Let's Discuss The Postulates :: Consistency of Probability Interpretation

We will now discuss the postulates of quantum mechanics.

I want to discuss the conditions under which the probability interpretation will be consistent.

Let \(A\) be a dynamical variable and the corresponding operator be denoted by \(\hat{A}\).
Let the eigen values and eigen vectors be denoted by \(\alpha_k\) and \(\ket{\alpha_k}\) respectively.
\begin{equation}
\hat{A} \, \ket{\alpha_k} = \alpha_k \ket{\alpha_k}.
\end{equation}

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If a system is known to be in state with value \(\alpha_1\) of \(A\) , what is the probability that a measurement of \(A\) will give some other value \(\alpha_2 \ne \alpha_1\)?

ANSWER ZERO:

Now let us ask :
The Answer is:

What guarantees that theory will always give the answer you expect?

It is given that a system is in state \(\ket{\psi}\).
In order to find the probabilities of different possible outcomes \(\alpha_n\) on measurement of \(A\), one needs to expand the vector \(\ket{\psi}\) and write it as superposition of eigenvectors \(\hat{A}\).

Let us ask:
The Answer is:

What ensures that such every vector can be written as linear combination of eigen vectors of \(\hat{A}\)?

The postulates give the prescription of computing the probabilities of different outcomes of measurement of \(A\).

Let us ask:
The Answer is:

That sum of all probabilities is one means \(\sum_n |c_n|^2 = |<\psi|\psi>|^2\).
How do we see this property?

If a system is in a state \(\ket{\psi}\) which satisfies \(|\innerproduct{\psi}{\alpha_m}|^2=1\), it means that outcome of measurement of \(A\) will definitely be the value \(\alpha_m\).

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Click for the proof

For consistency with the first part of the postulate, we must be able to prove that

the given condition, \(|\innerproduct{\psi}{\alpha_m}|^2=1\), implies that \(\ket{\psi}\) is an eigen vector of \(\hat{A}\) with eigen value \(\alpha_m\).

WHY ? How do I see the last statement?

You have to recall the full statement of Cauchy Schwarz inequality.

The Cauchy Schwarz inequality \[|<f|g>|^2 \le |f| |g|\] becomes equality if and only if \[|f> = \text{const} |g>\].

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We have several requirements coming from the consistency of probability interpretation, the content of the third postulate.

All the requirements on the eigen vectors of operator \(\hat{A}\) are met if \(\hat{A}\) is a self adjoint operator.

Let's Discuss The Postulates :: Consistency of Probability Interpretation

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