An electron with initial kinetic energy $5.0$ eV encoun­ters a barrier with height $U_0$ and width $0.60$ nm. What is the transmission coefficient if (a) $U_0=7.0$ eV; (b) $U_0=9.0$ eV; (c) $U_0=13.0$ eV?

Verified step by step guidance

1. Step 1: Understand the problem. The transmission coefficient describes the probability of an electron tunneling through a potential barrier. This is a quantum mechanical phenomenon governed by the Schrödinger equation. The key parameters are the electron's initial kinetic energy (5.0 eV), the barrier height (U₀), and the barrier width (0.60 nm). The transmission coefficient depends on these values and can be calculated using the formula for quantum tunneling.
2. Step 2: Recall the formula for the transmission coefficient in the case of a rectangular potential barrier. For an electron with energy E encountering a barrier of height U₀ and width L, the transmission coefficient T is approximately given by: $T=e^{-}kL$ where k is the decay constant inside the barrier, defined as: $k=\sqrt{2mħ²(U-E)}$ where m is the mass of the electron, ħ is the reduced Planck constant, U₀ is the barrier height, and E is the electron's energy.
3. Step 3: Substitute the given values into the formula for k. The mass of the electron (m) is approximately 9.11 × 10⁻³¹ kg, and the reduced Planck constant (ħ) is approximately 1.05 × 10⁻³⁴ J·s. Convert the energies (E and U₀) from eV to joules using the conversion factor 1 eV = 1.6 × 10⁻¹⁹ J. For each case (a, b, c), calculate k using the formula: $k=\sqrt{2mħ²(U-E)}$
4. Step 4: Calculate the transmission coefficient T for each case (a, b, c) using the formula: $T=e^{-}kL$ where L = 0.60 nm = 0.60 × 10⁻⁹ m. Substitute the values of k calculated in Step 3 into this formula for each case.
5. Step 5: Interpret the results. The transmission coefficient T will be a number between 0 and 1, where values closer to 1 indicate higher probabilities of tunneling. Compare the results for cases (a), (b), and (c) to observe how the barrier height U₀ affects the tunneling probability. Higher barrier heights generally result in lower transmission coefficients.

Key Concepts

Quantum Tunneling

Quantum tunneling is a phenomenon in quantum mechanics where a particle can pass through a potential energy barrier, even if its energy is less than the height of the barrier. This occurs due to the wave-like nature of particles, allowing for a probability of finding the particle on the other side of the barrier. The likelihood of tunneling is quantified by the transmission coefficient.

Transmission Coefficient

The transmission coefficient (T) is a measure of the probability that a particle will successfully tunnel through a barrier. It is calculated using the properties of the barrier and the particle's energy. A higher transmission coefficient indicates a greater likelihood of tunneling, while a lower coefficient suggests that tunneling is less probable.

Potential Energy Barrier

A potential energy barrier is a region in space where the potential energy is higher than the energy of a particle, effectively impeding its motion. In quantum mechanics, even if a particle's energy is lower than the barrier height, there is still a non-zero probability of tunneling through it. The height and width of the barrier significantly influence the transmission coefficient.