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# 2 - MIT Department of Chemistry 5.74 Spring 2004...

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MIT Department of Chemistry 5.74, Spring 2004: Introductory Quantum Mechanics II p. 13 Instructor: Prof. Andrei Tokmakoff QUANTUM DYNAMICS 1 The motion of a particle is described by a complex wavefunction ψ( r, t ) that gives the probability amplitude of finding a particle at point r at time t . If we know ψ () , how does it r , t 0 change with time? ? r , t ( ) r , t ( ) t > t 0 0 We will use our intuition here (largely based on correspondence to classical mechanics) We start by assuming causality: t precedes and determines t . ( ) 0 Also assume time is a continuous parameter: lim ( t ) = ( ) t 0 t t 0 Define an operator that gives time-evolution of system. t ( ) = Ut , t ( ) t ( ) 0 0 This “time-displacement operator” is similar to the “space-diplacement operator” ik(r r 0 ) r ) = e r 0 ) which moves a wavefunction in space. U does not depend on . It is a linear operator. if t 0 = a 1 + a 2 ϕ( t 0 ) ψ ϕ 1 ( t 0 ) ψ t = U t , t 0 ) ψ( t 0 ) ( = aU t t 0 ) + a U t t 0 ) 1 ( ϕ 1 ( t 0 ) 2 ( ϕ 2 ( t 0 ) = at t 1 a 2 ϕ + 1 ϕ 2 From Merzbacher, Sakurai, Mukamel

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p. 14 while a i t () typically not equal to a i ( 0 ) , a ( t ) = a ( t ) n n 0 n n Properties of U(t,t 0 ) Time continuity: Ut , t = 1 Composition property: 2 , t ( 2 , t ) = ( 1 , t 0 ) (This should suggest an exponential form). 0 1 1 , t ψ ( t ) = 2 , t ( t 0 ) 2 1 0 Note: Order matters! = 2 , t t 1 1 , t ( 0 , t ) = 1 0 U 1 ( t, t ) = U t , t ) inverse is time-reversal ( 0 0 Let’s write the time-evolution for an infinitesimal time-step, δ t. lim U t 0 + δ t, t )= 1 ( 0 t δ→ 0 We expect that for small δ t , the difference between 0 , t ) and Ut 0 t , t 0 ) will be linear ( 0 ( in δ t . (Think of this as an expansion for small t): 0 + δ t ) = Ut ) δ t ( 0 ( 0 i 0 is a time-dependent Hermetian operator. We’ll see later why the expansion must be complex. Also, Ut 0 t 0 ) is unitary. We know that U -1 U = 1 and also ( ( 0 t ) U t ) = ( 1 + Ω δ t )( 1 − Ωδ 1 ( 0 i i t ) 0 0 We know that Ut t ) = t ) . ( 0 ( ) ( 0
p. 15 Knowing the change of U during the period δ t allows us to write a differential equation for the time-development of Ut , t () . Equation of motion for U : 0 dU t, t 0 ) lim U t t, t ) − U t, t 0 ) ( ( 0 ( = t dt δ→ 0 δ t lim Ut t , ) − 1 t 0 ) ( ( = t 0 δ t

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2 - MIT Department of Chemistry 5.74 Spring 2004...

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