A proton and a deuteron (a particle with the same charge as...

A proton and a deuteron (a particle with the same charge as a proton, but twice the mass) attempt to penetrate a rectangular potential barrier of height $10 \mathrm{MeV}$ and thickness $10^{-14} \mathrm{~m} .$ Both particles have total energies of $3 \mathrm{MeV}$. (a) Use qualitative arguments to predict which particle has the highest probability of succeeding. (b) Evaluate quantitatively the probability of success for both particles.

Key Concepts

Quantum Tunneling

Quantum tunneling is the phenomenon where particles, due to their wave-like nature, have a non-zero probability of crossing energy barriers that exceed their kinetic energy. This counterintuitive process is a core concept in quantum mechanics and underpins many phenomena and applications, from nuclear fusion in stars to modern electronic devices.

Rectangular Potential Barrier

The rectangular potential barrier is a simplified model used in quantum mechanics to study how particles behave when encountering sharp changes in potential energy. It involves a barrier with a constant height and finite width, and solving the Schrödinger equation for this scenario reveals insights into how particles can partially transmit through or reflect from such barriers.

Transmission Probability

Transmission probability refers to the likelihood that a particle will successfully tunnel through a potential barrier. This probability is determined by factors including the difference between the barrier height and the particle's energy, the barrier's width, and the mass of the particle, with heavier particles generally exhibiting a lower tunneling probability.

WKB Approximation

The WKB approximation is a semi-classical method used to solve the Schrödinger equation in regions where the potential varies slowly, or remains constant in cases like a rectangular barrier. It provides an approximate expression for the exponential decay of the wave function within a barrier, which is key to calculating the tunneling, or transmission, probability.

Mass Dependence in Tunneling

The mass of a particle critically influences its tunneling probability. In quantum mechanics, a heavier particle has a more rapidly decaying wave function within a barrier, making it less likely to tunnel than a lighter particle with the same energy and encountering the same potential barrier. This dependence is essential when comparing tunneling phenomena across particles of different masses.

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