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 $U=5.00 \mathrm{eV}$ and $L=$ 950 $\mathrm{pm}$ as shown in Figure $\mathrm{P} 41.28$ . Classically, the electron cannot pass through the barrier because $E<U$ . Quantum--mechanically, however, the probability of tunneling is not zero. Calculate this probability, which is the transmission coefficient.

Key Concepts

Rectangular Potential Barrier

The rectangular potential barrier is a simplified model used in quantum mechanics to analyze tunneling. It represents a region with a constant potential energy that is higher than the particle's energy. This model allows for analytical solutions to the Schrödinger equation and helps illustrate key aspects of the tunneling phenomenon.

Exponential Decay in the Barrier

Within the potential barrier, the wave function of a tunneling particle exhibits exponential decay due to the barrier’s higher potential energy relative to the particle’s energy. This decay rate is dependent on the difference between the barrier height and the particle’s energy, as well as the barrier width, and plays a crucial role in determining the transmission probability.

Quantum Tunneling

Quantum tunneling is a phenomenon in quantum mechanics where a particle can penetrate and pass through a potential barrier even if its energy is less than the height of the barrier. This contrasts with classical mechanics, where a particle would be completely reflected. Tunneling is a direct consequence of the wave-like nature of particles and the probabilistic interpretation of quantum mechanics.

Transmission Coefficient

The transmission coefficient is a measure of the probability that a quantum particle will tunnel through a potential barrier. It is derived from the solutions to the Schrödinger equation and quantifies the likelihood of finding the particle on the opposite side of the barrier after encountering it.

Schrödinger Equation

The Schrödinger equation is the fundamental equation of non-relativistic quantum mechanics that describes how the quantum state of a physical system changes over time. In the context of tunneling, solving the time-independent Schrödinger equation provides wave functions that reveal how a particle's probability amplitude decays within a potential barrier.

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