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, the conservation of fluid is given by:

(∂P/∂t) + ∇·J = 0

By comparing our quantum derivation to this classical form, we identify:

• Position Probability Density (P): Ψ*Ψ
• Probability Current Density (J): (ℏ/2mi) [Ψ*∇Ψ - Ψ∇Ψ*]

Conclusion

The Probability Current Density (J) represents the flow of probability per unit area per unit time. Mastering this derivation is crucial for understanding the conservation of probability in any quantum system.

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