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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

The wavelength of an electron in an infinite potential is \(\alpha / 2,\) where \(\alpha\) is the width of the infinite potential well. Which state is the electron in? a) \(n=3\) b) \(n=6\) c) \(n=4\) d) \(n=2\)

Suppose a quantum particle is in a stationary state (energy eigenstate) with a wave function \(\psi(x, t) .\) The calculation of \(\langle x\rangle,\) the expectation value of the particle's position, is shown in the text. Calculate \(d\langle x\rangle / d t(\operatorname{not}\langle d x / d t\rangle)\).

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