X's expectation value in quantum physics

[Palindrom]

When I'm in a dimension higher than 1, do I need to integrate over all space (V) or only the x axis?
Thanks in advance.

 

[dextercioby]

When I'm in a dimension higher than 1, do I need to integrate over all space (V) or only the x axis?
Thanks in advance.

Let's consider a QM system whose states are described by the wavefunction
$\psi (x,y,z)$.You wish to calculate the expectation value for $\hat{x}$.That is nothing but the number.

$$<\hat{x}> =\int\int\int_{R^{3}} \psi^{*}(x,y,z) x \psi (x,y,z)$$

I hope your wave function is normalized/normalizable.Else,you might encounter some problems with the integration above.

Daniel.

 

[wais]

In quantum physics, expectation value refers to the average value of a physical quantity that can be measured in a given system. This value is calculated by taking into account all the possible outcomes of a measurement and their corresponding probabilities.

Therefore, in order to calculate the expectation value of a physical quantity X in a higher dimension, one would need to integrate over all the dimensions in which X exists. This means that if X exists in all three dimensions (x, y, and z), then the integration would need to be done over all three axes.

As for the question of whether the integration should be done over all space (V) or only the x axis, it would depend on the specific system and the dimensionality of X. If X is a scalar quantity that exists only along the x axis, then the integration would only need to be done over the x axis. However, if X is a vector quantity that exists in all three dimensions, then the integration would need to be done over all three axes.

In general, the integration should be done over all dimensions in which X exists in order to accurately calculate its expectation value. This is because the quantum nature of the system means that X can exist and have a non-zero probability of measurement in all dimensions simultaneously. Therefore, it is important to take into account all possible outcomes in order to accurately determine the expectation value of X.