The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. There are two forms of the Schrödinger equation: time-dependent and time-independent. I’ll provide a brief derivation of the time-dependent Schrödinger equation.

The time-dependent Schrödinger equation is given by: \[i\hbar \frac{\partial \Psi}{\partial t} = H \Psi,\] where: – \(i\) is the imaginary unit, – \(\hbar\) is the reduced Planck constant, – \(\Psi\) is the quantum state of the system (wave function), – \(t\) is time, – \(H\) is the Hamiltonian operator. The Hamiltonian operator is defined as: \[H = -\frac{\hbar^2}{2m} \nabla^2 + V,\] where: – \(m\) is the mass of the particle, – \(\nabla^2\) is the Laplace operator, – \(V\) is the potential energy. Now, let’s derive the time-dependent Schrödinger equation: 1. Start with the de Broglie relation for a particle with momentum \(p\): \[p = \hbar k,\] where \(k\) is the wave number. 2. The total energy \(E\) is the sum of kinetic and potential energies: \[E = \frac{p^2}{2m} + V.\] 3. Substitute the de Broglie relation into the expression for total energy: \[E = \frac{\hbar^2 k^2}{2m} + V.\] 4. The total energy operator \(\hat{E}\) is obtained by replacing \(k\) with the spatial derivative operator \(\nabla\): \[\hat{E} = -\frac{\hbar^2}{2m} \nabla^2 + V.\] 5. Apply the Schrödinger equation by replacing \(\hat{E}\) with \(i\hbar \frac{\partial}{\partial t}\) and rearrange to obtain the time-dependent Schrödinger equation: \[i\hbar \frac{\partial \Psi}{\partial t} = \hat{E} \Psi.\] This derivation provides the basis for the time-dependent Schrödinger equation, which describes the evolution of quantum states over time in terms of the Hamiltonian operator.