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Operators: Time-evolution Consider the Hamiltonian of a free-particle of charge q and mass m in an external electric field E in one-dimension: 1 Ĥ = -p² – qxE 2m' Using the general operator equation of motion, solve for the time-dependence of the position and momentum operators in terms of their initial values at time t = 0.

Operators: Time-evolution Consider the Hamiltonian of a free-particle of charge q and mass m in an external electric field E in one-dimension: 1 Ĥ = -p² – qxE 2m' Using the general operator equation of motion, solve for the time-dependence of the position and momentum operators in terms of their initial values at time t = 0.

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**Operators: Time-evolution** Consider the Hamiltonian of a free particle of charge \( q \) and mass \( m \) in an external electric field \( \mathcal{E} \) in one dimension: \[ \hat{H} = \frac{1}{2m} \hat{p}^2 - q \hat{x} \mathcal{E} \] Using the general operator equation of motion, solve for the time-dependence of the position and momentum operators in terms of their initial values at time \( t = 0 \).
Transcribed Image Text:**Operators: Time-evolution** Consider the Hamiltonian of a free particle of charge \( q \) and mass \( m \) in an external electric field \( \mathcal{E} \) in one dimension: \[ \hat{H} = \frac{1}{2m} \hat{p}^2 - q \hat{x} \mathcal{E} \] Using the general operator equation of motion, solve for the time-dependence of the position and momentum operators in terms of their initial values at time \( t = 0 \).
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