A proton (charge

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · Quantum dynamics and its formulations

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Observables, commutators and the uncertainty relations

Problem

A proton (charge +e=1.602×10−19 C+e = 1.602\times10^{-19}\ \mathrm{C}, mass mp=1.673×10−27 kgm_p = 1.673\times10^{-27}\ \mathrm{kg}) sits in a uniform electric field of magnitude E=5.00×106 V m−1\mathcal E = 5.00\times10^6\ \mathrm{V\,m^{-1}} along xx, giving the potential energy V(x)=−eExV(x) = -e\mathcal Ex. Derive Ehrenfest's equations for d⟨x⟩/dtd\langle x\rangle/dt and d⟨p⟩/dtd\langle p\rangle/dt for this potential, show that ⟨x⟩(t)\langle x\rangle(t) obeys exactly the classical equation of motion for a charge under a constant force, and find the resulting acceleration.

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