Lect6_TimePropagation

Lect6_TimePropagation - Lecture 6 Time Propagation Ordinary...

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Lecture 6: Time Propagation Outline: Ordinary functions of operators Powers Functions of diagonal operators Solving Schrödinger's equation Time-independent Hamiltonian The Unitary time-evolution operator Unitary operators and probability in QM Iterative solution Eigenvector expansion Ordinary Functions of Operators Let us define an `ordinary function’, f(x) , as a function that can be expressed as a power series in x , with scalar coefficients: When given an operator, A , as an argument, we define the result to be: Examples: THM: A functions of an operator is defined by its power series n n n x f x f ! = ) ( n n n A f A f ! = : ) ( ! " = = + + + + = + # + # = 0 3 2 7 5 3 ! .... ! 3 ! 2 ... ! 7 ! 5 ! 3 ) sin( n n A n A A A A I e A A A A A
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Powers of Operators An operator raised to the zero th power: Positive integer powers: Operator inversion: The operator A -1 is defined via: Negative powers: Fractional powers: ... : : : 3 2 1 etc AAA A AA A A A = = = I A = : 0 ( ) A A I A A = = ! ! ! : : 1 1 1 n n A A ) ( : 1 ! ! = ... : 2 / 1 2 / 1 etc A A A =
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Functions of Diagonal Operators Diagonal operators have the form: They can be expressed in Dirac notation as: Every operator is diagonal in the basis of its own eigenvectors They have the property: let C and D be diagonal matrices
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Lect6_TimePropagation - Lecture 6 Time Propagation Ordinary...

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