Wave-particle duality is a fundamental concept in quantum mechanics that describes the dual nature of particles, such as electrons and photons. It suggests that these particles exhibit both wave-like and particle-like characteristics, depending on how they are observed or measured. This concept challenges classical physics, which often treats particles and waves as distinct and separate phenomena. Here are the key aspects of wave-particle duality:

1. Particle-Like Behavior:
   • Particles are localized in space and have definite positions.
   • They have discrete and quantized properties, such as energy levels or momentum values.
   • When particles interact with detectors or undergo measurements, they behave as though they are discrete entities with specific properties.
2. Wave-Like Behavior:
   • Waves exhibit properties like interference, diffraction, and superposition. These phenomena involve the interaction and combination of different waves.
   • Waves are not localized to specific positions; they spread out over space and can occupy a range of positions simultaneously.
   • Waves are characterized by continuous, rather than discrete, properties.
3. Experimental Evidence:
   • Experiments, such as the double-slit experiment, provide strong evidence for wave-particle duality. In this experiment, particles like electrons or photons are directed at a barrier with two slits. When observed, they exhibit an interference pattern typical of waves, even though they are sent through the slits one at a time.
   • The photoelectric effect is another experiment where light (photons) behaves as particles. It involves the emission of electrons from a material when illuminated by light. The energy of the emitted electrons depends on the frequency (or color) of the light, as if they were individual particles.
4. Wave Functions:
   • In quantum mechanics, particles are described by mathematical functions known as wave functions or quantum wave functions. These wave functions provide a probabilistic description of a particle’s position, momentum, and other properties.
   • The square of the amplitude of the wave function (|ψ|^2) represents the probability density of finding the particle at a particular position.
5. Complementary Nature: Wave-particle duality suggests that it is impossible to simultaneously observe both the wave-like and particle-like aspects of a particle. The more accurately you measure one aspect (e.g., particle position), the less accurately you can know the other aspect (e.g., particle momentum).
6. Quantum Uncertainty: Heisenberg’s Uncertainty Principle is closely related to wave-particle duality. It states that there is a fundamental limit to how precisely we can simultaneously know certain pairs of properties, such as position and momentum, for a quantum particle.

Wave-particle duality is a profound and essential concept in quantum mechanics, highlighting the fundamentally probabilistic nature of the quantum world. It challenges our classical intuitions about the behavior of matter and energy and has important implications for the understanding of atomic and subatomic physics. This duality is not limited to electrons and photons; it applies to all particles in the quantum realm.