Which is the de-Broglie equation: (a) \(h=p \lambda\) (b) \(h=p \lambda^{-1}\) (c) \(h=\lambda p^{-1}\) (d) \(h=p+\lambda\)

One of the most groundbreaking ideas in physics is the principle of wave-particle duality. This concept is central to the field of quantum mechanics and fundamentally describes how particles at the microscopic scale, such as electrons and photons, can exhibit characteristics of both particles and waves. It was a realization that challenged classical physics and paved the way for new understanding.

For example, light can be understood as both a stream of particles known as photons and as electromagnetic waves. This duality is evident in experiments like the double-slit experiment, where light behaves like a wave when passing through two slits and creates an interference pattern, but also shows particle characteristics when individual photons are detected.

The wave aspect allows particles to exhibit properties such as interference and diffraction, while the particle aspect enables them to collide and transfer momentum. The de Broglie equation is a mathematical representation of this duality, linking a particle's wave-like behavior with its momentum.

Planck's constant is a fundamental quantity in quantum mechanics, symbolized by the letter 'h'. It's named after the physicist Max Planck, who introduced the idea in 1900. This constant plays a crucial role in quantifying the sizes of quanta, the smallest discrete units of energy involved in interactions like absorbing or emitting light.

Planck's constant has the value of approximately 6.626 x 10^{-34} joule seconds (Js). It appears in various quantum equations, indicating the quantized nature of physical quantities at small scales. Not only is it a component of the de Broglie equation, but it is also fundamental to the equation for photon energy, E=hf, where E is energy, h is Planck's constant, and f is the frequency of the photon. This concept helps explain why, at the atomic level, energy is transferred in discrete amounts rather than a continuous flow.

Momentum is a measure of the motion of a particle and is a fundamental concept in both classical and quantum physics. In classical mechanics, momentum is defined as the product of mass and velocity of an object (p=mv). However, in the context of quantum mechanics and wave-particle duality, the momentum of a particle also relates to its wave-like properties.

An important aspect of momentum in quantum mechanics is that, just like energy, it can also exhibit discrete values under certain conditions, such as when particles are confined to small spaces. This is where the de Broglie hypothesis comes into play, providing the connection between the momentum of a particle and its associated wavelength via the equation \(h=p\lambda\), with \(p\) being the momentum of the particle. Understanding particle momentum helps to grasp phenomena like electron orbitals and the behavior of particles in accelerators.