I sligthly confused about Dirac notation. As of now I always thought that
$$ |ψ,t⟩ = |ψ(x),t⟩ = |ψ(x,t)⟩ . $$
However, now I found out that
$$⟨x|ψ,t⟩=ψ(x,t).$$
What does this notation actually mean?
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What does this notation actually mean?
This is a ket labelled by $\psi$:
$$|\psi\rangle$$
This is a ket valued function of the time parameter $t$ labelled by $\psi(t)$
$$|\psi(t)\rangle$$
that returns a ket given a value of $t$.
The contraction of a bra and ket is a complex number
$$\langle \psi_1|\psi_2\rangle = c_{12}$$
The contraction of a bra and a ket valued function of time is complex valued function of time:
$$\langle \alpha|\psi(t)\rangle = \psi_\alpha(t)$$
Consider the ket valued function of the coordinate $x$
$$|x\rangle$$
which for a given $x$ coordinate, returns the eigenket of the position observable $\hat X$ with eigenvalue $x$
$$\hat X|x\rangle = x|x\rangle\,\quad \langle x |\hat X = x\langle x |$$
Then the contraction of the ket valued function of $t$, $|\psi(t)\rangle$, and the bra valued function of $x$, $\langle x|$, is a complex valued function of $x$ and $t$
$$\langle x|\psi(t)\rangle = \psi(x,t)$$
which is known as the (coordinate space) wavefunction.
I'm not sure what to make of something like $|\psi(x,t)\rangle$ unless, in this case, $x$ is considered a parameter like $t$
Hal Hollis
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