An electron having total energy E=4.50 eV approaches a...

An electron having total energy $E=4.50$ eV approaches a rectangular energy barrier with $U=5.00$ eV and $L=$ 950 $\mathrm{pm}$ as shown in Figure $\mathrm{P} 41.30 .$ Classically, the electron cannot pass through the barrier because $E<U$ . Quantum- mechanically, however, the probability of tunneling is not zero. (a) Calculate this probability, which is the transmission coefficient. (b) To what value would the width $L$ of the potential barrier have to be increased for the chance of an incident 4.50 -eV electron tunneling through the barrier to be one in one million?

Key Concepts

Potential Barrier

A potential barrier represents a region in space where the potential energy is higher than in surrounding regions. In the context of quantum mechanics, even when a particle’s energy is less than the height of this barrier, the particle’s wavefunction can still extend into and sometimes through the barrier, allowing for tunneling.

WKB Approximation

The WKB (Wentzel–Kramers–Brillouin) approximation is a semi-classical analytical method used to solve the Schrödinger equation in scenarios where the potential varies slowly. It is particularly useful for estimating the exponential decay of the wavefunction in a classically forbidden region, which directly relates to the tunneling probability.

Quantum Tunneling

Quantum tunneling is the phenomenon wherein a particle has a non-zero probability to penetrate and pass through a potential barrier, even if its energy is less than the barrier's height. This effect arises from the wave-like nature of particles in quantum mechanics and has no classical counterpart.

Transmission Coefficient

The transmission coefficient quantifies the probability that a particle will successfully tunnel through a potential barrier. It is determined by the behavior of the particle’s wavefunction as it encounters and decays within the barrier, and is a key concept in understanding quantum mechanical barrier penetration.

Transcript

00:01 So we're going to get the transmission coefficient.

00:03 To do that, we need c.

00:05 So c is equal the square root of 2 times the mass times u minus e divided by h bar, and h bar is equal to 6 .626 times 10 to the minus 34 joules seconds divided by 2 pi.

00:45 So this, c, this constant, we can go ahead and substitute in the values.

00:50 We know that this is going to be the mass of an electron.

01:01 And we have, let me go ahead and make some room and scroll down just a bit here.

01:22 And here for u minus e, we're going to have 5 .00 minus 4 .50 electron volts.

01:32 And in keeping the energy and units of electron volts, i'm going to have a conversion factor, 1 .60 times 10 to the minus 19 joules per electron bolt.

01:46 I just want to convert in the middle of the calculation to joules, given that the other quantities in this calculation are in si units, and then we'll divide by h -bar.

02:02 And then when carrying this out, we get 3 .620 times 10 to the 9th inverse meters.

02:36 Inverse meters are the units for c.

02:38 So then upon substitution into t, which is e to the minus 2c, we have e to the minus 2c, times 3 .620 times 10 to the ninth times 9 .5 times 10 to the minus 10.

03:09 And this comes out to 0 .0013, which is equal to 1 .03 times 10 to the minus 3.

03:23 And as a percent, if you multiply by 100 percent, this is equal to 1 .03, this is equal to 1 .03 times 10 to the minus 3.

03:29 Equal to 0 .13%.

03:32 So this is a small percent of a transmission, a 0 .103 percent chance...

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