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Chapter 37: Problem 5

Let \(\kappa\) be the magnitude of the wave number of a particle moving in one dimension with velocity \(v\). If the velocity of the particle is doubled, to \(2 v,\) then the wave number is: a) \(\kappa\) b) \(2 \kappa\) c) \(\kappa / 2\) d) none of these

Short Answer

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Question: When the velocity of a particle moving in one dimension is doubled, its wave number: a) stays the same b) is doubled c) is halved d) increases by a factor of 4 Answer: b) is doubled

Step by step solution

01

Recall the de Broglie wavelength equation

The de Broglie wavelength can be determined using the equation: \( \lambda = \dfrac{h}{m \cdot v} \) where \(\lambda\) is the wavelength, \(h\) is the Planck's constant, \(m\) is the mass of the particle, and \(v\) is its velocity.
02

Calculate the initial wave number

The wave number \(\kappa\) is the reciprocal of the wavelength. We can find the initial wave number using the following equation: \( \kappa = \dfrac{1}{\lambda} \) By substituting the de Broglie wavelength equation, we get: \( \kappa = \dfrac{m \cdot v}{h} \)
03

Determine the new wavelength after doubling the velocity

Now we need to find the new wavelength when the velocity is doubled to \(2v\). Using the de Broglie wavelength equation, we get: \( \lambda' = \dfrac{h}{m \cdot (2v)} = \dfrac{h}{2m \cdot v} \)
04

Calculate the new wave number

Now we need to find the new wave number. We can use the following equation: \( \kappa' = \dfrac{1}{\lambda'} \) Substituting the new wavelength, we get: \( \kappa' = \dfrac{1}{\dfrac{h}{2m \cdot v}} = \dfrac{2m \cdot v}{h} = 2 \kappa \) The new wave number when the velocity is doubled is \(2 \kappa\). Hence, the correct answer is option (b).

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Most popular questions from this chapter

A surface is examined using a scanning tunneling microscope (STM). For the range of the working gap, \(L\), between the tip and the sample surface, assume that the electron wave function for the atoms under investigation falls off exponentially as \(|\Psi|=e^{-\left(10.0 \mathrm{nm}^{-1}\right) a}\). The tunneling current through the STM tip is proportional to the tunneling probability. In this situation, what is the ratio of the current when the STM tip is \(0.400 \mathrm{nm}\) above a surface feature to the current when the tip is \(0.420 \mathrm{nm}\) above the surface?

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Consider an electron that is confined to a onedimensional infinite potential well of width \(a=0.10 \mathrm{nm}\), and another electron that is confined by an infinite potential well to a three-dimensional cube with sides of length \(a=0.10 \mathrm{nm} .\) Let the electron confined to the cube be in its ground state. Determine the difference in energy and the excited state of the one- dimensional electron that minimizes the difference in energy with the three- dimensional electron.
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