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Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the scale of atoms and subatomic particles. It challenges classical mechanics, especially when it comes to the motion of particles, introducing concepts such as wave-particle duality, in which particles exhibit properties of both particles and waves.

A significant point of quantum mechanics is its probabilistic nature. Unlike the deterministic predictions of classical physics, quantum mechanics only offers probabilities of where particles might be found. This is represented in mathematical frameworks like Schrödinger's equation. Additionally, the idea that particles can be in multiple states simultaneously until measured, known as superposition, further distinguishes quantum mechanics from classical physics.

Planck's equation is at the heart of quantum mechanics and is given by the formula \(E = hv\), where \(E\) represents the energy of a photon, \(h\) is Planck's constant, and \(v\) is the frequency of the electromagnetic wave associated with the photon.

The equation was derived by Max Planck, who proposed that energy is quantized and can only be emitted or absorbed in discrete units called 'quanta'. This groundbreaking idea led to the concept of the photon, the particle aspect of light. Planck's equation beautifully bridges the gap between the wave and particle nature of light, showing that energy is directly proportional to frequency, and thus connects the wave-like properties to the particle-like photons.

Photons are the quantum of light and other forms of electromagnetic radiation. They are unique in that they have no mass, carry energy, and travel at the speed of light in a vacuum.

The properties of photons include their particle-like ability to be counted and to impart momentum upon absorption or reflection, as described by the photoelectric effect. They are also described by their wave-like behaviors, such as being able to interfere and diffract, and their energy is solely dependent on their frequency, as explained by Planck's equation.

An interference pattern is created when waves, including those associated with particles like electrons and photons, overlap and combine. This pattern results from the superposition of the wave functions – areas of constructive interference where the waves enhance each other form bright bands, and areas of destructive interference lead to dark bands.

This phenomenon is commonly observed in experiments like the double-slit experiment, which showcases wave-particle duality. When photons pass through two closely spaced slits, they produce an interference pattern that cannot be explained if the photons were simply particles; instead, it reveals their wave-like character. However, remarkably, if the photons are observed, they appear to travel through one slit or the other, like particles, not creating the interference pattern. This enigmatic behavior highlights the dual nature of photons and is a definitive example of quantum mechanics at play.