Question

Consider a time-independent Hamiltonian H with eigenstates |phi_{n} angle, such that H|phi_{n} angle = E_{n}|phi_{n} angle. (a) Starting from the Schrödinger equation, i? d/dt |psi(t) angle = H|psi(t) angle, show that d/dt (langlepsi(t)|psi(t) angle) = 0, and hence the norm of the state vector |psi angle is conserved. (b) For an arbitrary operator A, prove that langlephi_{n}|[A, H]| phi_{n} angle = 0.

          Consider a time-independent Hamiltonian H with eigenstates |phi_{n}
angle, such that H|phi_{n}
angle = E_{n}|phi_{n}
angle.

(a) Starting from the Schr&ouml;dinger equation, i? d/dt |psi(t)
angle = H|psi(t)
angle, show that d/dt (langlepsi(t)|psi(t)
angle) = 0, and hence the norm of the state vector |psi
angle is conserved.

(b) For an arbitrary operator A, prove that langlephi_{n}|[A, H]| phi_{n}
angle = 0.

Consider a time-independent Hamiltonian H with eigenstates |phin
angle, such that H|phin
angle = En|phin
angle.

(a) Starting from the Schr ouml;dinger equation, i? d/dt |psi(t)
angle = H|psi(t)
angle, show that d/dt (langlepsi(t)|psi(t)
angle) = 0, and hence the norm of the state vector |psi
angle is conserved.

(b) For an arbitrary operator A, prove that langlephin|[A, H]| phin
angle = 0.

Added by Foster J.

Introduction to modern physics

Richtmyer F.K.,Kennard E.H., Cooper J.N. 6th Edition

Chapter 13

Instant Answer

Solved by Expert Lizandra Chagas

05/09/2022

Step 1

Step 1:** Starting from the Schrodinger equation, we have: $$\frac{d}{dt}(i\hbar \langle \psi(t) | \psi(t) \rangle) = \langle \psi(t) | H | \psi(t) \rangle - \langle \psi(t) | H | \psi(t) \rangle = 0$$ ** Show more…

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Consider a time-independent Hamiltonian H with eigenstates |̴̦n⟩, such that H|̴̦n⟩ = En|̴̦n⟩. (a) Starting from the Schrödinger equation, iħ d/dt |ψ(t)⟩ = H|ψ(t)⟩, show that d/dt (⟨ψ(t)|ψ(t)⟩) = 0, and hence the norm of the state vector |ψ⟩ is conserved. (b) For an arbitrary operator A, prove that ⟨φn|[A,H]|φn⟩ = 0.