{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

HW1_Solutions - HOMEWORK ASSIGNMENT 1 PHYS851 Quantum...

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

View Full Document Right Arrow Icon

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

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

Unformatted text preview: HOMEWORK ASSIGNMENT 1 PHYS851 Quantum Mechanics I, Fall 2008 1. [10 pts]What is the relationship between ( ψ | φ ) and ( φ | ψ ) ? What is the relationship between the matrix elements of ˆ M † and the matrix elements of ˆ M . Assume that H † = H what is ( m | H | n ) † in terms of ( m | H | n ) ? ( ψ | φ ) = ( φ | ψ ) ∗ . ( m | ˆ M † | n ) = ( n | ˆ M | m ) ∗ . ( m | ˆ H | n ) † = ( m | ˆ H | n ) ∗ . 2. Prove that ( AB ) † = B † A † , where A and B are both operators (Hint: Switch to a matrix formalism and use summation notation). What is ( φ | AB | ψ ) † ? ( ˆ A ˆ B ) † = ( A ∗ B ∗ ) T ( A ∗ B ∗ ) T mn = ( A ∗ B ∗ ) nm = ∑ k A ∗ nk B ∗ km = ∑ k ( B ∗ ) T mk ( A ∗ ) T kn = ∑ k B † mk A † kn = ( B † A † ) mn 3. [10 pts] The magnitude of a vector in 3-D real space is given by || r || ≡ √ vector r · vector r , also called the ’norm’ of the vector. For complex-valued vectors in N-dimensions this generalizes to || ψ || ≡ radicalbig ( ψ | ψ ) . Expand | ψ ) onto the basis of states {| n )} , n = 1 , 2 ,... N , and express its magnitude in terms of the components ψ n = ( n | ψ ) . || ψ || 2 = ( ψ | ψ ) = ∑ n ( ψ | n )( n | ψ ) = ∑ n ψ ∗ n ψ n = ∑ n | ψ n | 2 ∴ || ψ || = radicalbig ∑ n | ψ n | 2 4. [20 pts] Suppose you start out with a set of d linearly independent vectors; | e 1 ) , | e 2 ) ,... , | e d ) ; in a d-dimensional space, but they are not orthonormal. The-dimensional space, but they are not orthonormal....
View Full Document

{[ snackBarMessage ]}

Page1 / 2

HW1_Solutions - HOMEWORK ASSIGNMENT 1 PHYS851 Quantum...

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

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online