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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?

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    $\begingroup$ Try a very basic web search for "Dirac bra-ket notation" will answer your query. In fact as written the ket in your SE should really be $\vert\Psi(t)\rangle$ since the $\Psi(r,t)$ is actually a function in the position representation rather than a ket. $\endgroup$ – ZeroTheHero Mar 10 at 17:54
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    $\begingroup$ $|\Psi\rangle$ denotes a vector in a Hilbert space. $\endgroup$ – G. Smith Mar 10 at 18:02
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    $\begingroup$ How would you do "research effort" in any other way than by asking such a thing? $\endgroup$ – Pieter Mar 10 at 19:41
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    $\begingroup$ I strongly disagree with this closure - it is in no way a fit for the criteria that make questions off-topic as homework-like. As for the claim that this is lacking research effort, if OP has misread the notation to the extent shown here, no amount of googling will help, and there's a whole lot of QM textbook to get through before they would get to material that would help them understand. Moreover, it's not our place to be "surprised" at newcomers' basic questions (CC @Zero) - our role is to answer those questions. We were all confused beginners once. This is a perfectly legitimate question. $\endgroup$ – Emilio Pisanty Mar 11 at 10:30
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    $\begingroup$ @GiorgioP There is a robust tradition of voting to reopen when the consensus on closing is not so strong. $\endgroup$ – ZeroTheHero Mar 11 at 17:13
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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.

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  • $\begingroup$ Succinct and clear. Explains bra-ket notation well. The best explanation I've seen TBH. $\endgroup$ – ohwilleke Mar 10 at 19:54

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