Define Probability Current Density. Derive Equation of Continuity.
In quantum mechanics, particles are described by wavefunctions (\(\Psi\)). The probability of finding a particle in a certain region \(\Delta V\) at a particular time \(t\) is given by \(|\Psi|^2\), where \(|\Psi|^2\) represents the probability density.
The probability current density (\(J\)) is a vector quantity that describes the rate of flow of probability per unit area per unit time. It is denoted by \(\mathbf{J}\) and is given by the following expression:
\[ \mathbf{J} = \frac{\hbar}{2mi} \left( \Psi^* \nabla \Psi - \Psi \nabla \Psi^* \right) \]
Where:
- \(\hbar\) is the reduced Planck's constant (\(\hbar \approx 1.054571 \times 10^{-34}\) J·s).
- \(m\) is the mass of the particle.
- \(\nabla\) represents the gradient operator.
- \(\Psi^*\) is the complex conjugate of the wavefunction \(\Psi\).
The probability current density helps us understand how the probability distribution of a quantum system evolves with time and how particles move in quantum mechanics.
The Equation of Continuity is a fundamental principle in physics that states that the total amount of a conserved quantity within a given volume can only change if there is a net flow of that quantity into or out of the volume. It is commonly used in fluid dynamics, electromagnetism, and quantum mechanics.
In the context of quantum mechanics, the equation of continuity connects the change in probability density (\(|\Psi|^2\)) with the probability current density (\(\mathbf{J}\)).
The equation of continuity in quantum mechanics is expressed as:
\[ \frac{\partial |\Psi|^2}{\partial t} + \nabla \cdot \mathbf{J} = 0 \]
Where:
- \(\frac{\partial |\Psi|^2}{\partial t}\) is the time rate of change of the probability density.
- \(\nabla \cdot \mathbf{J}\) is the divergence of the probability current density.
Now, let's derive this equation using the probability current density (\(\mathbf{J}\)) mentioned above:
We start with the time-dependent Schrödinger equation:
\[ i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \Psi + V \Psi \]
where:
- \(\Psi\) is the wavefunction.
- \(V\) is the potential energy.
Now, let's take the complex conjugate of the Schrödinger equation:
\[ -i\hbar \frac{\partial \Psi^*}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \Psi^* + V \Psi^* \]
Next, let's multiply the first equation by \(\Psi^*\) and the second equation by \(\Psi\):
\[ \Psi^* \left( i\hbar \frac{\partial \Psi}{\partial t} \right) = -\frac{\hbar^2}{2m} \Psi^* \nabla^2 \Psi + V |\Psi|^2 \]
\[ \Psi \left( -i\hbar \frac{\partial \Psi^*}{\partial t} \right) = -\frac{\hbar^2}{2m} \Psi \nabla^2 \Psi^* + V |\Psi|^2 \]
Now, subtract the second equation from the first:
\[ i\hbar \left( \Psi^* \frac{\partial \Psi}{\partial t} - \Psi \frac{\partial \Psi^*}{\partial t} \right) = -\frac{\hbar^2}{2m} \left( \Psi^* \nabla^2 \Psi - \Psi \nabla^2 \Psi^* \right) \]
Now, we can express this in terms of the probability current density \(\mathbf{J}\):
\[ i\hbar \frac{\partial |\Psi|^2}{\partial t} = -\frac{\hbar^2}{2m} \nabla \cdot \mathbf{J} \]
Finally, rearrange the equation:
\[ \frac{\partial |\Psi|^2}{\partial t} = -\frac{1}{i\hbar} \frac{\hbar^2}{2m} \nabla \cdot \mathbf{J} \]
Remember that \(i^2 = -1\), so:
\[ \frac{\partial |\Psi|^2}{\partial t} = \frac{\hbar}{2mi} \nabla \cdot \mathbf{J} \]
And there we have the equation of continuity for quantum mechanics! This equation tells us that the rate of change of probability density in a region is equal to the negative divergence of the probability current density, ensuring that probability is conserved in a quantum system.
No comments: