How is Schrödinger's time-dependent wave equation derived and what are the energy and momentum operators?

In one line

Differentiating the free-particle wave function \(\Psi = Ae^{-i(Et - px)/\hbar}\) gives the operators \(E \to i\hbar\,\partial/\partial t\) and \(p \to -i\hbar\,\partial/\partial x\); putting them into \(E = p^2/2m + V\) gives \(i\hbar\,\partial\Psi/\partial t = -\frac{\hbar^2}{2m}\partial^2\Psi/\partial x^2 + V\Psi\).

Explained simply

The time-independent equation handles states of fixed energy. To see how a wave function changes with time (for example, a wave packet moving), we need an equation with time in it. The idea: write a matter wave in terms of energy and momentum, notice that taking a time derivative "pulls out" \(E\) and a space derivative "pulls out" \(p\), then plug these into the energy equation.

How it works

1. A free particle wave: \(\Psi = Ae^{-i(\omega t - kx)}\). Using \(E = \hbar\omega\) and \(p = \hbar k\): \(\Psi = Ae^{-i(Et - px)/\hbar}\).
2. Differentiate with time: \(\dfrac{\partial\Psi}{\partial t} = -\dfrac{iE}{\hbar}\Psi\), so \(E\Psi = i\hbar\dfrac{\partial\Psi}{\partial t}\).
3. Differentiate twice with \(x\): \(\dfrac{\partial^2\Psi}{\partial x^2} = -\dfrac{p^2}{\hbar^2}\Psi\), so \(p^2\Psi = -\hbar^2\dfrac{\partial^2\Psi}{\partial x^2}\).
4. Total energy: \(E = \dfrac{p^2}{2m} + V\). Multiply by \(\Psi\): \(E\Psi = \dfrac{p^2\Psi}{2m} + V\Psi\).
5. Substitute: \(i\hbar\dfrac{\partial\Psi}{\partial t} = -\dfrac{\hbar^2}{2m}\dfrac{\partial^2\Psi}{\partial x^2} + V\Psi\).
6. In short, \(\hat{H}\Psi = i\hbar\,\partial\Psi/\partial t\), where \(\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V\) is the Hamiltonian operator.
7. If \(V\) does not depend on time, \(\Psi = \psi(x)e^{-iEt/\hbar}\) separates it, and the equation reduces to the time-independent form.

Key formula

\(i\hbar\dfrac{\partial\Psi}{\partial t} = -\dfrac{\hbar^2}{2m}\dfrac{\partial^2\Psi}{\partial x^2} + V\Psi\). Operators: \(\hat{E} = i\hbar\,\partial/\partial t\), \(\hat{p} = -i\hbar\,\partial/\partial x\). \(\Psi(x,t)\): full wave function, \(\hbar = 1.055\times10^{-34}\) J s, \(i = \sqrt{-1}\).

Quick example

Check with \(\Psi = Ae^{-i(Et - px)/\hbar}\) and \(V = 0\): the left side gives \(E\Psi\), the right side gives \((p^2/2m)\Psi\). They are equal because, for a free particle, \(E = p^2/2m\). So the equation correctly reproduces the energy of a free particle.

Common mistakes

• Sign errors: the momentum operator is \(-i\hbar\,\partial/\partial x\), the energy operator is \(+i\hbar\,\partial/\partial t\).
• Forgetting the \(i\) on the left side; the equation is complex by nature.
• Writing \(\hbar^2/m\) instead of \(\hbar^2/2m\).

How MU asks it

Asked as "Derive Schrödinger's time-dependent wave equation" (5 marks), sometimes "Obtain the time-independent equation from the time-dependent one" (3 to 5 marks). Show each differentiation, identify the operators, substitute into the energy relation, and box the result.