An electron with kinetic energy E=5.00 eV is incident on a...

An electron with kinetic energy $E=5.00$ eV is incident on a barrier of width $L=0.200 \mathrm{nm}$ and height $U=10.0 \mathrm{eV}$ (Fig. P41.30). What is the probability that the electron (a) tunnels through the barrier? (b) Is reflected?

Key Concepts

Quantum Tunneling

This is the quantum mechanical phenomenon where a particle has a finite probability of crossing through a potential barrier even if its energy is lower than the barrier height. This defies classical expectations and is a direct consequence of the wave-like properties of particles as described by quantum mechanics.

Transmission and Reflection Coefficients

These coefficients represent the probabilities that a particle will either tunnel through (transmission coefficient) or be reflected (reflection coefficient) when it encounters a potential barrier. They are calculated using the continuity conditions of the wavefunction and its derivative at the boundaries of the barrier, reflecting the conservation of probability.

Schrödinger Equation for Potential Barriers

The Schrödinger equation provides the fundamental framework for analyzing the behavior of quantum particles. In the context of potential barriers, it is used to derive the wavefunctions in regions before, within, and after the barrier, which in turn allows for the computation of the transmission and reflection probabilities.

Exponential Decay inside the Barrier

Within a potential barrier where the particle’s kinetic energy is less than the potential energy, the wavefunction exhibits an exponential decay. This decay underpins the tunneling phenomenon by indicating that the probability amplitude decreases as the particle penetrates further into the barrier, yet still remains nonzero, allowing for a finite chance of exit on the other side.

Transcript

00:02 We'll start by getting the transmission coefficient.

00:06 Before we do that, we need a certain quantity that is needed for it.

00:11 So let's start with...

00:33 So notice we have h bar.

00:35 H bar is simply planck's constant divided by 2 pi.

00:39 Also, we have the energy and the potential in electron volts.

00:45 So we can actually keep them in electron bolts, as long as we have the proper conversion factor in our calculation.

00:52 So this quantity c is something that we'll need to calculate first before we get the transmission coefficient.

00:59 And we're going to go ahead and write down the mass for the electron.

01:03 So that is 9 .11 times 10 to the minus 31 kilograms.

01:12 So let's go ahead and substitute everything that we know for this problem.

01:19 So c is going to be equal to 2 times 9 .11 times 10 of the 1 .1.

01:26 Minus 31 kilograms.

01:32 Now we're going to have, normally i leave off units, but because we're working in electron volts, i'm going to leave them in this particular case for the energies.

01:43 So we have 10 .0 electron volts minus 5 .0 electron volts, 5 .00 electron volts.

01:53 We have three significant figures in this problem.

01:56 And we're going to have a conversion factor because we will need to convert to juilli.

02:01 Within the calculation.

02:04 So we have 1 .60 times 10 to the minus 19 joules per electron volt, because that is the conversion.

02:17 And this whole string here of quantities that are multiplying go under the square root.

02:26 And this will be divided by 6 .626 times 10 of the minus 34, which is the usual planx constant.

02:42 Which itself will be divided by 2 pi...

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