Question

I'm currently reading "The principles of quantum mechanics" by Dirac, and I'm having some trouble understanding some of his assumptions, because in the quantum mechanics course I'm following at university we are taking a different approach (a more modern one I guess).

I got to the point where Dirac explains why he decides to define the average value of the observable ξ in the state |x⟩ as ⟨x|ξ|x⟩. What I don't get is the following part, where he writes that the probability Pa of ξ having the value a is given by the average value of the Kronecker delta δξ a Pa=⟨x|δξa|x⟩

I don't understand if he's deciding to define the probability of ξ
having the value a in this way, or if he's deducing this formula.

In the following similar question: Why does the probability of obtaining a value of a measurement follow from Dirac's general assumption? By Symmetry writes:

The Projection Operator δεa is 1 for some particular |a⟩ and is 0 on any orthogonal state. The definition of the expectation in classical probability theory for this operator is given by⟨δεa⟩=1Pr(|x⟩=|a⟩)+0Pr(|x⟩⊥|a⟩)=Pr(|x⟩=|a⟩)
So, from this answer it seems to me that he's saying that Pa=⟨x|δξa|x⟩
by definition, but from Dirac's book I have the impression that he's deducing it from previous assumptions.

So my question is: is the formula for the probability of ξ having value a in the state |x⟩ just a definition or can it be deduced by previous assumptions in Dirac's book? If he's deducing it, how is he doing it?

Answer 1

The probability Pa of an observable having a certain value in quantum mechanics is derived from foundational principles such as the interpretation of wave functions and measurement operators. The expression ⟩x|δξa|x⟩ reflects the average probability of an eigenstate using Born's rule and the properties of the wave function.

Step-by-step explanation:

In Dirac's book on quantum mechanics, the probability Pa of an observable ξ having the value a in state |x⟩ is expressed as ⟩x| δξa |x⟩. Whether this is a definition or a deduction from prior assumptions is a subtle question. Dirac uses a theoretical framework where observables are represented by operators, and the measurement outcomes are eigenvalues of these operators.

The projection operator δξa acts as the mathematical representation of the measurement process, being 1 for the state corresponding to the eigenvalue a and 0 otherwise. Integrating this with Born's interpretation of the wave function, the outcome ⟩x| δξa |x⟩ represents the average probability for finding the system in the eigenstate corresponding to a.

Therefore, the expression for Pa can be seen as derived from the foundational postulates of quantum mechanics rather than a mere definition. It is a consequence of the properties of wave functions, the role of operators in quantum mechanics, and the fundamental probabilistic nature of quantum mechanical measurements as highlighted by Born's rule.