Quantum Mechanics Operator/eigenstates problem

[tsumi]

Homework Statement

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?

Homework 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.

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.

 

[BruceW]

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?

This is confusing. I can't work out what the question is meant to be. There is some system to begin with, right? And then a measurement of the variable represented by B is made on the system. And then after that, a measurement of the variable represented by A is made on the system? And we are meant to work out the probability that the second measurement will give zero. Did I get all this right? So what is the state meant to be at the beginning, before any measurements were made?

 

[tsumi]

It seams we are meant to work out the probability of after measuring the system with B and it becoming in state a0, getting the result 0 with the operator A. It's like a composed probability, something like that, one probability times the other.

The state at the beginning isn't told, I suppose it's not important, since you are told you are in state a0 after the measuring of B.

 

[BruceW]

(sorry I've taken a while to reply). So you mean that the system is measured with B, and then measured with A, we want to know the probability that this measurement with A will yield zero?

This will surely depend on the state at the beginning. For example, the state might have been an eigenstate of B, in which case the measurement of B will leave the state unchanged, and we know that the eigenstates of B each have a different overlap with the a0 state, and therefore will have a different probability of getting the zero result.

P.S. also, I had a go at calculating the eigenstates of B, and I think you have got them the wrong way around (i.e. -25 and 25 mixed up)?

 

[paranormal]

I understand your confusion and I will try my best to clarify the concepts involved in this problem. First of all, in quantum mechanics, the square of the wave function or the probability amplitude represents the probability of finding the system in a particular state. Therefore, when we take the square of <bλ|a0>, we are calculating the probability of the system collapsing into the state |a0> after being measured by the operator B. Similarly, when we take the square of <a0|bλ>, we are calculating the probability of the state |a0> collapsing into the eigenvalue 0 after being measured by the operator A.

Now, the reason why the probabilities are equal for both <bλ|a0> and <a0|bλ> is because these are two different measurements being performed on the same system. In other words, the probability of the system collapsing into the state |a0> after being measured by B is the same as the probability of the state |a0> collapsing into the eigenvalue 0 after being measured by A. This is because the state |a0> is an eigenstate of both operators B and A.

As for the actual values obtained in the probabilities, they are obtained by solving the eigenvalue equation for the operator B. In this case, we have two eigenvalues, -25 and 25, which correspond to the two eigenstates |bλ=-25> and |bλ=25>, respectively. These are obtained by substituting the given operator B into the eigenvalue equation B|bλ> = λ|bλ> and solving for λ.

I hope this helps clarify the concepts involved in this problem. If you have any further questions, feel free to ask. Good luck with your homework!