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Wave-particle duality is a fundamental concept in quantum mechanics, asserting that particles exhibit both wave and particle properties. This text delves into the dual nature of light, exploring its particle form as photons with quantized energy, and its wave phenomena like interference and diffraction. It also covers historical contributions from Planck, Einstein, and de Broglie, as well as the implications of Heisenberg's uncertainty principle for quantum measurements.

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## Concept of Wave-Particle Duality

### Definition of Wave-Particle Duality

Wave-particle duality is the concept that all particles exhibit both wave and particle characteristics

### Examples of Wave-Particle Duality

Light as a Prime Example

Light can be described as both a wave and a particle, known as a photon, in different phenomena

Dual Behavior in Composite Systems

Wave-particle duality also applies to composite systems like atoms and molecules

### Impact on Quantum Mechanics

The concept of wave-particle duality has been pivotal in advancing our understanding of quantum phenomena

## Wave-Like Nature of Light

### Characteristics of Light Waves

Light waves exhibit properties such as reflection, refraction, diffraction, and interference

### Mathematical Formulations

Planck's Law

Planck's law describes the intensity distribution of electromagnetic radiation emitted by a blackbody

Energy of Photons

The energy of a photon can be expressed as E = hf or E = hc/λ, depending on the given parameters

### Experimental Evidence

The wave-like nature of light is demonstrated through various phenomena, including reflection, refraction, diffraction, and interference

## Particle-Like Nature of Light

### Characteristics of Photons

Photons are massless particles that carry quantized energy proportional to their frequency

### Explanation of the Photoelectric Effect

Einstein's theory of photons explained the observation that electrons are emitted only when light reaches a certain threshold frequency

### Contributions of Physicists

The concept of wave-particle duality was shaped by the contributions of Max Planck, Albert Einstein, and Louis de Broglie

## Other Key Concepts in Quantum Mechanics

### Louis de Broglie's Hypothesis

De Broglie proposed that matter, like electrons, also exhibits wave-like properties

### Heisenberg Uncertainty Principle

The uncertainty principle states that there is a limit to the precision with which certain pairs of physical properties can be known simultaneously

### Probabilistic Nature of Quantum Measurements

The intrinsic probabilistic nature of quantum measurements is highlighted by the uncertainty principle

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