(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A quantic system can be found in the states |a0> and |a1>, and those are eigenstates of the operator A with the respective eigenvalues: 0 and 1. Consider the operator B defined by: B|a0> = 7|a0> - 24i|a1> and B|a1> = 24i|a0> - 7|a1>.

After measured by the operator B the system is in the state |a0> , right after that measure the observable described by A is measured. What is the probability of this measure yealding zero?

2. Relevant equations

I already found the eigenvalues of B they are -25 and +25.

And its eigenstates:

|bλ=-25> = (4/5)|a0> - i(3/5)|a1>

|bλ=25> = (3/5)|a0> + i(4/5)|a1>

Those are correct.

3. The attempt at a solution

Now, I am not sure on how to do this. Probability of |bλ=±25> colapsing on |a0> and then probability of |a0> colapsing on zero? How would you do it? I actually have a correction but I don't really understand it =\

The correction is as follows:

Pa0(bλ=25) = |<bλ=25|a0>|^2 = (4/5)^2

Pbλ25(a0=0) = |<a0|bλ=25>|^2 = (4/5)^2

Pa0(bλ=25,a0=0) = (4/5)^4

Pa0(bλ=-25) = |<bλ=-25|a0>|^2 = (3/5)^2

Pbλ25(a0=0) = |<a0|bλ=-25>|^2 = (3/5)^2

Pa0(bλ=-25,a0=0) = (3/5)^4

Total Probability = (4/5)^4 + (3/5)^4

It does make some sense, but some things I just don't get.

First, probability of |bλ=25> colapsing into |a0>, ok, but.. what is the square for? Doesn't the <bλ=25|a0> already do it? doesn't it mean "the probability of bλ colapsing in a0" ?

Second, probability of a0 colapsing into zero,ok , but then I don't understand why is it equal to the one above it, only it is the other way around.. isn't it the same thing?

Then you multiply the probabilities, do the same for the other eigenvalue and add, that makes sense.

Another thing I don't really understand very well is how the actual values, the results in front, are actually obtained.

I hope someone is able to help me. Thanks in advance for any attention.

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# Quantum Mechanics Operator/eigenstates problem

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