Define each of the following in terms of the state I> and, where appropriate, an

Question

Define each of the following in terms of the state I> and, where appropriate, an operator  with (non-degenerate) eigenstates I ai > corresponding to eigenvalues ai , using the Dirac notation:
i) the wavefunction (x) corresponding to I >
ii) the expectation value of  for state I >
iii) the equation giving the eigenvalues ai , in terms of  and I ai >
iv) orthonormality of I am > and I an >
v) I > in terms of the complete set of eigenstates, I ai >
vi) the probability of measuring an when in state I  >

vii) the Hermiticity of Â, expressed in terms of two general states of the system described by I  > and I  >

Explanation / Answer

i) the wavefunction (x) corresponding to I >
            <x|>=(x)

ii) the expectation value of for state I > is

< |A| >

iii)  the equation giving the eigenvalues ai , in terms of and I ai > is

                I ai > =aiI ai >

iv)  orthonormality of I am > and I an >
       <am|an > = mn

vi) the probability of measuring an when in state I >

          P=< |an| >   

vii)    the Hermiticity of , expressed in terms of two general states of the system described by I > and I > is

          < | I >=< | I >*

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