237module3

# 237module3 - Quantum Mechanics I Module 3 Getting Started 1...

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Quantum Mechanics I - Module 3 Getting Started 1. The ordinary differential equation ( x ) dx = [ ψ ( x )] 2 has the general solution ψ α ( x ) = 1 α - x where α can be any real (or complex) number. Examine the sum of two specific solutions, ψ 1 ( x ) = 1 1 - x , ψ 2 ( x ) = 1 2 - x . (a) Does the new function ψ 1+2 ( x ) = 1 1 - x + 1 2 - x satisfy the original differential equation? (b) From your knowledge of differential equations, specifically the word “linear”, what is it about the differential equation that causes solu- tions to be non-additive? 2. Fun with Dirac notation! Rewrite each of the expressions using integrals and wavefunctions. (a) h ψ | ψ i = 1 (b) A op | ψ i = a | ψ i (c) h ψ | x op | ψ i Discussion Questions 1. What is the goal of quantum mechanics so far as you can discern from this course? 2. What critical relationships connect the behavior of particles to the behav- ior of waves? 3. Describe what is meant by: (a) the Schroedinger Wave Equation (SWE).

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