Question

(i) Prove that \( \langle X \rangle \) remains unchanged under unitary transformation. (ii) The Hamiltonian operator \( H \) acting on a square integrable function \( \Psi(x, t) \) satisfies the Schrodinger equation \( ih \frac{\partial}{\partial t} \Psi(r, t) = H \Psi(r, t) \). Prove that for the probability conservation, the operator \( H \) must be Hermitian. (iii) Prove that the scalar product is invariant under unitary transformation.

          (i) Prove that \( \langle X \rangle \) remains unchanged under unitary transformation.
(ii) The Hamiltonian operator \( H \) acting on a square integrable function \( \Psi(x, t) \) satisfies the Schrodinger equation \( ih \frac{\partial}{\partial t} \Psi(r, t) = H \Psi(r, t) \). Prove that for the probability conservation, the operator \( H \) must be Hermitian.
(iii) Prove that the scalar product is invariant under unitary transformation.

(i) Prove that ⟨ X ⟩ remains unchanged under unitary transformation.
(ii) The Hamiltonian operator H acting on a square integrable function Ψ(x, t) satisfies the Schrodinger equation ih (∂)/(∂ t)Ψ(r, t) = H Ψ(r, t). Prove that for the probability conservation, the operator H must be Hermitian.
(iii) Prove that the scalar product is invariant under unitary transformation.

Added by Jessica C.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Sri K

05/30/2022

Step 1

Step 1: The Schrödinger equation is given by: \[i\hbar \frac{\partial \Psi}{\partial t} = H \Psi\] where \(\Psi\) is the wave function, \(t\) is time, \(H\) is the Hamiltonian operator, and \(\hbar\) is the reduced Planck's constant. Show more…

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G) The Hamiltonian operator acting on a square integrable function satisfies the Schrödinger equation. In order for conservation to hold, the operator H must be Hermitian.