ECE615_Lecture_16

ECE615_Lecture_16 - Schrdinger equation for quantum state s...

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Unformatted text preview: Schrdinger equation for quantum state s (1) Hamiltonian operator (2) energy eigenfunctions are orthonormal Lecture 16 s s i H = Density Matrix Formalism ( ) H H V t = + ( ) ( , ) ( ) ( ) s s n n n r t C t u r = a a ( ) ( ) n n n H u r E u r = a a * 3 ( ) ( ) m n mn u r u r d r = a a (2) into (1): (3) Multiply by and integrate, using simplification of Hamiltonian operator matrix elements: (4) is Schrdinger equation in terms of probability amplitudes. 1 ( ) ( ) ( ) ( ) ( ) s s n n n n n n i C t u r C t H u r = a a ( ) ( ) s s m mn n n i C t H C t = * ( ) m u r a * 3 ( ) ( ) mn m n H u r H u r d r = a a expectation value of observable quantity (A): (5) => Dirac delta notation in terms of probability amplitudes (6) Density matrix. Probability that system is in state s * 3 * A A | A | | A | s s s s d r s s = = = * A s s m n mn mn C C A = * 3 A |A| A mn m n m n u u u u d r = = (7)- classical probability (8) Expectation value of A (9) 2 * * ( ) s s nm m n m n s p s C C C C = = ( ) p s ensemble average...
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ECE615_Lecture_16 - Schrdinger equation for quantum state s...

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