MIT Department of Chemistry 5 74 Spring 2004 Introductory Quantum Mechanics II Instructor Prof Andrei Tokmakoff p 1 5 74 TIME DEPENDENT QUANTUM MECHANICS The time evolution of the state of a system is described by the time dependent Schr dinger equation TDSE r t H r t t i Most of what you have previously covered is time independent quantum mechanics where we mean that H is assumed to be independent of time H H r We then assume a solution of the form r t r T t i 1 H r r T t r T t t 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 Schr dinger Eqn H r r E r H r r E r H is operator corresponding to E Second eqn i Solution 1 T iE T t 0 E t T t t T t Aexp iEt Aexp i t p 2 So for a set of eigenvectors n r with corresponding eigenvalues En there are a set of corresponding eigensolutions to the TDSE n r t an n r exp i nt n En The probability density P r t r t dr r t r t may be time dependent for r t but is independent of time for the eigenfunctions n r t Therefore r are called stationary states However more generally a system may be represented as a linear combination of eigenstates r t cn n r t cn e i t n r n n n For such a case the probability density will oscillate with time coherence e g two eigenstates r t c1 1e i t c2 2 e i 1 2 t p t c1 1 c2 2 c1 c2 1 2 e i 2 1 t c2 c1 2 1e i 2 1 t 2 2 probability density oscillates as cos 2 1 t This is a simple example of coherence coherent superposition state Including momentum a wavevector of particle leads to a wavepacket 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 t U t t 0 t0 For the time independent Hamiltonian iH r t r t 0 t 1 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 T exp iHt 1 iHt 1 iHt 2 2 f H Multiplying eq 1 from the left by T 1 exp iHt we have t iHt r t 0 exp integrating t 0 t exp iHt iHt r t exp 0 r t 0 0 H t t0 r t 0 U t t0 r t0 r t exp For functions of an operator A Given a set of eigenvalues and eigenvectors of A i e A n an n you can show by expanding the function as a polynomial that f A n f an n p 4 n r t e En t t0 n r t0 or alternatively if we substitute the projection operator identity relationship U t t 0 e iH t t 0 n n n n e i n t t 0 n n En n This form is useful when n are characterized we ll develop U t t0 more later So now we can write our time developing wave function as n r t n e i n t t 0 e i n t t 0 n n r t 0 n n cn t 0 n Time evolution of a coupled two level system 2LS It is common to reduce or map problem onto a 2LS We then discard remaining degrees of freedom or incorporate them as a heat bath H H 0 H bath Let s discuss the time evolution of a 2LS with a time independent Hamiltonian Consider a 2LS with two unperturbed eigenstates a and b with eigenenergies a and b which are then coupled through an interaction Vab V a a H a a a b b b a Vab b b Vba a a Vba Vab b Since the Hamiltonian is Hermetian Hij H ji we suggest 2 b b p 5 Vab Vba Ve i H a i Ve Ve i b If we define the variables E a b 2 b a 2 Then we can solve for the eigenvalues of the coupled systems E 2 V 2 Because the expressions get messy we don t choose to find the eigenvectors for the coupled system using this expression Rather we use a substitution where we define tan 2 V 2 0 2 1 H E I i tan 2 e tan 2 e i 1 We now find that we can express the eigenvalues as E sec2 We now want to find the eigenstates of the Hamiltonian H where e g c a a cb b cos e i 2 a sin ei 2 b sin e i 2 a cos ei 2 b Orthonormal complete orthogonal a a b b 1 V p 6 Examine the limits a Weak coupling V 1 Here 0 and corresponds to a perturbed by the Vab interaction corresponds to b For 0 a b b Strong coupling V 1 Now 45 and the a b basis states are indistinguishable The eigenstates are symmetric and antisymmetric combinations 1 b a 2 Whether or corresponds to the symmetric or asymmetric combination depends on whether V is positive or negative For V 45 We can schematically represent the energies of these states a These eigenstates exhibit avoided crossing b p 7 The time evolution of this system is given by our time evolution operator U t t0 e t t0 e i t t0 Now a and b are not the eigenstates preparing a will lead to time evolution Let s prepare the system so that it is initially in state a t0 0 0 a What is the probability that it is found in state b at time t Pba t b t 2 b U t t 0 a 2 To evaluate this you need to know the transformation from the a b to the basis given above This gives Pba t where the Rabi Frequency R 1 V2 2 2 2 sin Rt V 2 V 2 R represents the frequency at which probability amplitude oscillates between a and b states Pba t V2 V2 2 0 Notice for V 0 t R t a b the stationary states and there is no time dependence For V then R V and P 1 after t 2 R 2V p 8 TIME INDEPENDENT HAMILTONIAN There are two types of values that we often calculate Correlation amplitude C t t measures the resemblance between the state of your system at time t and …