Inverse of Exponential - Introduction to Quantum Mechanics I | PHYS 622, Assignments of Quantum Mechanics

Download Inverse of Exponential - Introduction to Quantum Mechanics I | PHYS 622 and more Assignments Quantum Mechanics in PDF only on Docsity! Homework 11: Due December 3 1. Consider a Hamiltonian which is periodic in time )(ˆ)(ˆ τ+= tHtH . a. Show that nttUttUtntU ),(ˆ),(ˆ),(ˆ 0000 ττ +=+ b. Show that ),(ˆ 0ttU can be written in the form )(ˆ 000 0),(ˆ),(ˆ ttHiettUttU −−= where ),(ˆ),(ˆ 0000 ttUttU =+τ and Ĥ is a Hemitian operator given by ( ) τ τ ),(ˆlogˆ 00 ttUiH + = . The Log of the operator as defined as the inverse of the exponential (i.e. )ˆlog(ˆ AB = if and only if )ˆexp(ˆ BA = ) . c. Using the result in a. show that there exist solutions to the time-dependent Schrodinger equation of the form )()exp()( ttit nn ψεψ −= where )(tnψ is periodic, )()( tt nn ψτψ =+ , and nε is a constant (sometimes called a quasi-energy. This is the time analog of Bloch’s theorem. Sakurai chapter 4 : 1, 2 , 5, 6