1 - MIT Department of Chemistry 5.74, Spring 2004:...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
MIT Department of Chemistry 5.74, Spring 2004: Introductory Quantum Mechanics II p. 1 Instructor: Prof. Andrei Tokmakoff 5.74 TIME-DEPENDENT QUANTUM MECHANICS The time evolution of the state of a system is described by the time-dependent Schrödinger equation (TDSE): ψ r , t () i = = H ˆ r , t t Most of what you have previously covered is time-independent quantum mechanics, where we mean that H ˆ is assumed to be independent of time: H ˆ = H r ˆ ( ) . We then assume a solution of the form: ( r , t ) = ϕ r ( ) 1 () = H r () ( ) T t ˆ r i = Tt () ∂ t r Here the left-hand side is a function of t only, and the right-hand side is a function of r only. This can only be satisfied if both sides are equal to the same constant, E Time-Independent Schrödinger Eqn. H r = E H r = E r ˆ r ˆ r r H is operator corresponding to E Second eqn.: 1 T iE i = = E = 0 t t + = Solution: = A exp ( iEt / = ) = A exp ( i ω t )
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
p. 2 r So, for a set of eigenvectors ϕ () with corresponding eigenvalues E n , there are a set of n corresponding eigensolutions to the TDSE. r , t ψ = a ( r ) exp ( i ω t ) = E / = n n n n n n The probability density * P = ( r , t ) ( r , t ) d may be time dependent for ψ( r, t ) , but is independent of time for the eigenfunctions ψ ( r,t ) . r , t ( ) r = r , t ( ) n Therefore, r are called stationary states . However, more generally a system may be represented as a linear combination of eigenstates: i t ( r , t ) = c ( r , t ) = c e n ( r ) n n n n n n For such a case, the probability density will oscillate with time: coherence . e.g., two eigenstates i 1 t + c 2 2 e i 2 t ( r , t ) = c 1 1 e 2 2 * i ( 2 1 t ) + c * + i ( 2 ) t * pt = ψ ψ = c 1 + c 2 + c 1 c 2 * 2 e c 1 * 1 e 1 2 1 2 2 probability density oscillates as cos ( 2 ) t 1 This is a simple example of coherence (coherent superposition state). Including momentum (a wavevector) of particle leads to a wavepacket. 1
Background image of page 2
p. 3 TIME EVOLUTION OPERATOR More generally, we want to understand how the wavefunction evolves with time. The TDSE is linear in time. Since the TDSE is deterministic, we will define an operator that describes the dynamics of the system: ψ ( t ) = Ut , t 0 ( ) ( t 0 ) For the time-independent Hamiltonian: () + iH r , t r , t = 0 (1) t = To solve this, we will define an operator T = exp ( iHt / = ) , which is a function of an operator . A function of an operator is defined through its expansion in a Taylor series: 1 iHt 2 T = exp [ iHt = ] = 1 iHt + 2!
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 12

1 - MIT Department of Chemistry 5.74, Spring 2004:...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online