- #1

- 38

- 3

## Main Question or Discussion Point

I've started few days ago to study quantum physics, and there's a thing which isn't clear to me. I know that a quantum state is represented by a ray in a Hilbert space (so that ##k \left| X \right\rangle## is the same state of ##\left| X \right\rangle##). Suppose now to have these three states:

##\left| A \right\rangle = \frac{1}{\sqrt{2}} (\left| \uparrow \right\rangle + \left| \downarrow \right\rangle)##

##\left| B \right\rangle = \frac{1}{\sqrt{2}} (\alpha \left| \uparrow \right\rangle + \beta \left| \downarrow \right\rangle)##

##\left| C \right\rangle = \frac{e^{i \phi}}{\sqrt{2}} (\left| \uparrow \right\rangle + \left| \downarrow \right\rangle)##

(where ##\alpha , \beta \in \mathbb{C}## and ##\phi \in \mathbb{R}##)

Question is, are these states rapresenting the same quantum state?

##\left| A \right\rangle = \frac{1}{\sqrt{2}} (\left| \uparrow \right\rangle + \left| \downarrow \right\rangle)##

##\left| B \right\rangle = \frac{1}{\sqrt{2}} (\alpha \left| \uparrow \right\rangle + \beta \left| \downarrow \right\rangle)##

##\left| C \right\rangle = \frac{e^{i \phi}}{\sqrt{2}} (\left| \uparrow \right\rangle + \left| \downarrow \right\rangle)##

(where ##\alpha , \beta \in \mathbb{C}## and ##\phi \in \mathbb{R}##)

Question is, are these states rapresenting the same quantum state?