What does $|$ mean in the Schrödinger Equation? I saw the $|$ symbol in the Schrödinger Equation $$i\hbar\frac{\partial}{\partial{t}}|\Psi(r,t)\rangle=\hat{H}|\Psi(r,t)\rangle$$  But I don't know what the $|$ means.
What does $|$ mean in the Schrödinger Equation?
 A: The vertical bar means nothing by itself. The notation $|\psi\rangle$ is called a “ket” and indicates a vector in a Hilbert space, representing some quantum state. The corresponding notation $\langle\psi|$ is called a “bra” and denotes the Hermitian conjugate. Scalar products between two vectors are written as $\langle\phi|\psi\rangle$, with only one vertical bar.
Sometimes inside the bra or the ket you just see quantum numbers labeling the quantum state. For example, for the states of a hydrogen atom you may see $|n,l,m\rangle$ instead of $|\psi_{nlm}\rangle$, or for a spin state you may see just $|+\rangle$ or $|\uparrow\,\rangle$ to indicate "spin up". The idea is that you put inside whatever is sufficient to specify the quantum state.
Since bras and kets are vectors, you can operate on them with operators in Hilbert space, leading to notation like $\hat{H}|\psi\rangle$. When you take this vector-operated-on-by-an-operator and take its scalar product with another vector, you write it as $\langle\phi|\hat{H}|\psi\rangle$.
The Schrodinger equation tells you how the quantum state evolves, as a vector moving in Hilbert space that points in different directions at different points in time.
The bra-ket notation was invented by Dirac and is described in more detail in this Wikipedia article.
