- Only Answer Parts B I And B Ii The Last Two Questions 1 (95.04 KiB) Viewed 27 times
0 dt ME(t)e-i 71,Only answer Parts b.i and b.ii (the last two questions)
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Only answer Parts b.i and b.ii (the last two questions)
Only answer Parts b.i and b.ii (the last two questions)
. 2. Time evolution in two-level systems (a) A certain molecular system can be in one of two states: [10 marks) 11) 12) -0 ) 0 1 C) ( and the Hamiltonian is Eo -A II -A E 0) Compute the eigenvalues E4 of energy and the associated eigenstates E+). (i) If a system is prepared in a state E+), what is the state of the system at t> 0? If a system is prepared in a state 1) or 2), what is the state of the system at a later time? (ii) Suppose that the Hamiltonian is modified by the presence of a constant electric field, such that the new Hamiltonian is E, he -A II -A Εν - με where e and ji are constants. Compute the new eigenvalues and eigenvectors. Give the answer for the unit-norm eigenvectors in terms of a single parameter 0, where 0 is an angle to be determined. (b) The same molecule is now prepared in a state E+) and is placed in a cavity containing a time-dependent electric field e(t). Relative to the usual basis (i.e. (1,0) and (0,1)), the Hamiltonian is now Do -A 1 0 II + με(t) -A E such that the Schrödinger equation reads in(t)) = 11(0)|\-(t)). dt Since the eigenbasis of the unperturbed Hamiltonian is complete, the state (0)) can be written as ()) = COE+) + C2(0)|E_). Assuming this information answer the following two questions: () Obtain the ODEs satisfied by G and C2. [5 marks) () Using the trial solution C = n(t)e-i£44/h, C, = 72(t)e-iß_l/h, show that dy d92 ih- He(t)ekunde ih dt where wo=2A/h. [5 marks) (..) -(:. 0 dt ME(t)e-i 71,
. 2. Time evolution in two-level systems (a) A certain molecular system can be in one of two states: [10 marks) 11) 12) -0 ) 0 1 C) ( and the Hamiltonian is Eo -A II -A E 0) Compute the eigenvalues E4 of energy and the associated eigenstates E+). (i) If a system is prepared in a state E+), what is the state of the system at t> 0? If a system is prepared in a state 1) or 2), what is the state of the system at a later time? (ii) Suppose that the Hamiltonian is modified by the presence of a constant electric field, such that the new Hamiltonian is E, he -A II -A Εν - με where e and ji are constants. Compute the new eigenvalues and eigenvectors. Give the answer for the unit-norm eigenvectors in terms of a single parameter 0, where 0 is an angle to be determined. (b) The same molecule is now prepared in a state E+) and is placed in a cavity containing a time-dependent electric field e(t). Relative to the usual basis (i.e. (1,0) and (0,1)), the Hamiltonian is now Do -A 1 0 II + με(t) -A E such that the Schrödinger equation reads in(t)) = 11(0)|\-(t)). dt Since the eigenbasis of the unperturbed Hamiltonian is complete, the state (0)) can be written as ()) = COE+) + C2(0)|E_). Assuming this information answer the following two questions: () Obtain the ODEs satisfied by G and C2. [5 marks) () Using the trial solution C = n(t)e-i£44/h, C, = 72(t)e-iß_l/h, show that dy d92 ih- He(t)ekunde ih dt where wo=2A/h. [5 marks) (..) -(:.
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