This is a fundamental equation in quantum 力学 that describes how the quantum state of a physical system changes over time. It is a linear partial differential equation for the wavefunction, [latex]\Psi(x, t)[/latex]. The time-dependent version is [latex]i\hbar\frac{\partial}{\partial t}\Psi = \hat{H}\Psi[/latex], where [latex]\hat{H}[/latex] is the Hamiltonian operator, representing the total energy of the system.
The Schrödinger equation is the quantum mechanical counterpart to Newton’s second law in classical mechanics. While Newton’s law predicts the trajectory of a particle, the Schrödinger equation predicts the future behavior of a system’s wavefunction. The wavefunction, [latex]\Psi[/latex], is a complex-valued probability amplitude, and the square of its magnitude, [latex]|\Psi|^2[/latex], gives the probability density of finding the particle at a given position and time. The equation comes in two main forms: time-dependent and time-independent.
The time-dependent Schrödinger equation (TDSE), [latex]i\hbar\frac{\partial}{\partial t}\Psi(x, t) = \hat{H}\Psi(x, t)[/latex], describes a system evolving in time. The time-independent Schrödinger equation (TISE), [latex]\hat{H}\Psi(x) = E\Psi(x)[/latex], is used for systems in a stationary state, where the energy [latex]E[/latex] is constant. Solving the TISE for a given potential yields the allowed energy eigenvalues ([latex]E[/latex]) and the corresponding energy eigenfunctions ([latex]\Psi[/latex]), which represent the stable states of the system, such as the electron orbitals in an atom. The Hamiltonian operator [latex]\hat{H}[/latex] is constructed from the classical expression for the total energy (kinetic plus potential) by replacing classical variables with their corresponding quantum operators. For a single non-relativistic particle, [latex]\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V(x, t)[/latex].
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