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In Quantum Mechanics are there any difference between $\psi$ and $\Psi$ the two describing the wave function? From intuition I would think that $\psi$ only depends on $x$ (position) while $\Psi$ depends on both $x$ and time.

I have seen both: $\Psi(x)$, $\Psi(x,t)$, $\psi(x)$ and $\psi(x,t)$

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    $\begingroup$ It's quite strange to ask whether there's any convention when in your last sentence you point out yourself that there's no strict convention =) $\endgroup$
    – OON
    Dec 12, 2017 at 12:09
  • $\begingroup$ I have seen capital psi used for an operator valued distribution (of spacetime), i.e. a quantum field but I'm not aware of any consistent convention. Could you give examples of texts / papers where both are used and you think there might be some consistent usage? $\endgroup$ Dec 12, 2017 at 12:12
  • $\begingroup$ A common use is capital for a LC of base functions $\endgroup$
    – Alchimista
    Dec 12, 2017 at 12:21
  • $\begingroup$ Another convention would be to denote by $\psi $ the Weyl spinor and by $\Psi$ the Dirac one. $\endgroup$
    – DanielC
    Dec 12, 2017 at 13:13
  • $\begingroup$ Most commonly Greek capital $\Psi$ is for the full time-dependent but that’s not completely uniform. $\endgroup$ Dec 12, 2017 at 13:25

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No, it is just a matter of convention, which heavily depends on the context.

As an example, in basic quantum mechanics, when dealing with single- and many-particle states, $\psi$ tends to be used for the state of single particles while $\Psi$ for many-particle states. But this is really not a rule, and this notation is not consistently followed, so you just have to see in every given text what kind of notation and conventions are being followed.

Again, both $\psi$ and $\Psi$ can be dependent on $x$, $t$, both, or none, depending on the context and the meaning given to the symbols.

It really is just notation.

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This is just notation, and depending on the author and context it might mean quite different things.

In Quantum Field Theory, for example, $\Psi$ is usually used for the quantum field of a fermion, like the electron.

Now, even though it highly depends on the source and context, from what you describe on the question I think the convention in place (which I also have seem in some texts) is:

Assuming $\Psi$ is $\Psi(x,t)$ and $\psi$ is just $\psi(x)$, the difference between them is: $\psi$ is one certain possible quantum state in the so-called position representation.

It is directly the probability amplitude for position, so that $|\psi(x)|^2$ is the probability density that the particle is located near $x$ and it encodes all information about the system, if the system is at said state.

Now, while $\psi$ is just one possible quantum state, $\Psi(x,t)$ is actualy the time evolution of the system. For $t$ fixed, $\Psi(x,t)$ is exactly the "$\psi$" at time $t$, in other words, the quantum state at that time.

It is the solution to Schrodinger's time evolution equation

$$i\hbar \dfrac{\partial\Psi}{\partial t}=H\Psi,$$

with some initial state condition $\Psi(x,0)=\psi_0(x)$.

But again, this is just one possible notation which I've detailed. It certainly could mean something else in another context, written by another author.

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  • $\begingroup$ The $\Psi$ being time-dependent and the $\psi$ being time-independent is safely to be thought of as the "Messiah convention" (A. Messiah first published his treatise in French in 1958, subsequently translated to English in 1961). But it is also found in the 1935 work "Introduction to Quantum Mechanics" by Linus Pauling and E. Wilson and earlier in the book by Condon and Morse "Quantum Mechanics" in 1929. $\endgroup$
    – DanielC
    Dec 13, 2017 at 1:36

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