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knowlewj01
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Homework Statement
Suppose we have a spin 1/2 Particle in a prepared state:
[itex]\left|\Psi\right\rangle = \alpha \left|\uparrow\right\rangle + \beta\left|\downarrow\right\rangle[/itex]
where
[itex]\left|\uparrow\right\rangle \left|\downarrow\right\rangle[/itex]
are orthonormal staes representing spin up and spin down respectively.
also: [itex] \left|\alpha\right|^2 + \left|\beta\right|^2 = 1[/itex]
[itex]\alpha & \beta[/itex] are complex numbers.
find a state which is orthogonal to [itex]\left|\Psi\right\rangle[/itex]
Homework Equations
The Attempt at a Solution
I went about this first by saying that the inner product of two states which are orthogonal is 0, so propose that:
[itex] \left\langle\Phi\right|\left|\Psi\right\rangle = 0[/itex]
where
[itex]\left|\Phi\right\rangle = \gamma\left|\uparrow\right\rangle + \delta\left|\downarrow\right\rangle[/itex]
where [itex] \gamma & \delta[/itex] are complex numbers:
[itex] \therefore[/itex]
[itex] \left\langle\Phi\right|\left|\Psi\right\rangle = \alpha \gamma^* + \beta \delta^* = 0[/itex]
Not sure where to go from here, i must be missing something. anyone know what it is?
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