{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# hw1solutions - HOMEWORK ASSIGNMENT 1 PHYS852 Quantum...

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: HOMEWORK ASSIGNMENT 1 PHYS852 Quantum Mechanics I, Spring 2008 1. Let the states | 1 ) and | 2 ) form an orthonormal basis spanning a 2-d Hilbert space. Let H = a | 1 )( 1 | + b | 2 )( 2 | + c | 1 )( 2 | + d | 2 )( 1 | be the Hamiltonian of the system. What condition on c and d does the requirement that H be hermitian impose? Use this to eliminate d and then find the eigenvalues and normalized eigenvectors of the system in terms of the unspecified parameters a , b , and c . Answer: H † = H requires d = c ∗ , and also a ∗ = a and b ∗ = b . The leads to the eigenvalue equation ( a − ω )( b − ω ) − | c | 2 = ω 2 − ( a + b ) ω + ab − | c | 2 = 0. The solutions are ω 1 = 1 2 bracketleftBig a + b − radicalbig ( b − a ) 2 + 4 | c | 2 bracketrightBig , ω 2 = 1 2 bracketleftBig a + b + radicalbig ( b − a ) 2 + 4 | c | 2 bracketrightBig . with the normalized eigenvectors | ω 1 ) = ( b − ω 1 ) | 1 ) − c ∗ | 2 ) radicalbig ( b − ω 1 ) 2 + | c | 2 | ω 2 ) = c | 1 ) + ( ω 2 − a ) | 2 ) radicalbig ( ω 2 − a ) 2 + | c | 2 , 2. Consider a Hilbert space spanned by the basis set {| 1 ) , | 2 ) , | 3 )} . Let the hamiltonian be H = planckover2pi1 ω ( | 1 )( 1 | + 2 | 1 )( 2 | + 2 | 2 )( 1 | + | 2 )( 2 | + 2 | 2 )( 3 | + 2 | 3 )( 2 | + | 3 )( 3 | ) , where ω = 1 . 5kHz Find the energy eigenvalues and normalized eigenvectors of this Hamiltonian....
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

hw1solutions - HOMEWORK ASSIGNMENT 1 PHYS852 Quantum...

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

View Full Document
Ask a homework question - tutors are online