1. Photoelectric Effect Experiments and Observations

The photoelectric effect is the emission of electrons from a material when light shines on it. Experiments revealed key observations that classical wave theory could not explain: electrons are emitted only if the light's frequency exceeds a certain threshold, the emission is almost instantaneous, and the kinetic energy of emitted electrons depends on the light's frequency, not its intensity. These findings suggested that light energy is not continuous but comes in discrete packets.

2. Wave Theory Failure and Einstein's Photoelectric Equation

Classical wave theory predicted that light intensity should determine whether electrons are emitted and their energy, which contradicted experimental results. Albert Einstein explained the photoelectric effect by proposing that light consists of discrete energy packets called photons. His photoelectric equation, $h\nu = W + KE_{\text{max}}$, states that the energy of an incident photon ($h\nu$) is used to overcome the work function ($W$) of the material and the remaining energy appears as the maximum kinetic energy ($KE_{\text{max}}$) of the emitted electron. This confirmed the particle nature of light.

3. Particle Nature of Light: The Photon

The concept of the photon, introduced by Einstein, describes light as a stream of particles, each carrying a quantum of energy $E = h\nu$, where $h$ is Planck's constant and $\nu$ is the frequency. Photons also exhibit momentum, given by $p = h/\lambda$, where $\lambda$ is the wavelength. This duality—light behaving as both a wave and a particle—is a cornerstone of quantum mechanics. It explains phenomena like the photoelectric effect and the Compton effect (scattering of photons by electrons).

4. Wave Nature of Matter

Louis de Broglie hypothesized that if light can exhibit particle-like properties, then matter particles, such as electrons, should also exhibit wave-like properties. He proposed that a particle with momentum $p$ has an associated wavelength, known as the de Broglie wavelength ($\lambda$), given by $\lambda = \frac{h}{p}$. This concept of matter waves suggests that all matter has wave characteristics, though these are generally only significant for very small particles like electrons due to their small mass and momentum.

5. Davisson and Germer Experiment

The Davisson-Germer experiment provided experimental confirmation of de Broglie's hypothesis about the wave nature of matter. In this experiment, electrons were diffracted by a nickel crystal. The observed diffraction pattern, similar to X-ray diffraction patterns, showed that electrons behaved as waves with a wavelength consistent with de Broglie's relation $\lambda = h/p$. This groundbreaking experiment solidified the concept of wave-particle duality for matter.

6. Additional: Heisenberg's Uncertainty Principle

Heisenberg's Uncertainty Principle is a fundamental principle in quantum mechanics that sets a limit on the precision with which certain pairs of physical properties of a particle, such as position ($x$) and momentum ($p$), can be simultaneously known. It states that the product of the uncertainties in these measurements is at least Planck's constant divided by $2\pi$: $\Delta x \Delta p \ge \frac{\hbar}{2}$, where $\hbar = h/2\pi$. This principle highlights the inherent probabilistic nature of quantum measurements and the impossibility of simultaneously knowing exact values for certain complementary properties.