To expand the expression $\langle a_1 \phi_1 + a_2 \phi_2 | b_1 \psi_1 + b_2 \psi_2 \rangle$, we can distribute and use the properties of inner products in quantum mechanics.

Here’s the step-by-step expansion:

1. Distribute the terms: $\langle a_1 \phi_1 + a_2 \phi_2 | b_1 \psi_1 + b_2 \psi_2 \rangle = a_1^* b_1 \langle \phi_1 | \psi_1 \rangle + a_1^* b_2 \langle \phi_1 | \psi_2 \rangle + a_2^* b_1 \langle \phi_2 | \psi_1 \rangle + a_2^* b_2 \langle \phi_2 | \psi_2 \rangle$

Here’s a breakdown of each term:

• $a_1^* b_1 \langle \phi_1 | \psi_1 \rangle$: This is the inner product of $\phi_1$ and $\psi_1$, weighted by $a_1^*$ and $b_1$.
• $a_1^* b_2 \langle \phi_1 | \psi_2 \rangle$: This is the inner product of $\phi_1$ and $\psi_2$, weighted by $a_1^*$ and $b_2$.
• $a_2^* b_1 \langle \phi_2 | \psi_1 \rangle$: This is the inner product of $\phi_2$ and $\psi_1$, weighted by $a_2^*$ and $b_1$.
• $a_2^* b_2 \langle \phi_2 | \psi_2 \rangle$: This is the inner product of $\phi_2$ and $\psi_2$, weighted by $a_2^*$ and $b_2$.

Final expanded form:

$\langle a_1 \phi_1 + a_2 \phi_2 | b_1 \psi_1 + b_2 \psi_2 \rangle = a_1^* b_1 \langle \phi_1 | \psi_1 \rangle + a_1^* b_2 \langle \phi_1 | \psi_2 \rangle + a_2^* b_1 \langle \phi_2 | \psi_1 \rangle + a_2^* b_2 \langle \phi_2 | \psi_2 \rangle$

Would you like me to explain any specific part further, or would you like to proceed with another related concept?

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Related questions:

1. What is the physical significance of the inner product $\langle \phi_1 | \psi_1 \rangle$?
2. How do you calculate inner products in a basis of vectors?
3. What happens if $\phi_1$ and $\psi_1$ are orthogonal?
4. How can you use the above expansion to simplify a matrix element in quantum mechanics?
5. Can you apply this expansion to calculate probabilities in quantum systems?

Tip: Always keep in mind the complex conjugate when dealing with inner products involving coefficients like $a_1^*$ or $b_1^*$.