Suppose we have a state |ψ⟩ that is a linear combination of two other states |ψ⟩=a|1⟩+b|2⟩, where a and b are non-zero, but otherwise unspecified, and |1⟩ and |2⟩ are orthonormal (⟨1|2⟩=⟨2|1⟩=0, ⟨1|1⟩=⟨2|2⟩=1). What is ⟨1|ψ⟩?
b2
b
a
a∗

Question

Suppose we have a state |ψ⟩ that is a linear combination of two other states |ψ⟩=a|1⟩+b|2⟩, where a and b are non-zero, but otherwise unspecified, and |1⟩ and |2⟩ are orthonormal (⟨1|2⟩=⟨2|1⟩=0, ⟨1|1⟩=⟨2|2⟩=1). What is ⟨1|ψ⟩?
b2
b
a
a∗

anonymous · Answer

it would be a, i think..

anonymous · Answer

yah
$$ | \Psi \rangle = a|1 \rangle + b|2 \rangle = a \left(\begin{matrix} 1 \ 0\end{matrix}\right) + b\left(\begin{matrix}0 \ 1\end{matrix}\right) = \left(\begin{matrix}a \ b\end{matrix}\right) 
\ \
\ \
\langle 1 | \Psi \rangle = (1^* \ \ 0)\left(\begin{matrix}a \ b\end{matrix}\right) = a
$$