anonymous
  • anonymous
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
Physics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
Q: Find inner product
Michele_Laino
  • Michele_Laino
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: \[\Large \sigma _1^* = 0.2,\quad \sigma _2^* = 0.1 - 0.5i\]

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