Question

Two long, straight wires that lie along the same line have a separation at their tips of \(2.00 \mathrm{nm}\). The potential energy of an electron in the gap is about \(1.00 \mathrm{eV}\) higher than it is in the conduction band of the two wires. Conduction-band electrons have enough energy to contribute to the current flowing in the wire. What is the probability that a conduction electron in one wire will be found in the other wire after arriving at the gap?

Short answer

Answer: The probability that a conduction electron in one wire will be found in the other wire after arriving at the gap is approximately \(7.18 \times 10^{-4}\).

Step-by-step solution

Convert given quantities to SI units

First, we need to convert the given quantities to SI units. The width of the gap \(L = 2.00 \mathrm{nm} = 2.00 \times 10^{-9}\) meters. The potential energy difference between the gap and the conduction band \(U = 1.00 \mathrm{eV} = 1.00 \times 1.602 \times 10^{-19}\) Joules

Calculate the constant K

The constant \(K\) depends on the mass of the electron, its potential energy difference and the reduced Planck's constant, \(h\). The formula for \(K\) is given by: \(K = \frac{\sqrt{2mU}}{\hbar}\) Where \(m\) is the mass of the electron, which is \(9.11 \times 10^{-31}\) kg, and \(\hbar\) is the reduced Planck's constant, which is approximately \(1.055 \times 10^{-34} \mathrm{Js}\). Substitute the values for the mass and potential energy difference to get: \(K = \frac{\sqrt{2(9.11 \times 10^{-31}\mathrm{kg})(1.00 \times 1.602 \times 10^{-19}\mathrm{J})}}{1.055 \times 10^{-34} \mathrm{Js}}\)

Calculate the tunneling probability

Now that we have the value of \(K\), we can calculate the tunneling probability using the formula: \(P = e^{-2KL}\) Substitute the values for \(K\) and \(L\) into the formula: \(P = e^{-2\left(\frac{\sqrt{2(9.11 \times 10^{-31}\mathrm{kg})(1.00 \times 1.602 \times 10^{-19}\mathrm{J})}}{1.055 \times 10^{-34} \mathrm{Js}}\right)(2.00 \times 10^{-9}\mathrm{m})}\)

Evaluate the probability

Evaluate the expression to find the tunneling probability: \(P \approx 7.18 \times 10^{-4}\) Thus, the probability that a conduction electron in one wire will be found in the other wire after arriving at the gap is approximately \(7.18 \times 10^{-4}\).

Key concepts

Conduction Band

In the realm of solid-state physics, the conduction band is paramount when discussing electronic properties of materials. It represents an energy range where electrons are free to move within a material, thereby allowing electrical conductivity. Think of it as a track for electrons to race along, with sufficient energy making them move swiftly to contribute to an electric current.

More specifically, in metals, the conduction band is partially filled, meaning some electrons have natural access to these energies even at room temperature. In semiconductors and insulators, the conduction band is typically empty at absolute zero. For them to conduct electricity, electrons must gain enough energy, such as from an external source like heat or light, to jump up from the valence band (where they are usually 'stuck') to the conduction band – this is known as the band gap.

Understanding the conduction band is crucial to solving problems related to electronic conduction and hence is a vital concept in the scenario of electron tunneling probability.

Electron Tunneling Probability

One of the most fascinating phenomena in quantum mechanics is undoubtedly electron tunneling, where particles like electrons can pass through barriers that would be insurmountable according to classical physics. This effect is due to the probabilistic nature of quantum mechanics, where particles are described by wavefunctions that can extend beyond potential barriers.

Calculating the tunneling probability of an electron allows us to predict how likely it is for the electron to appear on the other side of a barrier – like the gap between two wires in this exercise. The process of finding this probability involves knowing the electron's wavefunction and the characteristics of the potential barrier, such as its height (the potential energy) and width. In the case of the exercise, the narrower the gap and the lower the potential energy barrier, the higher the probability of the electron making it to the other side.

The probability is exponentially related to the width of the gap and the square root of the mass and potential energy. This relationship emphasizes the incredibly delicate balance of factors that govern electron tunneling in materials and is crucial for designing nanoscale electronic devices.

Potential Energy in Quantum Mechanics

The concept of potential energy in quantum mechanics plays a pivotal role when we talk about forces acting on particles at microscopic levels. Potential energy is the energy stored by an object due to its position relative to other objects or fields it interacts with. In the context of quantum mechanics, instead of witnessing particles behaving predictably as we would expect in a Newtonian framework, we encounter probabilities and wavefunctions.

In quantum systems, the potential energy is often described by potential wells or barriers. For example, in the exercise, an electron must overcome a potential energy 'hill' to move from one wire to the other. Quantum mechanics allows us to calculate the probability of scenarios that would be impossible in classical mechanics, such as 'tunneling' through this hill. The ability to calculate this potential and predict particle behavior in quantum systems is essential for understanding a wide range of phenomena, from the behavior of atoms and molecules to the workings of transistors in electronics.