Tag: Solutions

Problem 1.11 [This problem generalizes Example 1.2.] Imagine a particle of mass and energy in a potential well , sliding frictionlessly back and forth between the classical turning points ( and in Figure 1.10). Classically, the probability of finding the particle in the range (if, for example, you took a snapshot at a random time…

Problem 1.10 Consider the first 25 digits in the decimal expansion of (3, 1, 4, 1, 5, 9, … ). (a) If you selected one number at random, from this set, what are the probabilities of getting each of the 10 digits?(b) What is the most probable digit? What is the median digit? What is…

Problem 1.8 Suppose you add a constant to the potential energy (by “constant” I mean independent of as well as ). In classical mechanics this doesn’t change anything, but what about quantum mechanics? Show that the wave function picks up a time-dependent phase factor: . What effect does this have on the expectation value of…

Problem 1.7 Calculate . Answer: . This is an instance of Ehrenfest’s theorem, which asserts that expectation values obey the classical laws. Solution: Problem 1.7

Problem 1.6 Why can’t you do integration-by-parts directly on the middle expression in Equation 1.29—pull the time derivative over onto , note that , and conclude that ? Solution: Problem 1.6

Problem 1.5 Consider the wave function where , , and are positive real constants. (We’ll see in Chapter 2 for what potential (V) this wave function satisfies the Schrödinger equation.) (a) Normalize .(b) Determine the expectation values of and .(c) Find the standard deviation of . Sketch the graph of , as a function of…

Problem 1.4 At time a particle is represented by the wave function where , , and are (positive) constants.(a) Normalize (that is, find , in terms of and ).(b) Sketch , as a function of .(c) Where is the particle most likely to be found, at ?(d) What is the probability of finding the particle…

Problem 1.3 Consider the gaussian distribution where , , and are positive real constants. (The necessary integrals are inside the back cover.)(a) Use Equation 1.16 to determine .(b) Find , , and .(c) Sketch the graph of . Solution: Problem 1.3