An electron with initial kinetic energy $6.0$ eV encounters a barrier with height $11.0$ eV. What is the probability of tunneling if the width of the barrier is (a) $0.80$ nm and (b) $0.40$ nm?

Verified step by step guidance

1. Step 1: Recognize that this is a quantum tunneling problem, where the probability of tunneling through a potential barrier is given by the formula: \( T = e^{-2 \kappa L} \), where \( \kappa = \sqrt{\frac{2m(U - E)}{\hbar^2}} \). Here, \( m \) is the mass of the electron, \( U \) is the barrier height, \( E \) is the electron's energy, \( \hbar \) is the reduced Planck's constant, and \( L \) is the width of the barrier.
2. Step 2: Convert the given energies from electron volts (eV) to joules (J) using the conversion factor \( 1 \text{ eV} = 1.602 \times 10^{-19} \text{ J} \). Calculate \( U - E \) in joules, where \( U = 11.0 \text{ eV} \) and \( E = 6.0 \text{ eV} \).
3. Step 3: Calculate \( \kappa \) using the formula \( \kappa = \sqrt{\frac{2m(U - E)}{\hbar^2}} \). Use the mass of the electron \( m = 9.11 \times 10^{-31} \text{ kg} \) and \( \hbar = 1.055 \times 10^{-34} \text{ J·s} \). Substitute the value of \( U - E \) from Step 2 into this equation.
4. Step 4: For part (a), substitute \( L = 0.80 \text{ nm} = 0.80 \times 10^{-9} \text{ m} \) into the tunneling probability formula \( T = e^{-2 \kappa L} \). For part (b), repeat the calculation with \( L = 0.40 \text{ nm} = 0.40 \times 10^{-9} \text{ m} \).
5. Step 5: Simplify the expressions for \( T \) for both cases (a) and (b) to express the tunneling probabilities in terms of \( \kappa \) and \( L \). This will give the final expressions for the tunneling probabilities without numerical evaluation.

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 non-zero probability of finding the particle on the other side of the barrier. The probability of tunneling decreases exponentially with increasing barrier width and height.

Kinetic Energy and Potential Energy

Kinetic energy is the energy possessed by an object due to its motion, while potential energy is the stored energy based on an object's position in a force field, such as gravitational or electric fields. In this context, the electron's initial kinetic energy (6.0 eV) is compared to the potential energy barrier (11.0 eV) it encounters, determining the likelihood of tunneling through the barrier.

Barrier Width and Height in Tunneling

The width and height of a potential barrier significantly influence the tunneling probability of a particle. A wider barrier or a higher barrier reduces the probability of tunneling, as the particle's wave function decays exponentially within the barrier. The relationship can be quantitatively described using the Schrödinger equation, which provides a mathematical framework for calculating tunneling probabilities based on these parameters.