Relation between a measurement and the operators

In summary, the conversation discusses the application of two operators, M1 and M2, on a given state |p>, with M1 representing the measurement of outcome |0> and M2 representing the measurement of outcome |1>. The operators satisfy the completeness equation and can be used to calculate the probability of obtaining a specific outcome. However, the conversation also raises the question of whether a measurement can only contain a subset of operators or if it must contain a set of operators that satisfies the completeness equation. It is suggested that a measurement must contain a complete set of operators and the conversation ends with a clarification of the operator representing the quantity being measured and its corresponding eigenvectors.
  • #1
ouacc
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Now, here is the problem. (Capital letters indicate operators, lower letters are states, * indicates Hermitian conjugate)
Say we know that state | p > = cos(a) |0> + sin(a) |1> (0<a<PI, a is in R)
Two operators : M1= |0><0| , M2=|1><1|, apperatantly they satisfy the completeness equation. M1*M1+M2*M2 = I

According to some textbooks, if we perform ONE opeartor on |p>, we should have TWO outcomes |0> or |1>, with certain probabilities. Here is the details:

If I use one operator on the state, say M1,
then: the state AFTER the operation is: M1 |p> / sqrt(<p | M*M |>p) = cos(a) |0> / cos(a) = |0> ...(1)
the probability of this happening is PM1= <p | M*M |>p = cos(a)^2 ...(2)

NOW, according to (1), it is impossible to get state (outcome) |1> after the operation.
SO, where do the after-state (outcome) |1> go?
I can get the outcome of |1> only with the other operator M2.
M2 |p> / sqrt(<p | M*M |>p) = sin(a) |1> / sin(a) = |1> ...(3)
with the probability PM2=sin(a)^2.....(4)

To me, it seems that :
The language of "use operator on state |p>" is not correct.
A measurement can NOT contain ONLY a subset of the operators (in this case, contains only M1, not M2),
A measurement have to be composed of a SET of operators which satisfies completeness equation.


Is this the case in real-world experiments? We can not say "perform M1 on p", but have to say "perform the measurement which has M1 and M2 on p" or "measure the spin in the |0> to |1> axis"?

BTW, put it in another way. If we keep M1 unchanged, but change M2 to | 1/sqrt(2)( |0> - |1>) > < 1/sqrt(2)( <0| - <1|)|, (they do not satisfy the compleness equation). SO, a measurement containing ONLY M1 and M2 can not be made in real life. Is this right?
 
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  • #2
I'd say that if the operator representing the quantity that you are measuring is M, and it's eigenvectors are |0> (eigenvalue = M1) and |1> (eigenvalue = M2) (and that's the complete set of eigenvectors for that operator), then for any initial state | p > = cos(a) |0> + sin(a) |1>, the projection operators acting on that state can give you the probability amplitude of a measurement of M giving any particular value (M1 or M2).
 
  • #3


I can confirm that your understanding is correct. The measurement operators M1 and M2 must satisfy the completeness equation in order for them to represent a valid measurement. This means that they must be able to account for all possible outcomes of the measurement, in this case, the states |0> and |1>. If we only use one of the operators, we can only obtain one of the outcomes and the other outcome is not accounted for. This is why it is important to have a set of operators that satisfy the completeness equation in order to perform a valid measurement.

In real-world experiments, we cannot simply say "perform M1 on p" as this would not be a complete measurement. Instead, we must specify the set of operators being used, such as "measure the spin in the |0> to |1> axis" or "perform the measurement which has M1 and M2 on p". This ensures that all possible outcomes are accounted for and the measurement is valid.

Additionally, your example of changing M2 to |1/sqrt(2)(|0> - |1>)><1/sqrt(2)(<0| - <1|)| shows that not all sets of operators will satisfy the completeness equation, and therefore cannot be used for a valid measurement. This further emphasizes the importance of using a complete set of operators in measurements.

In conclusion, your understanding of the relation between a measurement and the operators is accurate. A valid measurement must be composed of a set of operators that satisfy the completeness equation in order to account for all possible outcomes. This is important to keep in mind when designing and conducting experiments.
 

1. What is the relation between a measurement and an operator?

The measurement of a physical quantity is directly related to the operator that represents that quantity. The operator acts on a quantum state to produce the possible outcomes of a measurement.

2. How do operators affect the outcome of a measurement?

The outcome of a measurement is determined by the operator that corresponds to the physical quantity being measured. The operator acts on the quantum state to produce the possible measurement results.

3. Can operators be used to predict the outcome of a measurement?

Yes, operators can be used to predict the probabilities of obtaining different measurement results. The eigenvalues of the operator correspond to the possible measurement outcomes, and the square of the corresponding eigenstates gives the probability of obtaining that outcome.

4. Are there different types of operators for different physical quantities?

Yes, there are different operators for different physical quantities. For example, there are operators for position, momentum, energy, and spin. Each operator is specific to the physical quantity it represents and has its own set of eigenvalues and eigenstates.

5. What is the significance of the relation between a measurement and an operator in quantum mechanics?

The relation between a measurement and an operator is fundamental in quantum mechanics. It allows us to predict the outcomes of measurements and understand the behavior of quantum systems. It also plays a crucial role in the mathematical formulation of quantum mechanics and the development of quantum algorithms for practical applications.

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