Consider a Hamiltonian describing a particle of charge q and mass m in a one-dimensional harmonic oscillator well in an oscillating external electric field of magnitude E: 1 H = 2m (a) Evaluate the equations of motion for operators p(t) and â(t) and express your answer in terms of their initial values at t = 0, namely â&(0) and p(0). (b) Evaluate the commutator of the position operator & (t) at different times, i.e. [ê(t1), ¤ (t2)] for t1 # t2 and thus show that operators that commute at the same time, may not commute at different times.
Consider a Hamiltonian describing a particle of charge q and mass m in a one-dimensional harmonic oscillator well in an oscillating external electric field of magnitude E: 1 H = 2m (a) Evaluate the equations of motion for operators p(t) and â(t) and express your answer in terms of their initial values at t = 0, namely â&(0) and p(0). (b) Evaluate the commutator of the position operator & (t) at different times, i.e. [ê(t1), ¤ (t2)] for t1 # t2 and thus show that operators that commute at the same time, may not commute at different times.
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![Consider a Hamiltonian describing a particle of charge q and mass m in a one-dimensional harmonic oscillator well
in an oscillating external electric field of magnitude E:
1
+
2
mw²â² – qEâ cos(wot)
2m
(a) Evaluate the equations of motion for operators p(t) and â (t) and express your answer in terms of their initial
values at t = 0, namely â& (0) and ôp(0).
(b) Evaluate the commutator of the position operator â (t) at different times, i.e.
[â(t1), î (t2)] for tị # t2
and thus show that operators that commute at the same time, may not commute at different times.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb460c0c-d029-4e90-a450-1d82490780a1%2Fb1366c70-f2b8-4b9f-9646-c1841972b0bf%2Fasmf3ui_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider a Hamiltonian describing a particle of charge q and mass m in a one-dimensional harmonic oscillator well
in an oscillating external electric field of magnitude E:
1
+
2
mw²â² – qEâ cos(wot)
2m
(a) Evaluate the equations of motion for operators p(t) and â (t) and express your answer in terms of their initial
values at t = 0, namely â& (0) and ôp(0).
(b) Evaluate the commutator of the position operator â (t) at different times, i.e.
[â(t1), î (t2)] for tị # t2
and thus show that operators that commute at the same time, may not commute at different times.
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