5. (10 pt) Suppose that we have two operators A and B such that ⟨α | A | α⟩ = ⟨α | B | α⟩ for al

Question

5. (10 pt) Suppose that we have two operators A and B such that $\langle \alpha | A | \alpha \rangle = \langle \alpha | B | \alpha \rangle$ for all $|\alpha\rangle$ in the Hilbert space under consideration. Prove that A = B. (Recall the following definition: A = B if and only if $\langle \beta | A | \alpha \rangle = \langle \beta | B | \alpha \rangle$ in all kets $|\alpha\rangle$, $|\beta\rangle$ in the Hilbert space.)

          5. (10 pt) Suppose that we have two operators A and B such that $\langle \alpha | A | \alpha \rangle = \langle \alpha | B | \alpha \rangle$ for all $|\alpha\rangle$ in the Hilbert space under consideration. Prove that A = B.
(Recall the following definition: A = B if and only if $\langle \beta | A | \alpha \rangle = \langle \beta | B | \alpha \rangle$ in all kets $|\alpha\rangle$, $|\beta\rangle$ in the Hilbert space.)

5. (10 pt) Suppose that we have two operators A and B such that ⟨α | A | α⟩ = ⟨α | B | α⟩ for all |α⟩ in the Hilbert space under consideration. Prove that A = B.
(Recall the following definition: A = B if and only if ⟨β | A | α⟩ = ⟨β | B | α⟩ in all kets |α⟩, |β⟩ in the Hilbert space.)

Added by Charles M.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Nick Johnson

05/15/2024

Step 1

Step 1: We are given that (Ala) = (a Bla) for all a) in the Hilbert space. Show more…

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5. (10 pt) Suppose that we have two operators A and B such that (Ala) = (a Bla) for all a) in the Hilbert space under consideration. Prove that A = B. (Recall the following definition: A = B if and only if (8|Ala) = (B|Ala) in all kets (a), (β) in the Hilbert space.)