Time-Independent Schrödinger Equation

The full time-dependent Schrödinger equation (TDSE) for a particle in a time-independent potential $V(x)$ reads

i\hbar \frac{\partial \Psi(x,t)}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \Psi(x,t)}{\partial x^2} + V(x)\,\Psi(x,t).

When the potential does not depend on time, there is a powerful simplification: separation of variables. Assume the wave function factors as

\Psi(x,t) = \psi(x)\,\varphi(t).

Substituting and dividing both sides by $\psi(x)\,\varphi(t)$ :

i\hbar \frac{1}{\varphi}\frac{d\varphi}{dt} = \frac{1}{\psi}\left[-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\,\psi\right].

The left side depends only on $t$ ; the right side depends only on $x$ . Because they are equal for all $x$ and $t$ , each side must equal the same constant. Calling that constant $E$ (its physical meaning will emerge immediately), the equation splits into two independent ordinary differential equations.

The time equation

Setting the left side equal to $E$ :

i\hbar \frac{d\varphi}{dt} = E\,\varphi.

This first-order ODE has the solution

\varphi(t) = e^{-iEt/\hbar}.

The full wave function therefore has the form $\Psi(x,t) = \psi(x)\,e^{-iEt/\hbar}$ . The time-dependent factor is a pure oscillation at angular frequency $\omega = E/\hbar$ , consistent with the de Broglie–Planck relation $E = \hbar\omega$ .

The spatial equation: the TISE

Setting the right side equal to $E$ gives the time-independent Schrödinger equation (TISE):

-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\,\psi(x) = E\,\psi(x).

In operator notation this is simply

\hat{H}\,\psi = E\,\psi,

where $\hat{H} = -\hbar^2 \partial^2/(2m\,\partial x^2) + V(x)$ is the Hamiltonian. This is an eigenvalue equation: $\psi(x)$ is an eigenfunction of $\hat{H}$ and $E$ is the corresponding eigenvalue. The constant $E$ is therefore the total energy of the state, which is how the separation constant acquires its name.

Why the separation is exact

The factored ansatz $\Psi = \psi(x)\,\varphi(t)$ is only consistent when $V$ has no explicit time dependence. If $V$ depended on $t$ , the right side of the separated equation would also depend on $t$ , and the argument that each side must be a constant would fail. For time-dependent potentials one must work with the full TDSE directly.

Stationary states and probability densities

A solution of the TISE produces a stationary state. Notice that the time dependence of the full wave function $\Psi(x,t) = \psi(x)\,e^{-iEt/\hbar}$ appears only as a phase factor. When computing the probability density,

|\Psi(x,t)|^2 = \psi^*(x)\,e^{+iEt/\hbar}\cdot\psi(x)\,e^{-iEt/\hbar} = |\psi(x)|^2,

the oscillating phase cancels and the result is independent of time. Expectation values of time-independent observables are likewise constant — nothing you can measure from outside the system changes. The state is called "stationary" in exactly this sense: the probability distribution is frozen even though the wave function is still oscillating.

General solutions via superposition

The TISE typically yields a discrete (or continuous) set of eigenfunctions $\psi_n(x)$ with energies $E_n$ . Because the TDSE is linear, any linear combination of solutions is also a solution. The general wave function is

\Psi(x,t) = \sum_n c_n\,\psi_n(x)\,e^{-iE_n t/\hbar},

where the coefficients $c_n$ are set by the initial condition $\Psi(x,0)$ . When the system is in a superposition of two or more energy eigenstates with different energies, the cross-terms in $|\Psi|^2$ oscillate at frequencies $\omega_{mn} = (E_m - E_n)/\hbar$ — these are the quantum beating frequencies that determine spectral lines.