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Phys321Lec10 - Formalism of Quantum Mechanics Dirac...

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Formalism of Quantum Mechanics Dirac Notation We can use a shorthand notation for the normalization integral The state is called a ket. The complex conjugate of the ket is called a bra More generally, the scalar product of two states is written as Note that the Dirac notation is independent of the representation of the states. However, for calculation purposes, one must pick a particular representation. I = ! ( r , t ) " 2 d r = ! * ( r , t ) ! ( r , t ) " d r = ! ! ! 1 ! 2 = ! 1 * ( r , t ) ! 2 ( r , t ) " d r = ! 1 * ( p , t ) ! 2 ( p , t ) " d p ! ! Prove that ! 1 ! 2 = ! 2 ! 1 *
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Formalism of Quantum Mechanics Hilbert space If Where N is a finite real number, then we say that the function ψ is square-integrable The vector space which includes all square-integrable functions is called Hilbert space Wave functions must be square integrable in order to be normalizable. Therefore wave functions live in Hilbert space ! ! = N
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Formalism of Quantum Mechanics Dynamical variables and linear operators • Every dynamical variable is associated with a linear operator which can operate on wave functions.
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