# Tunneling effect or barrier penetrability

• logearav
In summary, the wave function of a free particle is continuous at the barrier and decays exponentially inside the barrier. The reduced probability for the particle after coming out of the barrier is due to the loss of energy, which has a probability of being reflected or transmitted just like the particle itself. The concepts of probability and energy are distinct and knowing the probability of finding a particle in a region does not necessarily provide information about its energy.
logearav
The wavefunction associated with a free particle must be continuous at the barrier and will show an exponential decay inside the barrier. Why it shows an exponential decay inside the barrier? After coming out of the barrier, there is reduced probability for the particle. When there is no loss of energy after coming out of the barrier why there is reduced probability?

The wave function goes as e^ikx (plus or minus depending on direction). We have that k is real for E>V, and k is imaginary for E<V. Outside of the barrier, the wave function oscillates, but inside the barrier E<V, so we have that the wave function is exponential decay.

As for the energy, if the barrier is loss-less, then the energy has a % probability to be reflected or transmitted just like the particle. The energy is a property of the particle after all.

logearav said:
The wavefunction associated with a free particle must be continuous at the barrier and will show an exponential decay inside the barrier. Why it shows an exponential decay inside the barrier? After coming out of the barrier, there is reduced probability for the particle. When there is no loss of energy after coming out of the barrier why there is reduced probability?

Where does the probability appear in the expression for the particle energy? Those two concepts are distinct. Knowing the probability that a particle will be found in a region of space does not necessarily tell you anything about its energy.

## 1. What is the tunneling effect or barrier penetrability?

The tunneling effect or barrier penetrability is a quantum phenomenon where a particle is able to pass through a potential energy barrier that would be classically impossible to overcome. This effect occurs due to the wave-like nature of particles, allowing them to exist in multiple places at once and "tunnel" through the barrier.

## 2. How does the tunneling effect occur?

The tunneling effect occurs when the particle's wave function extends into the barrier region, allowing for a small probability of the particle to exist on the other side of the barrier. This probability decreases exponentially with the width and height of the barrier, but it is never zero.

## 3. What factors affect the tunneling effect?

The tunneling effect is affected by the energy and mass of the particle, as well as the width and height of the barrier. A higher energy or smaller mass particle is more likely to tunnel through a barrier, and a wider or taller barrier reduces the chances of tunneling.

## 4. What are some real-life applications of the tunneling effect?

The tunneling effect has many practical applications, including in electronics and microscopy. For example, it is used in tunneling diodes and scanning tunneling microscopes to manipulate and observe materials at the atomic level.

## 5. Are there any limitations to the tunneling effect?

While the tunneling effect is a fundamental phenomenon in quantum mechanics, it is not observable on a macroscopic scale. Additionally, the probability of tunneling decreases rapidly with the size and complexity of the barrier, making it difficult to control and predict in certain scenarios.

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