Quantum Mechanics: Self-Adjoint Operators and Stationary States, Slides of Chemistry

Download Quantum Mechanics: Self-Adjoint Operators and Stationary States and more Slides Chemistry in PDF only on Docsity! 1.2: Extension to N-particle system a) Consider N particles: i = 1,2,3,…,N Now: );,...,,(ˆˆˆˆ 21 trrrVTVTH N i i rrr +=+= ∑ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ + ∂ ∂ −=∇−== 2 2 2 2 2 22 2 2 2 222 1ˆ iiii i i i i i zyxmm p m T hhr Solve to get ),,...,,( 21 trrr N rrr Ψ b) Ψ must be bounded, continuous, single-valued, continuous derivatives with respect to all coordinates. c) Integrals over all space: N21N21N21 ...dzdzdz...dydydy...dxdxdx=τd d) Operators depend on all coordinates and all momenta and time docsity.com e) τdtrrr N 2 21 ),,...,,( rrr Ψ = probability that particle i is found between xi and xi +dxi,, yi and yi +dyi , zi and zi +dzi at time t f) τdrrrMrrrM NN ),...,,(ˆ),...,,( 2121 * rrrrrr ∫∫ ΨΨ= implies integration over all coordinates. docsity.com Called stationary states because all expectation values of the time-independent Operators are constants with respect to time and because |Ψ|2 is independent of time. Examples covered in C374 were the particle in a box, simple harmonic oscillator The rigid rotor, and the H-atom. docsity.com 1.4: Self-adjoint (Hermitian) operators These operators are special because they have real eigenvalues and therefore, correspond to physical observables. Designate an adjoint operator as: +M̂ This corresponds to the complex conjugate of a transposed matrix (Later) ( )*ˆˆ TMM =+ Must satisfy the following defining equation: ( ) ( ) τφχτφχ dMdM ** ˆˆ ∫∫ += A self-adjoint operator means MM ˆˆ =+ Therefore: ( ) ( ) τφχτφχ dMdM ** ˆˆ ∫∫ = docsity.com Theorem: Quantum mechanical operators corresponding to real physical properties of a system are self-adjoint. jjj mM ψψ =ˆFrom postulate II where mj is real. ( ) mdmdMMM ====⇒ ∫∫ τψψτψψ ** ˆ Now: ( ) ( ) ( )[ ]**** ˆ∫== τψψ dMMM If operator is self-adjoint ( ) [ ] mdmdM ==⎥⎦⎤⎢⎣⎡⇒ ∫∫ *** **ˆ τψψτψψ ( ) MM =∴ * as it must be since M corresponds to a physicalobservable. docsity.com