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Unformatted text preview: Chem 120A Dirac Notation 02/01/06 Spring 2006 Lecture 7 READING: See Supplemental Materials on blackboard.berkeley.edu Dirac notation for probability amplitudes Consider a simple single-slit experiment: x y Source Detector (s a) (a) (a y) a Figure 1: An experiment for considering probability. Particles are emitted from the source with random velocities. The particles then strike the detector at some point along y . The detector counts the number of strikes at each point. We defined the total probability amplitude of a particle striking the detector at a point y as the product of the amplitudes for each segment along the path: ( s y ) = ( s a ) ( a ) ( a y ) . (1) Another way of expressing this quantity is to use Dirac bra(c)ket notation. In this notation, we denote the probability amplitude of a particle leaving the source, s , and arriving at a point y on the detector as: s y = ( s y ) (2) probability of a particle leaving the source and hitting the detector at y Each of the terms in Equation 1 can also be written in this way, so that in Dirac notation, we express the total probability amplitude, ( s y ) , as s y = s a a a y , (3) where a is either 0 when the particle does not get through the slit, or 1 when it does. Chem 120A, Spring 2006, Lecture 7 1 At this point, you might wonder why this type of notation is useful. You will see that this notation gives us a more clear and physically relevant way of viewing a complicated system. Another very important reason why Dirac notation is used in quantum mechanics is that it helps to clarify the meaning of the complex conjugate. We have already seen that probability amplitudes can carry a complex term corresponding to the phase of the amplitude. The complex conjugate transpose in Dirac notation is defined as follows. (What is meant by complex conjugate transpose will be clear later in the notes.) [ s y ] T s y s y y s (4) the probability amplitude of a particle leaving the detector at y and arriving at the source In this way, we see that the complex conjugate of a given probability amplitude is just the amplitude for the reverse process, and that taking the complex conjugate actually corresponds to a physically meaningful process! We will return to this, but first, we will describe the analogy of Dirac notation to geometrical vectors. Analogy to vectors Consider a vector in two dimensions: This vector, V must be described in a method that defines both its y x V magnitude and direction. There are an infinite number of ways for expressing this information; these are associated with defining various axes x and y , such that the two axes are orthogonal. Suppose we choose x and y as they are shown in Figure . We can describe the vector by the unit vectors that lie along x and y , i and j respectively:...
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