# Notation in Quantum Mechanics

When we write equations in QM, in certain places, the wave function is represented as $\psi(x,t)$, which is the wave function in position space, and in some other places, it is written as $\Psi(t)$. What is the difference between the two?

There is no significance in the choice between upper- and lower-case $\psi$ (or $\Psi$) to denote a system's wavefunction. The two are used interchangeably and it is the author's discretion to use either symbol. (On the other hand, of course, one shouldn't use the two symbols interchangeably within the same text; if both are used they would refer to different objects.)
There is also little to say, in general, about whether the position $x$ is included as an argument to the wavefunction or not. Some authors may, for example, choose to denote by $\Psi(t)\in\mathcal H$ the Hilbert space state vector, which in the position representation is given by the function $x\mapsto\psi(x,t)$, but this is relatively rare. In general this is a case-by-case matter, but most of the modern literature uses Dirac notation, in which the state vector $|\psi\rangle\in\mathcal H$ and its wavefunction $\langle x|\psi\rangle=\psi(x,t)\in\mathbb C$ are distinct objects, shown emphatically distinct by the notation.
Finally, these uses of notation are usually a source of confusion when the wavefunction is split into spatial and temporal parts, such as when setting up a separation of variables scheme for solving the TDSE via the TISE. In such cases, one postulates that the wavefunction $\Psi(x,t)$ of a particle takes the form $$\Psi(x,t)=\psi(x)e^{-i E t/\hbar},$$ in which the temporal and spatial dependences are separated; here if $\psi$ satisfies the TISE then $\Psi$ will satisfy the TDSE. In such situations it's important to keep in mind that $\psi$, whilst a convenient calculational tool, does not represent the state of the system: $\Psi$ does. Beware of notation in such situations, and always check what definition each symbol has been given in the particular text you're reading.