Math Problem Statement

bra a1 phi1 + a2 phi2 ket b1 psi1 b2 psi2 expand this

Solution

To expand the expression a1ϕ1+a2ϕ2b1ψ1+b2ψ2\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: a1ϕ1+a2ϕ2b1ψ1+b2ψ2=a1b1ϕ1ψ1+a1b2ϕ1ψ2+a2b1ϕ2ψ1+a2b2ϕ2ψ2\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:

  • a1b1ϕ1ψ1a_1^* b_1 \langle \phi_1 | \psi_1 \rangle: This is the inner product of ϕ1\phi_1 and ψ1\psi_1, weighted by a1a_1^* and b1b_1.
  • a1b2ϕ1ψ2a_1^* b_2 \langle \phi_1 | \psi_2 \rangle: This is the inner product of ϕ1\phi_1 and ψ2\psi_2, weighted by a1a_1^* and b2b_2.
  • a2b1ϕ2ψ1a_2^* b_1 \langle \phi_2 | \psi_1 \rangle: This is the inner product of ϕ2\phi_2 and ψ1\psi_1, weighted by a2a_2^* and b1b_1.
  • a2b2ϕ2ψ2a_2^* b_2 \langle \phi_2 | \psi_2 \rangle: This is the inner product of ϕ2\phi_2 and ψ2\psi_2, weighted by a2a_2^* and b2b_2.

Final expanded form:

a1ϕ1+a2ϕ2b1ψ1+b2ψ2=a1b1ϕ1ψ1+a1b2ϕ1ψ2+a2b1ϕ2ψ1+a2b2ϕ2ψ2\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?


Related questions:

  1. What is the physical significance of the inner product ϕ1ψ1\langle \phi_1 | \psi_1 \rangle?
  2. How do you calculate inner products in a basis of vectors?
  3. What happens if ϕ1\phi_1 and ψ1\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 a1a_1^* or b1b_1^*.

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Math Problem Analysis

Mathematical Concepts

Quantum Mechanics
Inner Products
Linear Algebra

Formulas

Inner product expansion formula
Distributive property

Theorems

Properties of inner products in quantum mechanics

Suitable Grade Level

Undergraduate