Chapter6_16

# Chapter6_16 - 1> and | Ψ 2> is< Ψ 1 | Ψ 2> like...

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PHY3063 R. D. Field Department of Physics Chapter6_16.doc University of Florida Dirac “Bracket” Notation (1) It is very convenient to make the following definitions Ψ Ψ >≡ Ψ Ψ < dx t x t x ) , ( ) , ( | 1 * 2 1 2 , and Ψ Ψ >≡ Ψ Ψ < dx t x O t x O op ) , ( ) , ( | | 1 * 2 1 2 . Note that | Ψ 1 > is called the “Ket” and < Ψ 2 | is called the “Bra” . “Ket-Vectors”: We associate with each dynamical state of the system a “Ket-vector”, | Ψ > . The “Kets” form a linear vector space as follows: | Ψ 1 > is like the vector 1 V r | Ψ 2 > is like the vector 2 V r | Φ > = a| Ψ 1 > + b| Ψ 2 > is also a “Ket-vector” like 2 1 V b V a U r r r + = . Dual Space and “Bra-Vectors”: The is an “antilinear” correspondence between “Ket-Space” and “Bra-Space” as follows: | Ψ > < Ψ | a| Ψ > a * < Ψ | “Bra-Space” is adjoint space dual to “Ket-Space” with (a| Ψ >) a * < Ψ | Scalar Product: The scalar product between the “Ket-vectors” | Ψ
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Unformatted text preview: 1 > and | Ψ 2 > is < Ψ 1 | Ψ 2 > like the dot product of two vectors 2 1 V V r r ⋅ < Ψ 1 | Ψ 2 > is a complex number The scalar product < Ψ 2 | Ψ 1 > is referred to as the “overlap” between the two states. Norm of a State: The “norm” of the “Ket-vector” | Ψ > is the scalar product of | Ψ > with itself < Ψ | Ψ > like the square of a vector V V r r ⋅ < Ψ | Ψ > ≥ 0 (positive definite real number) Normalized wavefunctions have < Ψ | Ψ > = 1 and two wavefunctions are said to be “orthogonal” if their overlap is zero, < Ψ 2 | Ψ 1 > = 0 . An “orthonormal” set of wavefunctions has the property that ij j i δ >= Ψ Ψ < | . “Ket-Vector” Space: | Ψ i > “Bra-Vector” Space: < Ψ i | Dual (“Adjoint”) Like Ψ (x,t)! Like Ψ ∗ (x,t)!...
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## This note was uploaded on 05/29/2011 for the course PHY 3063 taught by Professor Field during the Spring '07 term at University of Florida.

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