Give an explaination an expression of trial wave function,hamilton operator and expectation value
Added by Tim K.
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Please give an exact answer, clear steps and compute an expectation value
Timothy J.
(a) Write down (i) the Hamiltonian operator, Ĥ (where Ĥ = T̂ + V̂), for a general potential V(x). (ii) the operator for momentum, p̂. (b) Evaluate the commutator [Ĥ, p̂] 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 observable represented by some general operator Ô is given by d⟨O⟩/dt = ⟨i/ħ [Ĥ, Ô]⟩ where Ĥ is the Hamiltonian operator you wrote down in part (a) and [Ĥ, Ô] is the commutator of the operators Ĥ and Ô. Use this result and the commutator you obtained in part (b) to show that d⟨p⟩/dt = -⟨dV(x)/dx⟩, which is special case of Ehrenfest's Theorem. What is the significance of this result?
Adi S.
1. A particle is represented (at time t = 0) by the wave function, Ψ(x, 0) = { A(a² - x²) : -a ≤ x ≤ +a, 0 : otherwise. a) Determine the normalization constant A. b) What is the expectation value of x (at t = 0)? c) What is the expectation value of p (at t = 0)? Note: you cannot use p = md⟨x⟩/dt because you only know ⟨x⟩ at t = 0. d) What is the expectation value of x² (at t = 0)? e) What is the expectation value of p² (at t = 0)? f) What is the uncertainty in x (σx) (at t = 0)? g) What is the uncertainty in p (σp) (at t = 0)? h) Check that your results in (f) and (g) are consistent with the uncertainty principle.
David M.
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