Explain the concept of wave-particle duality and provide an example.

Wave-particle duality is a fundamental concept in quantum mechanics that challenges our classical understanding of particles and waves. It proposes that particles, such as electrons and photons, exhibit both particle-like and wave-like properties depending on the experimental conditions. This dual nature of matter and energy is a cornerstone of quantum physics.

Wave-Like Properties:
- When particles are observed under certain conditions, they behave like waves. This means they can exhibit interference patterns, diffraction, and superposition, similar to what we observe with classical waves like light or sound.

Particle-Like Properties:
- Conversely, particles also exhibit particle-like properties. This includes the ability to be localized in space, have discrete energies, and interact with matter as discrete entities.

Example:
One classic example of wave-particle duality is the double-slit experiment, often conducted with electrons or photons. Here's how it works:

1. Setup: Imagine a barrier with two narrow slits and a screen behind it. Electrons (or photons) are fired one by one at the barrier.

2. Expectation: If electrons were purely particles, you would expect them to pass through one slit or the other and create two distinct bands on the screen opposite the slits.

3. Observation: However, what's observed is an interference pattern on the screen, akin to the pattern produced by overlapping waves. This suggests that the electrons are behaving like waves, interfering with each other as they pass through both slits simultaneously.

4. Particle-Like Behavior: Interestingly, when researchers attempt to observe which slit each electron goes through, the interference pattern disappears. The act of measurement collapses the wave-like behavior, and the electrons behave more like particles, passing through one slit or the other.

This experiment demonstrates the wave-particle duality of electrons. In the absence of measurement, they exhibit wave-like properties, creating an interference pattern. When we attempt to measure their path, their behavior becomes more particle-like, and the interference pattern vanishes.

Wave-particle duality challenges our classical intuitions but is a fundamental aspect of quantum mechanics. It highlights that particles at the quantum level do not conform to classical concepts, and their behavior is inherently probabilistic and context-dependent. This concept is at the heart of many quantum phenomena and has profound implications for our understanding of the quantum world.