The barrier length in quantum tunneling

[Alan Ezra]

Hi,

In transmission coefficient T= exp(-2sqrt(2m(U-E)/hbar^2)L), L, as I interpret it, is the distance of the potential barrier. I am wondering if I have N particles all with kinetic energy E, approaching the barrier, can I integrate the transmission coefficient over a distance from infinity to the potential barrier(the charged radius?), and times the number of particles N, to find out how many of them can penetrate the barrier?

Thank you so much for helping me

best regards
alan

 

[BvU]

Hello Alan,

I checked here and found a different expression. Turns out your $T$ is a simplification for sqrt(2m(U-E)/hbar^2)L >> 1 (and you left out a factor 4E(U-E0)/U² ).

Point is that this is already the probability for transmission, so no need to integrate.

 

[jtbell]

Point is that this is already the probability for transmission

Actually $|T|^2$ is the probability for a single particle to make it through the barrier. But as you say, no need to integrate, provided all the particles have the same energy, and therefore the same $T$.

 

[BvU]

T is already a square. Not to be squared again.

 

[Alan Ezra]

Hello Alan,

I checked here and found a different expression. Turns out your $T$ is a simplification for sqrt(2m(U-E)/hbar^2)L >> 1 (and you left out a factor 4E(U-E0)/U² ).

Point is that this is already the probability for transmission, so no need to integrate.

Hi BvU,

Thanks for the help! So should I use the expression

and multiply it by the number of particles to find out the number of particles tunneled through? And so the expression I gave, the simplified one took a approximation for sinh^2?? Is that right? Thanks.

 

[jtbell]

T is already a square.

Right you are. I got confused between t and T=|t|².