85109HW3Solutions

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

This preview shows pages 1–4. Sign up to view the full content.

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

View Full Document

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: 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 −...
View Full Document

{[ snackBarMessage ]}

### Page1 / 15

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online