In quantum mechanics, just as electric charge is conserved, probability is also a conserved quantity. The Probability Current Density (or Probability Flux) is a vector quantity describing the flow of probability in space, ensuring that the disappearance of a particle from one region is accounted for by its appearance in another.
Figure 1: Visualization of probability flux \( \mathbf{J} \) leaving a volume element.
Derivation of Probability Current Density \( \mathbf{J} \)
1. The Conservation of Probability
The derivation begins with the Continuity Equation. If \( \rho(\mathbf{r}, t) \) represents the probability density (probability per unit volume), then the conservation of probability is expressed as:
\[ \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = 0 \]
Here, \( \mathbf{J} \) is the probability current density vector we wish to derive. This equation states that the rate of decrease of probability in a volume equals the net flux of probability flowing out of its surface.
2. Step-by-Step Derivation
Step A: Time Derivative of Density
The probability density is defined as \( \rho = |\psi|^2 = \psi^* \psi \). We differentiate this with respect to time:
\[ \frac{\partial \rho}{\partial t} = \frac{\partial}{\partial t} (\psi^* \psi) = \psi^* \frac{\partial \psi}{\partial t} + \psi \frac{\partial \psi^*}{\partial t} \quad \dots(1) \]
Step B: Utilizing the Schrödinger Equation
We use the time-dependent Schrödinger equation to find expressions for the time derivatives.
Equation for \( \psi \):
\[ i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \psi + V\psi \]
Rearranging for \( \frac{\partial \psi}{\partial t} \):
\[ \frac{\partial \psi}{\partial t} = \frac{1}{i\hbar} \left( -\frac{\hbar^2}{2m} \nabla^2 \psi + V\psi \right) \] \[ \frac{\partial \psi}{\partial t} = \frac{i\hbar}{2m} \nabla^2 \psi - \frac{i}{\hbar} V\psi \quad \dots(2) \]
Equation for \( \psi^* \):
Taking the complex conjugate of the Schrödinger equation (assuming \( V \) is real):
\[ -i\hbar \frac{\partial \psi^*}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \psi^* + V\psi^* \] \[ \frac{\partial \psi^*}{\partial t} = -\frac{i\hbar}{2m} \nabla^2 \psi^* + \frac{i}{\hbar} V\psi^* \quad \dots(3) \]
Step C: Substitution and Simplification
Substitute equations (2) and (3) back into equation (1):
\[ \frac{\partial \rho}{\partial t} = \psi^* \left( \frac{i\hbar}{2m} \nabla^2 \psi - \frac{i}{\hbar} V\psi \right) + \psi \left( -\frac{i\hbar}{2m} \nabla^2 \psi^* + \frac{i}{\hbar} V\psi^* \right) \]
Grouping the terms:
\[ \frac{\partial \rho}{\partial t} = \frac{i\hbar}{2m} (\psi^* \nabla^2 \psi - \psi \nabla^2 \psi^*) - \frac{i}{\hbar} \psi^* V \psi + \frac{i}{\hbar} \psi V \psi^* \]
The terms containing potential \( V \) cancel each other out. We are left with:
\[ \frac{\partial \rho}{\partial t} = \frac{i\hbar}{2m} \left( \psi^* \nabla^2 \psi - \psi \nabla^2 \psi^* \right) \]
Step D: Applying Vector Identity
We use the vector identity: \( \nabla \cdot (f \nabla g - g \nabla f) = f \nabla^2 g - g \nabla^2 f \). Applying this to our equation:
\[ \psi^* \nabla^2 \psi - \psi \nabla^2 \psi^* = \nabla \cdot (\psi^* \nabla \psi - \psi \nabla \psi^*) \]
Substituting this back into the time derivative equation:
\[ \frac{\partial \rho}{\partial t} = \frac{i\hbar}{2m} \nabla \cdot \left( \psi^* \nabla \psi - \psi \nabla \psi^* \right) \]
Rearranging to match the form \( \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = 0 \):
\[ \frac{\partial \rho}{\partial t} + \nabla \cdot \left[ -\frac{i\hbar}{2m} (\psi^* \nabla \psi - \psi \nabla \psi^*) \right] = 0 \]
3. Final Result
By comparing the derived equation with the continuity equation, we identify the term in the brackets (with a sign adjustment to match the divergence theorem convention) as the Probability Current Density.
\[ \mathbf{J} = \frac{i\hbar}{2m} (\psi \nabla \psi^* - \psi^* \nabla \psi) \]
Alternatively written as: \[ \mathbf{J} = \text{Re} \left( \psi^* \frac{\hat{p}}{m} \psi \right) \]
4. Physical Interpretation
- Flow of Probability: Just as electric current is the flow of charge, \( \mathbf{J} \) is the flow of probability.
- Fluid Analogy: If we view the probability cloud as a fluid, \( \rho \) is the fluid density and \( \mathbf{J} = \rho \mathbf{v} \) represents the flux, where \( \mathbf{v} \) is the velocity field of the probability flow.
- Stationary States: For real wavefunctions (stationary states with no time dependence in phase), \( \psi = \psi^* \), and consequently \( \mathbf{J} = 0 \). There is no net flow of probability in a standing wave.
Figure 2: Current density is zero for real wavefunctions (left) but non-zero for travelling waves (right).