- #1

jumi

- 28

- 0

## Homework Statement

Let a one-qubit system be in the state [itex]\left|ψ\right\rangle[/itex] = [itex]\frac{\sqrt{15}\left|0\right\rangle + i\left|1\right\rangle}{4}[/itex]. If we perform a measurement to see whether the qubit is in the state [itex]\left|x_{+}\right\rangle[/itex] = [itex]\frac{\left|0\right\rangle + \left|1\right\rangle}{\sqrt{2}}[/itex] or in the orthogonal state [itex]\left|x_{-}\right\rangle[/itex] = [itex]\frac{\left|0\right\rangle - \left|1\right\rangle}{\sqrt{2}}[/itex], what is the probability of each of these two outcomes?

## The Attempt at a Solution

I know the given state can be written as [itex]\left|ψ\right\rangle[/itex] = [itex]\frac{\sqrt{15}}{4}[/itex][itex]\left|0\right\rangle[/itex] + [itex]\frac{i}{4}[/itex][itex]\left|1\right\rangle[/itex]

So therefore α = [itex]\sqrt{15}[/itex]/4, and β = i/4. And [itex]\left|α^{2}\right|[/itex] + [itex]\left|β^{2}\right|[/itex] = 1 (right?). But those are only for states |0> or |1>, right?

So I basically have no idea how to do this.

Can anybody help me or put me in the right direction?

Thanks in advance.