Measured result is equal to expectation value

[BUI TUAN KHAI]

Can I ask a basic question. This was a question in a test, I could not solve this.
When is it true that the result of a single measurement for a dynamical variable is equal to the expectation value of the operator corresponding to that dynamical variable?
Thank you for your help.
Sincerely yours.
Khai.

 

[Nugatory]

In general the result of a single measurement will not be equal to the expectation value, but if you make a large number of measurements the average is likely to be close the expectation value; the more measurements you make, the closer the average will be.

That's "in general". In the specific case of a system prepared in an eigenstate of the observable that you are measuring, the result will in principle be the corresponding eigenvalue every time and the expectation value is exactly equal to that eigenvalue.

 

[BUI TUAN KHAI]

As I think,
What we observe from the measurement is the eigenvalue.
According to the question, it means that the eigenvalue = expectation value ?
If we have an observable A, and a physical state |a>, we have A|a> = a|a>.
<A> = <a|A|a> = a<a|a> = a
Thus, this case only happens if:

• |a> is the eigenket of A (we measure A in physical state, which is also the eigenstate of A)
• A has real eigenvalue

 

[BUI TUAN KHAI]

In general the result of a single measurement will not be equal to the expectation value, but if you make a large number of measurements the average is likely to be close the expectation value; the more measurements you make, the closer the average will be.

That's "in general". In the specific case of a system prepared in an eigenstate of the observable that you are measuring, the result will in principle be the corresponding eigenvalue every time and the expectation value is exactly equal to that eigenvalue.

Thank you for you explanation.

 

[DrClaude]

• A has real eigenvalue

If A is an observable, it can only have real eigenvalues.