Wave Nature of Matter and De Broglie's Equation

De Broglie’s hypothesis stated that there is symmetry in nature and that if light and radiation behave as both particles and waves, matter too will have both the particle and wave nature.

Through de Broglie’s relationship, we now had a wave theory of matter. The ‘Lambda (λ)’ here represents the wavelength of the particle and ‘p’ represents the momentum of the particle. Here h is Planck’s constant, m is the mass and v is the velocity of the particle. The significance of the de Broglie relationship is that it proved mathematically that matter can behave as a wave. In layman’s terms, the de Broglie equation says that every moving particle – microscopic or macroscopic –has its own wavelength.

For microscopic objects, the wave nature of matter is observable.

For larger objects, the wavelength gets smaller with the increasing size of the object, quickly becoming so small as to become unnoticeable which is why macroscopic objects in real life don’t show wave-like properties. Even the cricket ball you throw has a wavelength that is too small for you to observe. The wavelength and the momentum in the equation are connected by the Planck’s constant.

Heisenberg’s Uncertainty Equation:

Say you measure the momentum of the particle accurately so that ‘delta p (Δp)’ is zero. To satisfy the equation above, the uncertainty in the position of the particle, ‘delta x (Δx)’ has to be infinite. From de Broglie’s equation, we know that a particle with a definite momentum has a definite wavelength ‘Lambda’. A definite wavelength extends all over space all the way to infinity. By Born’s Probability Interpretation this means that the particle is not localized in space and therefore the uncertainty of position becomes infinite.

In real life though, the wavelengths have a finite boundary and are not infinite and thus both the position and momentum uncertainties have a finite value. De Broglie’s equation and Heisenberg’s Uncertainty Principle are apples of the same tree.