2003AugQualQUANT - Ben Sauerwine Practice for Qualifying...

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

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