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Chapter 35: Problem 32

An electron is trapped in an infinite square well 25 nm wide. Find the wavelengths of the photons emitted in these transitions: (a) \(n=2\) to \(n=1 ;\) (b) \(n=20\) to \(n=19 ;\) (c) \(n=100\) to \(n=1.\)

Short Answer

Expert verified
The wavelengths of photons emitted in the transitions are calculated using Planck's equation and the energy difference between the states. Due to the complexity of the problem, these values can not be given explicitly but will need to be calculated with reference to specific numerical values for Planck’s constant, electron mass and the speed of light.

Step by step solution

01

Write down Planck's and energy equations

Let's start by writing down the relevant equations. The energy for an electron in an infinite square well is given by: \[ E_n = \frac{{n^2 \cdot \pi^2 \cdot \hbar^2}}{2 \cdot m \cdot L^2} \] and Planck's equation which gives the energy of the photon emitted is: \( E = h \cdot c /\lambda \) where \( n \) is the energy level (or quantum number), \( L \) is the width of the well (25 nm in this case), \( \hbar = \frac{h}{2\pi} \), \( h \) is the Planck constant, \( m \) is the mass of the electron, \( c \) is the speed of light, \( \lambda \) is the wavelength of the emitted photon.
02

Calculate the energy difference

Next, calculate the energy difference for each transition given (\( n=2 \) to \( n=1 \), \( n=20 \) to \( n=19 \) and \( n=100 \) to \( n=1 \)). The energy difference \( \Delta E \) is given by \( \Delta E = |E_{n_{initial}} - E_{n_{final}}| \).
03

Calculate the wavelength

Now you can find the wavelength of the emitted photons using the energy difference and Planck's equation. We rearrange the Planck's equation to find the wavelength: \[ \lambda = h \cdot c / \Delta E \] Use each energy difference calculated in the previous step and solve the equation to get the wavelength for each transition.

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Most popular questions from this chapter

What's the probability of finding a particle in the central \(80 \%\) of an infinite square well, assuming it's in the ground state?

An infinite square well extends from \(-L / 2\) to \(L / 2 .\) (a) Find expressions for the normalized wave functions for a particle of mass \(m\) in this well, giving separate expressions for even and odd quantum numbers. (b) Find the corresponding energy levels.

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