Orthonormality
Considering relations 1.14, 1.21 and 1.23 then follows the two important properties for the probability amplitude:
1. Letting |a) = |2,) in 1.14 and considering 1.23 then the normalization condition:
From the normalization condition, 1.24 is rewritten as
The asterisk denotes the complex conjugate and from this equation
Both sums are equal for any 2 and so follows the reciprocity relation:
This enable us to have
This implies that the distribution over 2 for given fi coincides with the distribution over pi for given 2. For the probability interpretation to be feasible it is necessary that the following completeness relation be satisfied:
In the state|2j), the observable physical quantity does not have a defined value. We can only talk about its probability P(2, —> /r„):
This is because the probability of one of the independent events is the sum of probabilities.
2. Suppose now in 1.14 we let |a) = |24) for i ф к then from 1.23 follows
From the orthogonality condition 1.23 of the vectors say, |24) and ||/„) then
We swap A and // with their indices in this equation and we have
The orthonormality condition 1.24 permits to rewrite 1.32 and 1.33 in a compact form. So, a set of vectors is orthonormal if they are mutually orthogonal and are each normalized:
These last equations permit us to confirm the superposition principle in 1.14.
Orthogonality
We consider the distribution of some physical measurement over X where we select some arbitrary probability amplitudes (i,| a), (i2| a) and so on satisfying the following normalization condition:
Here, a is some state. From equation 1.21 results two important corollaries from the definition of the probability amplitude:
which may be multiplied by an arbitrary phase factor expj/r/;} that we neglect. This relation in 1.22 can be rewritten
where Sik is the symmetric Kronecker ^-symbol. Property 1.23 expresses the orthogonality of the basis vectors |2,)and |/4) for i ф k The eigenbasis states with different energies are orthogonal. It is instructive to note that two vectors are orthogonal or perpendicular if their inner product vanishes. This implies that their projections onto each other vanish.
