(a) Write down: (i) The Hamiltonian operator, H (where H = T + V), for a general potential V (x). (ii) The operator for momentum, p. (b) Evaluate the commutator [H, pl of these operators and comment on the significance of the result. (c) How would your result for (b) be affected if V (x) = 0, i.e. for a free particle? (d) It can be shown that the time evolution of the expectation value (O) of an observ- able represented by some general operator Ô is given by (O)P dt where Ĥ is the Hamiltonian operator you wrote down in part (a) and [H,Ô] is the commutator of the operators H and Ô. Use this result and the commutator you obtained in part (b) to show that d (p) dV(x)' dt dr which is special case of Ehrenfest's Theorem. What is the significance of this result?
(a) Write down: (i) The Hamiltonian operator, H (where H = T + V), for a general potential V (x). (ii) The operator for momentum, p. (b) Evaluate the commutator [H, pl of these operators and comment on the significance of the result. (c) How would your result for (b) be affected if V (x) = 0, i.e. for a free particle? (d) It can be shown that the time evolution of the expectation value (O) of an observ- able represented by some general operator Ô is given by (O)P dt where Ĥ is the Hamiltonian operator you wrote down in part (a) and [H,Ô] is the commutator of the operators H and Ô. Use this result and the commutator you obtained in part (b) to show that d (p) dV(x)' dt dr which is special case of Ehrenfest's Theorem. What is the significance of this result?
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![(a) Write down:
(i) The Hamiltonian operator, Ĥ (where Ĥ = Î + V), for a general potential
V (x).
(ii) The operator for momentum, p.
(b) Evaluate the commutator [Ĥ, p) of these operators and comment on the significance
of the result.
(c) How would your result for (b) be affected if V (r) = 0, i.e. for a free particle?
(d) It can be shown that the time evolution of the expectation value (O) of an observ-
able represented by some general operator Ô is given by
0 - (a.0)
(O)P
dt
where Ĥ is the Hamiltonian operator you wrote down in part (a) and [Ĥ‚Ô] is the
commutator of the operators H and Ô.
Use this result and the commutator you obtained in part (b) to show that
(a).
dV(x)\
(d) p
dt
dr
which is special case of Ehrenfest's Theorem. What is the significance of this
result?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F502966a9-668d-4d7a-a9d2-e1a141c1bffd%2Fe57183c0-b910-4667-aff8-7c96dc940c0c%2Fmvb9re_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(a) Write down:
(i) The Hamiltonian operator, Ĥ (where Ĥ = Î + V), for a general potential
V (x).
(ii) The operator for momentum, p.
(b) Evaluate the commutator [Ĥ, p) of these operators and comment on the significance
of the result.
(c) How would your result for (b) be affected if V (r) = 0, i.e. for a free particle?
(d) It can be shown that the time evolution of the expectation value (O) of an observ-
able represented by some general operator Ô is given by
0 - (a.0)
(O)P
dt
where Ĥ is the Hamiltonian operator you wrote down in part (a) and [Ĥ‚Ô] is the
commutator of the operators H and Ô.
Use this result and the commutator you obtained in part (b) to show that
(a).
dV(x)\
(d) p
dt
dr
which is special case of Ehrenfest's Theorem. What is the significance of this
result?
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