Is it vacum state is orthogonal or not?

[razif]

Homework Statement

$$\left\langle0_{x}|0_{y}\right\rangle$$ is this orthogonal or not?

Homework Equations

for $$\left\langle1_{x}|1_{y}\right\rangle$$ we already know that this state is orthogonal to each others because 1 state at x-axis while the others in y-axis

for $$\left\langle1_{x}|1_{x}\right\rangle$$ we know that this state is not orthogonal to each others because they were in same axis

for $$\left\langle0_{x}|0_{x}\right\rangle$$ we can say that this state not orthogonal to each other because both vacuum state in same x-axis

The Attempt at a Solution

but for $$\left\langle0_{x}|0_{y}\right\rangle$$ can we say that this state is not orthogonal to each others?because both of them are vacuum state although it written that the vacuum state at x-axis and vacuum state at y-axis, if yes this is not orthogonal, can someone point me the reason. I just assumed that because both state are in vacuum state and they can be at any axis without changing any state at that axis.

 

[Jhero]

I would like to clarify that the statement \left\langle0_{x}|0_{y}\right\rangle is not orthogonal is incorrect. In quantum mechanics, orthogonal states are defined as states that have a zero inner product, which means they are perpendicular to each other. In this case, the vacuum state at x-axis and the vacuum state at y-axis are indeed orthogonal because they are in different axes and therefore have a zero inner product.

The reason why \left\langle0_{x}|0_{x}\right\rangle is not orthogonal is because both states are in the same axis, and therefore have a non-zero inner product. This is also true for \left\langle1_{x}|1_{x}\right\rangle, where both states are in the same axis and have a non-zero inner product.

In summary, the vacuum state at x-axis and the vacuum state at y-axis are orthogonal because they are in different axes, while the vacuum state at x-axis and the vacuum state at x-axis are not orthogonal because they are in the same axis. I hope this clarifies any confusion.