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lecture4p2_lecture5 - Chem 120a 01/26/07 Spring 07 Dirac...

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Chem 120a Dirac Notation: Bras and Kets cont. 01/26/07 Spring 07 Lecture 5 Reading:
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To prove that v = iv i + jv j , we go back to the concept of the “completeness relation” introduced in lecture 3. We start with j By performing a sneaky, clever operation, we get i iv + j jv v = ( i i + j j ) v Since i and j span the 2D space we’re working in, the completeness relation is i j j = 1 Plugging this in, v = (1) Huzzah! Creatures as i i are known as “ket-bras”. They are operators! i i = the projection operator for i , i ˆ P j j = the projection operator for j , j ˆ P What is this projection operator mumble-jumble? They project the ket v onto basis vector i i ˆ P i i v = v x
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ab More on inner product Recall that a = [ a *] T = () ** xy aa x y b b ⎛⎞ ⎜⎟ ⎝⎠ = (a x *)(b x ) + (a y *)(b y ) = b a * What does physically mean? Æ It maps b onto a set of complex #’s The bra is thus a secondary quantity defined by the inner product. Linearity For kets, For bras Addition: c = a + b Scalar Multiplication: d = c a d = a c* Normalization v = v x 0 + v y 1 Normalization condition : vv = |v x | 2 + |v y | 2 = |v| 2 = 1 Rescaling, the normalized ket
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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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lecture4p2_lecture5 - Chem 120a 01/26/07 Spring 07 Dirac...

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