A particle of mass moving in one dimension has free Hamiltonian and is additionally…

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Quantum dynamics and its formulations · Quantum dynamics and its formulations

The Structure of Quantum Mechanics, Ch Time evolution as a unitary group, and the three pictures

Problem

A particle of mass mm moving in one dimension has free Hamiltonian H^0=p^2/2m\hat H_0=\hat p^2/2m and is additionally subject to a weak, spatially uniform, time-dependent applied force described by V^(t)=−f(t)x^\hat V(t)=-f(t)\hat x, where f(t)f(t) is a given real function of time (for example an oscillating field). The full Hamiltonian is H^(t)=H^0+V^(t)\hat H(t)=\hat H_0+\hat V(t). (a) Define the interaction-picture state by ∣ψ(t)⟩I=U^0(t)†∣ψ(t)⟩S|\psi(t)\rangle_I=\hat U_0(t)^\dagger|\psi(t)\rangle_S, where U^0(t)=e−iH^0t/ℏ\hat U_0(t)=e^{-i\hat H_0t/\hbar} and ∣ψ(t)⟩S|\psi(t)\rangle_S obeys iℏ d∣ψ⟩S/dt=(H^0+V^(t))∣ψ⟩Si\hbar\,d|\psi\rangle_S/dt=(\hat H_0+\hat V(t))|\psi\rangle_S. Starting from this definition, derive iℏddt∣ψ(t)⟩I=V^I(t)∣ψ(t)⟩I,V^I(t)=U^0(t)†V^(t)U^0(t),i\hbar\frac{d}{dt}|\psi(t)\rangle_I=\hat V_I(t)|\psi(t)\rangle_I,\qquad \hat V_I(t)=\hat U_0(t)^\dagger\hat V(t)\hat U_0(t), stating which property of U^0\hat U_0 (unitarity, being generated by H^0\hat H_0, or the time-independence of H^0\hat H_0) is used at each step. (b) Using x^H(t)=x^+p^t/m\hat x_H(t)=\hat x+\hat p t/m, already available for the free particle alone, write V^I(t)\hat V_I(t) explicitly in terms of x^\hat x, p^\hat p, mm, tt and f(t)f(t). (c) State in one or two sentences what the interaction picture is good for here and why solving the equation in part (a) is generally easier than solving the original Schrodinger equation directly, without claiming that the interaction picture is itself an approximation.

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