JNTU B.Tech I Year ENGINEERING PHYSICS - II Previous Question Paper (June - 2022 - R16)

The total energy E of a particle is the sum of its kinetic energy and potential energy V, given by E = p²/2m + V. Substituting the quantum mechanical operators for energy and momentum, where E = iℏ(∂/∂t) and p = -iℏ∇, we obtain the operator form Eψ = (p²/2m + V)ψ. Applying this to a wave function ψ(r,t), we get iℏ(∂ψ/∂t) = (-ℏ²/2m)∇²ψ + Vψ. This is the time-dependent Schrodinger wave equation, which describes how the quantum state of a physical system changes with time.

Heisenberg’s uncertainty principle states that it is impossible to simultaneously determine both the exact position (x) and the exact momentum (p) of a particle with absolute precision. The product of the uncertainties is given by ΔxΔp ≥ ℏ/2, where ℏ is the reduced Planck constant. This implies that the more precisely one property is measured, the less precisely the other can be known.

A classic example is the diffraction of an electron passing through a narrow slit. To determine the position of the electron, the slit width Δx must be small. However, as the slit narrows, the diffraction effect increases, spreading the electron beam and increasing the uncertainty in its momentum Δp. Another example is an electron in an atom; because it is confined to a small region (small Δx), its momentum must have a large uncertainty (large Δp), which explains why electrons do not fall into the nucleus.