Hi there!(adsbygoogle = window.adsbygoogle || []).push({});

As you might have already guessed, I'm referring primarily to the 'geometrical' difference (is there such geometry in Hilbert space?) between ##n##-dimensional state vectors

[tex] | \psi \rangle = \left( \begin{matrix} \psi_1 \\ \psi_2 \\ \vdots \\ \psi_n \end{matrix} \right) [/tex]

and their corresponding basis vectors

[tex] \langle \psi | = \left( \begin{matrix} \psi_1^* & \psi_2^* & \cdots & \psi_n^* \end{matrix} \right). [/tex]

What would these (particularly the latter) look like on a graph? (What kind of graph could one even represent these on?)

Additionally, if the inner product of two vector spaces represents the projection of one vector onto another, the inner product of these two vector spaces would equal 1, meaning they are parallel (which is true). But what does that mean for two non-equal states?For example,

[tex] \langle \phi | \psi \rangle [/tex]

where ##|\phi\rangle \ne |\psi\rangle##. How does this inner product represent the probability amplitude of the wavefunction from two separate states? I've been imagining this as an ##n##-dimensional generalisation of the dot (scalar) product of two vectors thus far!

I'm looking mainly for conceptualisations/visualisations (personal interpretations invited) of how this process works but also corrections on where I'm misunderstanding the theory.

Thanks in advance,

Harry

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# Visualising the Conjugate Transposition of a Vector

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