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HW1_Solutions

# HW1_Solutions - HOMEWORK ASSIGNMENT 1 PHYS851 Quantum...

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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....
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HW1_Solutions - HOMEWORK ASSIGNMENT 1 PHYS851 Quantum...

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