Use qualitative arguments and construct the Schroedinger equation in one dimension for a particle subjected to a potential V(x,t).

1. Start with the time-dependent Schroedinger equation for a single particle in one dimension:
i \hbar \frac{\partial \psi(x,t)}{\partial t} = \hat{H} \psi(x,t)

2. Define the Hamiltonian operator \( \hat{H} \) as the sum of the kinetic and potential energy operators:
\hat{H} = \hat{T} + \hat{V}
where the kinetic energy operator is:
\hat{T} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2}
and the potential energy operator is represented by \( V(x,t) \):
\hat{V} = V(x,t)

3. Substitute the Hamiltonian operator into the time-dependent Schroedinger equation:
i \hbar \frac{\partial \psi(x,t)}{\partial t} = \left( -\frac{\hbar^2}{2m} \frac{\partial^2 \psi(x,t)}{\partial x^2} + V(x,t) \psi(x,t) \right)

4. This yields the final form of the Schroedinger equation:
i \hbar \frac{\partial \psi(x,t)}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi(x,t)}{\partial x^2} + V(x,t) \psi(x,t)

The answer is:
i \hbar \frac{\partial \psi(x,t)}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi(x,t)}{\partial x^2} + V(x,t) \psi(x,t)