How to Find the Final State and Probability After Measuring \(L_x^2\)?

[liorda]

Homework Statement

consider the state $$(\frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}})$$ in $$L_z$$ basis. If $$L_x^2$$ is measured and the result of 0 is obtained, find the final state after the measurement. How probable is this result?

The Attempt at a Solution

I'm not sure if the state is superposition of the known ground states of L_x in L_z representation. How to find the state of the system after the measurement? And should I sum the probabilities of getting each of that state eigenvalues?

thanks.

 

[liorda]

any moderator, please move my question to quantum physics forum. thanks.

 

[baba89]

I would say that the concept of superposition of basis states is a fundamental principle in quantum mechanics. It states that a quantum system can exist in multiple states at the same time, and the final state is a combination of all these states. In the given example, the state (\frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}}) is a superposition of the basis states in the L_z representation. This means that the system has equal probabilities of being in the states \frac{1}{2} and -\frac{1}{2} at the same time, with a weight of \frac{1}{\sqrt{2}} for the state \frac{1}{2}.

In order to find the state after the measurement of L_x^2, we need to use the projection operator formalism. The final state after the measurement will be the projection of the initial state onto the eigenstate of L_x^2 with eigenvalue 0. This can be calculated by taking the inner product of the initial state with the eigenstate of L_x^2 with eigenvalue 0. The probability of obtaining this result can be calculated by taking the square of the magnitude of this inner product.

It is important to note that in quantum mechanics, the probability of obtaining a particular result is not additive. This means that we cannot simply sum the probabilities of getting each of the state eigenvalues. Instead, we need to use the projection operator formalism to calculate the probability of obtaining a particular result.

In summary, the state after the measurement of L_x^2 will be a superposition of the basis states in the L_z representation, with a probability determined by the projection of the initial state onto the eigenstate with eigenvalue 0. The probability of obtaining this result cannot be calculated by simply summing the probabilities of getting each state eigenvalue, but rather by using the projection operator formalism.