Derivation of Probability Current Density and Equation of Continuity

Derivation of Probability Current Density In Quantum Mechanics, the motion of a material particle is associated with a wave function. If the probability of finding a particle in a bounded region decreases with time, the probability of finding it outside must increase by the same amount. This is described by the Probability Current Density . 1. The Schrödinger Basis We start with the Time-Dependent Schrödinger Equation (TDSE): iℏ (∂Ψ/∂t) = -(ℏ²/2m)∇²Ψ + VΨ --- (i) Taking the Complex Conjugate of equation (i): -iℏ (∂Ψ*/∂t) = -(ℏ²/2m)∇²Ψ* + VΨ* --- (ii) 2. Mathematical Manipulation To find the rate of change, we perform the following: Multiply eq (i) by Ψ* from the left. Multiply eq (ii) by Ψ from the left. Subtract the resulting equations. Using the UV derivation method for ∇·(Ψ*∇Ψ - Ψ∇Ψ*), we simplify the expression to relate it to the Equation of Continuity . 3. The Equation of Continuity In hydrodynamics, t...
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