Why is the expectation value of the commutator not zero in method 1?

[alizeid]

I'm trying to extract the expected value of a |+45> polarized photon in the HV- base. I do it in two ways, one is correct and the other I get wrong answer. What am I doing wrong in the first method that is wrong?

method 1

method 2

thanks in advance

 

[Orodruin]

You computed the expectation value of different operators. Where did the minus sign come from in your second method?

Also, please type out your expressions instead of attaching them as images.

 

[alizeid]

You computed the expectation value of different operators. Where did the minus sign come from in your second method?

Also, please type out your expressions instead of attaching them as images.

why are they different operators? according to the book, they are the same

 

[Orodruin]

Then you either copied the book wrong or the book is wrong.

 

[alizeid]

why are they different operators? according to the book, they are the same

The real problem is

I get the expectation value of the commutator to zero, even for operator | 45>. This can not be correct. I can not solve this problem, can you help me?

 

[Orodruin]

The problem formulation is of little use unless you define the $|45\rangle$ basis in terms of the HV one and define what is meant by $\mathcal P_{HV}$. Since you have not told us what textbook you are using and we most likely will not have direct access to it, this is crucial information.

even for operator | 45>

$|45\rangle$ is a state, not an operator.

 

[alizeid]

The problem formulation is of little use unless you define the $|45\rangle$ basis in terms of the HV one and define what is meant by $\mathcal P_{HV}$. Since you have not told us what textbook you are using and we most likely will not have direct access to it, this is crucial information.

$|45\rangle$ is a state, not an operator.

 

[Orodruin]

So what is the problem?

 

[alizeid]

So what is the problem?

The expectation value of the commutator should NOT be zero. the commutator do not commut therefore the answer is wrong

 

[Orodruin]

The expectation value of the commutator should NOT be zero. the commutator do not commut therefore the answer is wrong

No it isn’t. In fact, the expectation of the commutator must be zero for the given state! Anything else would violate the uncertainty relation.