2003AugQualQUANT

# 2003AugQualQUANT - Ben Sauerwine Practice for Qualifying...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Ben Sauerwine Practice for Qualifying Exams Problem Source: CMU August 2003 Qualifying Exam A three-level quantum system, which might describe a spin-1 particle, for example, has a Hamiltonian H ˆ whose matrix representation in the orthonormal basis { 3 , 2 , 1 } is given by ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − = ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = 2 2 1 1 3 ˆ 3 2 ˆ 3 1 ˆ 3 3 ˆ 2 2 ˆ 2 1 ˆ 2 3 ˆ 1 2 ˆ 1 1 ˆ 1 ˆ ω ω ω ω i i H H H H H H H H H H h Here, , 2 1 > > ω ω are two angular frequencies. (a) A general solution to the Schrodinger equation ( ) ( ) t H t dt d i Ψ = Ψ ˆ can be written as ( ) ( ) ( ) ( ) 3 2 1 3 2 1 t c t c t c t + + = Ψ . Write down the set of coupled differential equations satisfied by the (possibly complex) coefficients ( ) t c 3 , 2 , 1 in terms of 2 1 , ω ω . ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t c t c i t c t c i t c i t c i t c i g e t c i t c i t c t c t c H t c t c t c i t c t c t c H t c t c t c dt d i 3 2 3 2 2 1 1 2 2 1 1 2 1 1 3 2 1 3 2 1 3 2 1 3 2 1 ., . 1 1 3 2 1 ˆ 3 2 1 3 2 1 ˆ 3 2 1 ω ω ω ω ω h & h h & h & h & & & & = + = − = − = + + = + + + + = + + (b) Assuming that ( ) t Ψ is normalized to unity, find the probability in terms of that an appropriate measurement at time t will find the system in:...
View Full Document

## This note was uploaded on 04/07/2012 for the course PHYSICS 767 taught by Professor Dr.jaouni during the Spring '12 term at Abu Dhabi University.

### Page1 / 4

2003AugQualQUANT - Ben Sauerwine Practice for Qualifying...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online