formula for b is:

b = sqrt(2m(U_b - E) / h^2)

where m is the mass of the proton, U_b is the height of the barrier, E is the energy of the proton, and h is the Planck's constant. Using the values you have provided, we get:

b = sqrt(2 * 1.673 * 10^-27 kg * (10 * 10^6 eV - 3 * 10^6 eV) / (6.626 * 10^-34 J s)^2)

b = 1.17 * 10^23 m^-1

Now, plugging this value of b into the formula for T, we get:

T = e^(-2 * 1.17 * 10^23 m^-1 * 10 fm)

T = 9.02 * 10^-6

This matches the answer given in the book. So, the mistake in your calculation was in the value of b. I hope this helps clarify your doubts. Keep up the good work!