Question

A particle of energy $E=5\mathrm{eV}$ approaches an energy barrier of height $U=8\mathrm{eV}$. Quantum mechanically there is a finite probability that the particle tunnels through the barrier. If the barrier height is slowly decreased, the probability that the particle will reflect from the barrier will a) decrease. b) increase. c) not change.

Short answer

Answer: The probability that the particle will reflect from the barrier will decrease.

Step-by-step solution

Remember the relationship between transmission and reflection probabilities

In quantum mechanics, a particle can either be reflected or transmitted (tunnel) when it encounters a barrier. Their probabilities should add up to 1, i.e. $P_{reflection}+P_{transmission}=1$.

Compare the initial energy and barrier height

The particle has an energy E = 5 eV, and it encounters an energy barrier with a height of U = 8 eV. Since the particle energy is less than the barrier height, there's a higher chance of reflection and a smaller chance of transmission (tunneling).

Visualize the decrease in barrier height

As the barrier height is reduced from 8 eV to a smaller value, the particle has a higher probability of tunneling through the barrier, as its energy can overcome the barrier's height more easily.

Determine the effect on reflection probability

Since the transmission probability increases with the decrease in barrier height, the reflection probability should decrease accordingly. This is because the sum of reflection and transmission probabilities should be equal to 1. So, the answer is: a) The probability that the particle will reflect from the barrier will decrease.

Key concepts

Energy Barrier

In quantum mechanics, an energy barrier is a potential energy increase that a particle encounters, which it needs to overcome to continue on its path. These barriers can be thought of as hills that a particle must climb over, requiring energy. For example, in this scenario, our particle has a given energy of 5 eV and is approaching a barrier with a height of 8 eV. This means that the barrier is too high for the particle’s energy in terms of classical physics decisions.
However, quantum mechanics allows for the particle to tunnel through this barrier, even if its energy is less than the height of the barrier. This unique property is a fundamental aspect of quantum theory. The concept of tunneling through an energy barrier is especially important in fields like quantum chemistry and semiconductor physics, where it explains phenomena like electron tunneling in semiconductors.

Reflection Probability

When a particle encounters an energy barrier, there are two possibilities: it might reflect back or tunnel through to the other side. The reflection probability quantifies the likelihood of the particle bouncing back instead of penetrating the barrier.
For a particle with less energy than the barrier height, like our example, the reflection probability starts high. This is because the barrier essentially stops the particle and reflects it back, akin to throwing a ball against a wall. However, as the barrier height is lowered, the reflection probability decreases. This occurs because the difference between the particle energy and the barrier is minimized, effectively granting the particle a higher chance of passing through.
Ultimately, reflection and transmission probabilities must add up to 1. So, when the reflection probability decreases, it directly increases the transmission probability.

Transmission Probability

Transmission probability is the chance that a particle will successfully tunnel through an energy barrier. This phenomenon is counter-intuitive from a classical physics standpoint, which considers energy barriers as impenetrable if they're higher than a particle's energy.
Yet, in the quantum realm, particles can exhibit behavior that allows them to "borrow" energy for a short time to tunnel through barriers. As illustrated in our scenario, as the barrier height decreases, the particle finds it increasingly easier to tunnel through. This happens because the barrier offers less resistance to the particle's energy.
Thus, when the barrier's height is reduced closer to or below the energy level of the particle, the transmission probability rises. This reflects a higher chance that the particle quantum tunnels past the barrier, underpinning many practical applications, such as understanding how microscopic particles cross energy barriers in chemical reactions or in electronics.