Question

We can express two functions |f> and |g> in terms of two orthonormal functions ψ1 and ψ2 as
|f>=(0.3+0.25i)|ψ1>+0.8|ψ2>,|g>=0.2|ψ1>+(0.1+0.5i)|ψ2>
Q: Find inner product

Answer 1

Q: Find inner product <g|f>

Answer 2

I rewrite your state vector as below:
\[\Large \begin{gathered}
\left| f \right\rangle = {z_1}\left| {{\psi _1}} \right\rangle + {z_2}\left| {{\psi _2}} \right\rangle \hfill \\
\left| g \right\rangle = {\sigma _1}\left| {{\psi _1}} \right\rangle + {\sigma _2}\left| {{\psi _2}} \right\rangle \hfill \\
\end{gathered} \]
where:
\[\Large \begin{gathered}
{z_1} = 0.3 + 0.25i,\quad {z_2} = 0.8 \hfill \\
{\sigma _1} = 0.2,\;\quad {\sigma _2} = 0.1 + 0.5i \hfill \\
\end{gathered} \]
Now, if the states\[\Large \left| {{\psi _1}} \right\rangle ,\quad \left| {{\psi _2}} \right\rangle \]
are orthonormal states, anmely:
\[\Large \left\langle {{\psi _1}\left| {{\psi _2}} \right\rangle = 0} \right.,\quad \left\langle {{\psi _1}} \right.\left| {{\psi _1}} \right\rangle = \left\langle {{\psi _2}} \right.\left| {{\psi _2}} \right\rangle = 1\]
then we can write:
\[\Large \begin{gathered}
\left\langle g \right.\left| f \right\rangle = \left( {\sigma _1^*\left\langle {{\psi _1}} \right| + \sigma _2^*\left\langle {{\psi _2}} \right|} \right)\left( {{z_1}\left| {{\psi _1}} \right\rangle + {z_2}\left| {{\psi _2}} \right\rangle } \right) = \hfill \\
\hfill \\
= \sigma _1^*{z_1} + \sigma _2^*{z_2} = ...? \hfill \\
\end{gathered} \]
where: \[\Large \sigma _1^* = 0.2,\quad \sigma _2^* = 0.1 - 0.5i\]