JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech I Year ENGINEERING PHYSICS II Previous Question Paper (June - 2022 - R16)

The total energy of a particle is the sum of its kinetic energy and potential energy, given by E = p²/2m + V. Substituting the quantum mechanical operators for energy and momentum, E = iℏ(∂/∂t) and p = -iℏ∇, we obtain the operator equation Eψ = (p²/2m + V)ψ. Substituting the operators into this expression yields the time-dependent Schrödinger wave equation:

Here, ℏ is the reduced Planck's constant (h/2π), m is the mass of the particle, V is the potential energy, and ∇² is the Laplacian operator. This equation describes how the wave function ψ(r,t) evolves over time in a given potential field.

Heisenberg’s uncertainty principle states that it is impossible to determine simultaneously both the position (x) and the momentum (p) of a particle with absolute precision. The product of the uncertainties in these measurements is always greater than or equal to a constant value:

This principle arises because the act of measuring a particle's position requires interaction with a photon, which imparts momentum to the particle, thereby changing its velocity. Similarly, measuring momentum requires techniques that disturb the particle's precise location.

A classic example is the diffraction of an electron through a slit. If an electron passes through a narrow slit of width Δx, its position is localized. However, due to diffraction, the electron spreads out, introducing an uncertainty in its momentum Δp. As the slit width Δx decreases, the diffraction pattern becomes wider, increasing the uncertainty in the electron's momentum, confirming the inverse relationship defined by the principle.