Question

Question 2 Simple harmonic oscillator time evolution (a) (1 point) Apply the \(\hat{x}(t)\) solution for the time-dependence of the \(\hat{x}\) operator of the simple harmonic oscilla- tor to find the time dependence of the correlation amplitude \(C_0(t) = \langle \hat{x}(t)\hat{x}(0)\rangle\) for the ground state \(|0\rangle\). (b) (1 point) Alternatively, apply the relation \(\hat{x}(t) = \hat{U}^+\hat{x}(0)\hat{U}\) and act with the \(\hat{U}^+ = e^{i\hat{H}t}\) and \(\hat{U} = e^{-i\hat{H}t}\) operators directly on the states to obtain \(C_0(t)\). (c) (1 point) Find \(C_1(t)\) for the 1st excited state \(|1\rangle\), using a method of your choice. (d) (1 point) Calculate and plot \(|C_0(t)|\) and \(|C_1(t)|\) and interpret the results.

          Question 2 Simple harmonic oscillator time evolution
(a) (1 point) Apply the \(\hat{x}(t)\) solution for the time-dependence of the \(\hat{x}\) operator of the simple harmonic oscilla-
tor to find the time dependence of the correlation amplitude \(C_0(t) = \langle \hat{x}(t)\hat{x}(0)\rangle\) for the ground state \(|0\rangle\).
(b) (1 point) Alternatively, apply the relation \(\hat{x}(t) = \hat{U}^+\hat{x}(0)\hat{U}\) and act with the \(\hat{U}^+ = e^{i\hat{H}t}\) and \(\hat{U} = e^{-i\hat{H}t}\)
operators directly on the states to obtain \(C_0(t)\).
(c) (1 point) Find \(C_1(t)\) for the 1st excited state \(|1\rangle\), using a method of your choice.
(d) (1 point) Calculate and plot \(|C_0(t)|\) and \(|C_1(t)|\) and interpret the results.

Question 2 Simple harmonic oscillator time evolution
(a) (1 point) Apply the x̂(t) solution for the time-dependence of the x̂ operator of the simple harmonic oscilla-
tor to find the time dependence of the correlation amplitude C0(t) = ⟨x̂(t)x̂(0)⟩ for the ground state |0⟩.
(b) (1 point) Alternatively, apply the relation x̂(t) = Û^+x̂(0)Û and act with the Û^+ = e^iĤt and Û = e^-iĤt
operators directly on the states to obtain C0(t).
(c) (1 point) Find C1(t) for the 1st excited state |1⟩, using a method of your choice.
(d) (1 point) Calculate and plot |C0(t)| and |C1(t)| and interpret the results.

Added by Paula A.

Question

Please give Ace some feedback