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I'm trying to understand what Braket state vectors are... and I'm not sure if I completely understand what's going on. Could someone clear this up? Here's what I (think) I understand now.

A ket is pretty much a state vector (written with different notation). A state vector is pretty much a vector corresponding to a particular observable. It has everything you need to know about the system. If you want to find (statistical) information regarding a particle, suppose the total energy, then you have to solve the eigenvalue equation

[tex]\hat{H} \psi = E \psi[/tex]

where [tex]\hat{H}[/tex] represents the operator regarding the observable and [tex]E[/tex] represents the eigenvalue. Here, [tex]E[/tex] is the mean value observed for the observable (in this case, the mean energy observed).

What I am completely missing is the concept of adding together state vectors... how exactly does this work? And most importantly, what does this mean? What can we do with them? I'm really sorry if I don't understand some of what seem to be the easier concepts, but I learned everything that I know from online sources only. :uhh:

In addition, I am sort of unsure as to how the entire complex number idea works with braket state vectors... am I understanding correctly that ket state vectors (and bra vectors) are in some N-dimensional (N isn't necessarily finite o.0?) complex vector space? If this is the case, what is the use of having bra vector if we already have ket vectors, which are just the conjugate transposes of bra vectors? I guess that they are used for inner and outer products, but could someone give me an actual example or a link to an introduction example where these are actually used to solve problems?

A ket is pretty much a state vector (written with different notation). A state vector is pretty much a vector corresponding to a particular observable. It has everything you need to know about the system. If you want to find (statistical) information regarding a particle, suppose the total energy, then you have to solve the eigenvalue equation

[tex]\hat{H} \psi = E \psi[/tex]

where [tex]\hat{H}[/tex] represents the operator regarding the observable and [tex]E[/tex] represents the eigenvalue. Here, [tex]E[/tex] is the mean value observed for the observable (in this case, the mean energy observed).

What I am completely missing is the concept of adding together state vectors... how exactly does this work? And most importantly, what does this mean? What can we do with them? I'm really sorry if I don't understand some of what seem to be the easier concepts, but I learned everything that I know from online sources only. :uhh:

In addition, I am sort of unsure as to how the entire complex number idea works with braket state vectors... am I understanding correctly that ket state vectors (and bra vectors) are in some N-dimensional (N isn't necessarily finite o.0?) complex vector space? If this is the case, what is the use of having bra vector if we already have ket vectors, which are just the conjugate transposes of bra vectors? I guess that they are used for inner and outer products, but could someone give me an actual example or a link to an introduction example where these are actually used to solve problems?

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