c120a_lecture9

c120a_lecture9 - Chem 120A Operators and change of basis in...

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Unformatted text preview: Chem 120A Operators and change of basis in Quantum Mechanics 02/05/07 Spring 2007 Lecture 9 Change of basis Suppose in our space, we have two sets of complete orthonormal basis: {| i i} and {| α i} , i.e. we know that h i | j i = δ i j and ∑ i | i ih i | = ˆ I h α | β i = δ αβ and ∑ α | α ih α | = ˆ I . (1) Note that ˆ I is an identity matrix. We can convert from basis {| α i} to basis {| i i} in the following manner | α i = I | α i = ∑ i | i ih i | ! | α i = ∑ i h i | α i| i i = ∑ i u i α | i i = ∑ i ( ˆ U ) i α | i i . (2) Here we define h i | α i ≡ u i α ≡ ( ˆ U ) i α . (3) Also | i i = ∑ α h α | i i| α i = ∑ α u * i α = ∑ α ( ˆ U † ) α i , (4) where h α | i i = h i | α i * = u * i α = ( ˆ U † ) α i . (5) Entries of matrix ˆ U are u i α . Matrix ˆ U is unitary if and only if ˆ U † ˆ U = ˆ U ˆ U † = ˆ I . Now we check that matrix ˆ U with entries u i α is indeed unitary....
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This note was uploaded on 02/19/2010 for the course CHEM 120A taught by Professor Whaley during the Spring '07 term at Berkeley.

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c120a_lecture9 - Chem 120A Operators and change of basis in...

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