The linearity of the Schrodinger equation is one of its most important features. It is the mathematical basis for the principle of superposition in quantum mechanics, which allows for phenomena like interference of wave functions.
Step 1: Write down the general form of the time-dependent Schrodinger equation.
\[
i\hbar \frac{\partial \Psi(\mathbf{r}, t)}{\partial t} = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r}, t) \right] \Psi(\mathbf{r}, t)
\]
where \(\Psi\) is the wave function.
Step 2: Analyze the properties of the equation.
- **Linearity:** The equation is linear because the wave function \(\Psi\) and its derivatives appear only to the first power. There are no terms like \(\Psi^2\) or \(\Psi \frac{\partial \Psi}{\partial t}\). This means that if \(\Psi_1\) and \(\Psi_2\) are solutions, then any linear combination \(c_1\Psi_1 + c_2\Psi_2\) is also a solution. This is the principle of superposition.
- **Order in time:** The equation contains a first derivative with respect to time, \(\frac{\partial}{\partial t}\). It is first-order in time.
- **Order in space:** The equation contains the Laplacian operator, \(\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\), which involves second derivatives with respect to spatial coordinates. It is second-order in space.
Step 3: Evaluate the given options.
1. non-linear differential equation: Incorrect.
2. linear differential equation: Correct.
3. second order equation in time: Incorrect. It is first-order in time.
4. first order equation in space: Incorrect. It is second-order in space.
