MIT5_74s09_lec02

# MIT5_74s09_lec02 - MIT OpenCourseWare http:/ocw.mit.edu...

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MIT OpenCourseWare http://ocw.mit.edu 5.74 Introductory Quantum Mechanics II Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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2-1 Andrei Tokmakoff, MIT Department of Chemistry, 2/13/2007 2.1. TIME-DEPENDENT HAMILTONIAN Mixing of eigenstates by a time-dependent potential For many time-dependent problems, most notably in spectroscopy, we often can partition the time-dependent Hamiltonian into a time-independent part that we can describe exactly and a time-dependent part HH V t + ( ) = 0 (2.1) Here H 0 is time-independent and Vt ( ) is a time-dependent potential, often an external field. Nitzan, Sec. 2.3., offers a nice explanation of the circumstances that allow us to use this approach. It arises from partitioning the system into internal degrees of freedom in H 0 and external degrees of freedom acting on H 0 . If you have reason to believe that the external Hamiltonian can be treated classically, then eq. (2.1) follows in a straightforward manner. Then there is a straightforward approach to describing the time-evolving wavefunction for the system in terms of the eigenstates and energy eigenvalues of H 0 . We know Hn E = n . (2.2) 0 n The state of the system can be expressed as a superposition of these eigenstates: c t n (2.3) ψ ( t ) = n ( ) n The TDSE can be used to find an equation of motion for the expansion coefficients ct = k ( t ) (2.4) k ( ) Starting with i = H (2.5) t = k ( ) =− i kH t (2.6) () t = i inserting nn = 1 kHnc t (2.7) n = n n
2-2 substituting eq. (2.1) we have: ct () i k =− kH Vt ( + ()) nct 0 n t = n (2.8) i E V t c t n δ kn + kn n = n or, + i Ec t i V t c t . (2.9) k = t = kk = kn n n If we make a substitution iE t = ( ) = e m bt ( ) , (2.10) m m which defines a slightly different expansion coefficient, we can simplify considerably. Notice 2 2 that = ct 0 In practice what we are doing is pulling out the . Also, b k () = c k ( 0 ) . bt k k “trivial” part of the time-evolution, the time-evolving phase factor for state m . The reasons will become clear later when we discuss the interaction picture. It is easy to calculate k and then add in the extra oscillatory term at the end. Now eq. (2.9) becomes e iE k t = b k i Vt e iE n t (2.11) kn = n t = n or i = b t k = n kn e i ω nk t b n (2.12) V This equation is an exact solution. It is a set of coupled differential equations that describe how probability amplitude moves through eigenstates due to a time-dependent potential. Except in simple cases, these equations can’t be solved analytically, but it’s often straightforward to integrate numerically.

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2-3 Resonant Driving of Two-level System As an example of the use of these equations, let’s describe what happens when you drive a two- level system with an oscillating potential. Vt ( ) = V cos ω t V = ft ( ) (2.13) Note: This is what you expect for an electromagnetic field interacting with charged particles, i.e.
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## This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.

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MIT5_74s09_lec02 - MIT OpenCourseWare http:/ocw.mit.edu...

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