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Unformatted text preview: 39 Wave Functions and Uncertainty Recommended class days: 2 Background Information Classical particles have a well-defined position at all instants of time and are described by a trajectory x ( t ). The experimental evidence for wave-particle duality should have convinced students that atomic-level particles do not have a well-defined position. Hence they cannot be described by a classical trajectory. This chapter raises the issue of how to describe the behavior of atomic particles. The double-slit experiment and the analogy between electrons and photons lead us to the idea of a wave function. This chapter looks at wave functions from a pictorial and graphical representation. The law of wave functionsthe Schrdinger equationis deferred to Chapter 40. It is important to emphasize that the wave function is a new hypothesis . Only subsequent experimental tests will reveal if there is any merit to the hypothesis. The wave function is an abstract concept, although no more abstract than the concept of a field. Not surprisingly, it takes quite some time before students begin to feel comfortable with this idea or can use it to reason about physical situations. The focus of this chapter is to connect the wave function with experimental outcomes and measurable probabilities, thus linking this new idea with physical reality. The probability ideas of quantum mechanics are especially difficult for many students. It takes a number of examples for them to catch on to measuring the probability of finding a particle in an interval x at position x . To keep the idea clear, this text writes explicitly Prob(in x at x ). The concept of probability density is especially difficult. The analogy of linear mass density along a string of variable diameter is helpful to many students. Student Learning Objectives To introduce the wave function as the descriptor of particles in quantum mechanics. To provide the wave function with a probabilistic interpretation. To understand the wave function through pictorial and graphical exercises. To introduce the idea of normalization. To recognize the limitations on knowledge imposed by the Heisenberg uncertainty principle. Pedagogical Approach Double-slit interference is the experiment that leads most naturally to the idea of a wave function. Both photons and electrons make wave-like interference patterns that are built up particle by particle. For light interference, a straightforward argument links the probability of a photon landing in an interval x at position x to the amplitude function A ( x ) of the classical light wave across the screen. Because electrons make the same pattern, we can postulate that there is some new wave-like 39-1 39-2 Instructors Guide function ( x ) that is associated with the probability of an electron landing in an interval x at position x . Our goal, in this chapter and the next, is to learn: What are the properties of this function ( x )?...
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- Spring '10