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Representing a vector in terms of eigenkets in continuos basis(stuck here,guys help)
I was reading Dirac and there is this formula which bothers me,
[tex]|P>= \int{|\right{\xi'd}\rangle{d\xi'}} + \sum{|\right{\xi^{r}b}\rangle}[/tex]
Where [tex]|\right\xi'\rangle[/tex] denotes the eigenket corresponding to the eigenvalue [tex]\xi'[/tex]
Now the integral term is bothering me.
Does it mean, that the integral is summing over the entire range of eigenvalues? But than what is the use of the summation term?
And if it is so than there is a contradiction cause a few pages later Dirac says that "If we have an eigenket [tex]|\right\xi'\rangle[/tex] dependent on other eigenkets of xi' then these other eigenkets must belong to the same eigenvalue xi' " What does this mean?? He has given a proof, i am able to understand it. But the start point of the proof is bothering me. From what i get is that in the integral term the |xi'> denotes some specific eigenket integrated over the whole range of eigenvalues. But the why that specific eigenket and not some other eigenket let say |xi''>
And eigenkets and eigenvalues go hand in hand, don't they?? As in you can't define let say some eigenket xi' without metioning the eigenvalue xi'.
I was reading Dirac and there is this formula which bothers me,
[tex]|P>= \int{|\right{\xi'd}\rangle{d\xi'}} + \sum{|\right{\xi^{r}b}\rangle}[/tex]
Where [tex]|\right\xi'\rangle[/tex] denotes the eigenket corresponding to the eigenvalue [tex]\xi'[/tex]
Now the integral term is bothering me.
Does it mean, that the integral is summing over the entire range of eigenvalues? But than what is the use of the summation term?
And if it is so than there is a contradiction cause a few pages later Dirac says that "If we have an eigenket [tex]|\right\xi'\rangle[/tex] dependent on other eigenkets of xi' then these other eigenkets must belong to the same eigenvalue xi' " What does this mean?? He has given a proof, i am able to understand it. But the start point of the proof is bothering me. From what i get is that in the integral term the |xi'> denotes some specific eigenket integrated over the whole range of eigenvalues. But the why that specific eigenket and not some other eigenket let say |xi''>
And eigenkets and eigenvalues go hand in hand, don't they?? As in you can't define let say some eigenket xi' without metioning the eigenvalue xi'.
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