Chapter2_10 - 1 > and | 2 > is...

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PHY4604 R. D. Field Department of Physics Chapter2_10.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 &gt; and | 2 &gt; is &lt; 1 | 2 &gt; like the dot product of two vectors 2 1 V V r r &lt; 1 | 2 &gt; is a complex number The scalar product &lt; 2 | 1 &gt; is referred to as the overlap between the two states. Norm of a State: The norm of the Ket-vector | &gt; is the scalar product of | &gt; with itself &lt; | &gt; like the square of a vector V V r r &lt; | &gt; 0 (positive definite real number) Normalized wavefunctions have &lt; | &gt; = 1 and two wavefunctions are said to be orthogonal if their overlap is zero, &lt; 2 | 1 &gt; = 0 . An orthonormal set of wavefunctions has the property that ij j i &gt;= &lt; | . Ket-Vector Space: | i &gt; Bra-Vector Space: &lt; i | Dual (Adjoint) Like (x,t)! Like (x,t)!...
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