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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.
 Spring '07
 Whaley
 Physical chemistry, pH

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