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Chapter 6: Q3CQ (page 223)
Why is the topic of normalization practically absent from Sections 6.1 and 6.2?
We don鈥檛 normalize with multiple particles as with a single particle.
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As we learned in example 4.2, in a Gaussian function of the formis the standard deviation or uncertainty in position.The probability density for gaussian wave function would be proportional tosquared:. Comparing with the timedependentGaussian probability of equation (6-35), we see that the uncertainty in position of the time-evolving Gaussian wave function of a free particle is given by
. That is, it starts atand increases with time. Suppose the wave function of an electron is initially determined to be a Gaussian ofuncertainty. How long will it take for the uncertainty in the electron's position to reach, the length of a typical automobile?
Show that if you attempt to detect a particle while tunneling, your experiment must render its kinetic energy so uncertain that it might well be "over the top."
For wavelengths greater than about, the dispersion relation for waves on the surface of water is
(a) Calculate the phase and group velocities for a wave ofwavelength.
(b) Will the wave spread as it travels? Justify your answer.
Given the situation of exercise 25, show that
(a) as , reflection probability approaches 1 and
(b) as , the reflection probability approaches 0.
(c) Consider the limit in which the well becomes infinitely deep and infinitesimally narrow--- that is and data-custom-editor="chemistry" but the product U0L is constant. (This delta well model approximates the effect of a narrow but strong attractive potential, such as that experienced by a free electron encountering a positive ion.) Show that reflection probability becomes:
A ball is thrown straight up at . Someone asks 鈥淚gnoring air resistance. What is the probability of the ball tunneling to a height of?鈥� Explain why this is not an example of tunneling as discussed in this chapter, even if the ball were replaced with a small fundamental particle. (The fact that the potential energy varies with position is not the whole answer-passing through nonrectangular barriers is still tunnelirl8.)
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