ECE615_Lecture_16

ECE615_Lecture_16 - Schrödinger equation for quantum state...

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Unformatted text preview: Schrödinger equation for quantum state s (1) Hamiltonian operator (2) energy eigenfunctions are orthonormal Lecture 16 ˆ s s i H ψ ψ = ɺ ℏ Density Matrix Formalism ˆ ˆ ˆ ( ) H H V t = + ( ) ( , ) ( ) ( ) s s n n n r t C t u r ψ = ∑ a a ˆ ( ) ( ) n n n H u r E u r = a a * 3 ( ) ( ) m n mn u r u r d r δ = ∫ a a (2) into (1): (3) Multiply by and integrate, using simplification of Hamiltonian operator matrix elements: (4) is Schrödinger equation in terms of probability amplitudes. 1 ( ) ˆ ( ) ( ) ( ) ( ) s s n n n n n n i C t u r C t H u r = ∑ ∑ a a ɺ ℏ ( ) ( ) s s m mn n n i C t H C t ⇒ = ∑ ɺ ℏ * ( ) m u r a * 3 ˆ ( ) ( ) 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 * ˆ ˆ ˆ A A | A | | A | s s s s d r s s ψ ψ ψ ψ = = = ∫ * A s s m n mn mn C C A = ∑ * 3 ˆ 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 ρ = = ∑ ( ) p s ensemble average...
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This note was uploaded on 01/10/2011 for the course ECE 615 taught by Professor Shalaev during the Fall '10 term at Purdue.

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ECE615_Lecture_16 - Schrödinger equation for quantum state...

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