85109HW3Solutions - PHYS851 Quantum Mechanics I Fall 2009...

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Unformatted text preview: PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 3: Solutions Fundamentals of Quantum Mechanics 1. [10pts] The trace of an operator is defined as Tr { A } = ∑ m ( m | A | m ) , where {| m )} is a suitable basis set. (a) Prove that the trace is independent of the choice of basis. Answer: Let {| m )} and {| e m )} be two independent basis sets for our Hilbert space. We must show that ∑ m ( e m | A | e m ) = ∑ m ( m | A | m ) . Proof: summationdisplay m ( e m | A | e m ) = summationdisplay mm ′ m ′′ ( e m | m ′ )( m ′ | A | m ′′ )( m ′′ | e m ) (1) = summationdisplay mm ′ m ′′ ( m ′′ | e m )( e m | m ′ )( m ′ | A | m ′′ ) (2) = summationdisplay m ′ m ′′ ( m ′′ | m ′ )( m ′ | A | m ′′ ) (3) = summationdisplay m ′ m ′′ δ m ′ m ′′ ( m ′ | A | m ′′ ) (4) = summationdisplay m ′ ( m ′ | A | m ′ ) (5) = summationdisplay m ( m | A | m ) (6) (b) Prove the linearity of the trace operation by proving Tr { aA + bB } = aTr { A } + bTr { B } . Answer: Tr { aA + bB } = summationdisplay m ( m | aA + bB | m ) (7) = summationdisplay m ( a ( m | A | m ) + b ( m | B | m ) ) (8) = a summationdisplay m ( m | A | m ) + b summationdisplay m ( m | B | m ) (9) = aTr { A } + bTr { B } (10) 1 (c) Prove the cyclic property of the trace by proving Tr { ABC } = Tr { BCA } = Tr { CAB } . Answer: First, if Tr { ABC } = Tr { BCA } then it follows that Tr { BCA } = Tr { CAB } , so we need only prove the first identity. Tr { ABC } = summationdisplay m ( m | ABC | m ) (11) = summationdisplay mm ′ m ′′ ( m | A | m ′ )( m ′ | B | m ′′ )( m ′′ | C | m ) (12) = summationdisplay mm ′ m ′′ ( m ′′ | C | m )( m | A | m ′ )( m ′ | B | m ′′ ) (13) = Tr { CAB } (14) 2 2. Consider the system with three physical states {| 1 ) , | 2 ) , | 3 )} . In this basis, the Hamiltonian matrix is: H = 1 2 i 1 − 2 i 2 − 2 i 1 2 i 1 (15) Find the eigenvalues { ω 1 ,ω 2 ,ω 3 } and eigenvectors {| ω 1 ) , | ω 2 ) , | ω 3 )} of H . Assume that the initial state of the system is | ψ (0) ) = | 1 ) . Find the three components ( 1 | ψ ( t ) ) , ( 2 | ψ ( t ) ) , and ( 3 | ψ ( t ) ) . Give all of your answers in proper Dirac notation. Answer: The eigenvalues are solutions to det | H − planckover2pi1 ωI | = 0 (16) Taking the determinate in Mathematica gives 4 ω + 4 ω 2 − ω 3 = 0 (17) which factorizes as ω ( ω 2 − 4 ω − 4) = 0 (18) which has as its solutions ω 1 = 2(1 − √ 2) (19) ω 2 = 0 (20) ω 3 = 2(1 + √ 2) (21) the corresponding eigenvectors are | ω 1 ) = 1 2 ( | 1 ) + √ 2 i | 2 ) + | 3 ) ) (22) | ω 2 ) = 1 √ 2 ( −| 1 ) + | 3 ) ) (23) | ω 3 ) = 1 2 ( | 1 ) − √ 2 i | 2 ) + | 3 ) ) (24) The components of | ψ ( t ) ) are found via | ψ ( t ) ) = e − iHt | ψ (0) |) , giving ( 1 | ψ ( t ) ) = 1 4 parenleftBig 2 + e − i 2(1 − √ 2) t + e − i 2(1+ √ 2) t parenrightBig (25) ( 2 | ψ ( t ) ) = i 2 √ 2 parenleftBig e −...
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85109HW3Solutions - PHYS851 Quantum Mechanics I Fall 2009...

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