About incompatibility and interference terms

[xboy]

In Modern Quantum Mechanics page 44-46, Sakurai compares two experiments:
1. Here three observables A,B,C are measured. The probability of obtaining the results a,b and c respectively are then :(|<c|b>|^2)(|<b|a>|^2). Summing up over all values of b we'll find $$\Sigma$$[tex]_{}b/tex](<c|b>|)^2(|<b|a>|)^2
2. But if we don't measure B,the probability of getting c if we have already got a becomes|<c|a>|^2

So the two probabilities are different.According to Sakurai the two probabilities are equal is [A,B]=0 or [B,C]=0.
I'm trying to prove this but haven't been able to find a good proof yet.All help will be appreciated.

 

[eljose79]

Thank you for bringing up this interesting topic from Sakurai's Modern Quantum Mechanics. The two experiments mentioned in the post are indeed different, and the equality of their probabilities depends on the commutativity of the observables involved.

To prove this, let's first define the probabilities mathematically. In the first experiment, we have three observables A, B, and C with the corresponding eigenstates |a>, |b>, and |c>, respectively. The probability of obtaining the results a, b, and c is given by:

P(a,b,c) = |<c|b>|^2|<b|a>|^2

Summing over all possible values of b, we get:

P(a,c) = ∑_b |<c|b>|^2|<b|a>|^2

On the other hand, in the second experiment, we have only two observables A and C, and the probability of obtaining c if we have already obtained a is given by:

P(a,c) = |<c|a>|^2

Now, to show that these two probabilities are equal, we need to prove that:

∑_b |<c|b>|^2|<b|a>|^2 = |<c|a>|^2

Expanding the left-hand side, we get:

∑_b |<c|b>|^2|<b|a>|^2 = ∑_b |<c|b>|^2|<a|b><b|a>| = ∑_b |<c|a><a|b><b|a>|^2

Using the identity |<c|a><a|b> = <c|b> (which can be derived from the completeness relation), we can rewrite this as:

∑_b |<c|a><a|b>|^2 = ∑_b |<c|a>|^2|<a|b>|^2

Now, if we assume that the observables A and B commute, i.e. [A,B] = 0, then we can rewrite the above as:

∑_b |<c|a>|^2|<a|b>|^2 = |<c|a>|^2∑_b |<a|b>|^2

Using the completeness relation again,

 

[denian]

The concept of incompatibility and interference terms is a fundamental aspect of quantum mechanics that helps explain the behavior of particles at the quantum level. In the context of the two experiments described by Sakurai, incompatibility refers to the fact that the two probabilities are different when B is measured in the first experiment and not measured in the second experiment. This is due to the fact that in quantum mechanics, the act of measurement can affect the state of the system being measured.

Interference terms, on the other hand, refer to the mathematical terms that arise when multiple observables are measured simultaneously. In the first experiment, the interference term is given by the sum over all values of b, while in the second experiment, there is no interference term since B is not measured.

Sakurai's statement that the two probabilities are equal only when [A,B]=0 or [B,C]=0 is known as the compatibility condition. This means that if the operators A and B commute (i.e. [A,B]=0), then they can be measured simultaneously without affecting each other's results. Similarly, if B and C commute, they can also be measured simultaneously without interference.

To prove this, you can use the commutation relations [A,B]=AB-BA and [B,C]=BC-CB and substitute them into the expression for the probability in the first experiment. You will find that the sum over all values of b will simplify to just one term, which is equal to the probability in the second experiment. This shows that the two probabilities are equal when [A,B]=0 or [B,C]=0.

In summary, incompatibility and interference terms are important concepts in quantum mechanics that help explain the results of experiments involving multiple observables. The compatibility condition, as stated by Sakurai, provides a condition for when the two probabilities will be equal. Further exploration and understanding of these concepts will help deepen our understanding of the strange and fascinating world of quantum mechanics.