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Chapter 35: Problem 31

An electron is confined to a cubical box. For what box width will a transition from the first excited state to the ground state result in emission of a 950 -nm infrared photon?

Short Answer

Expert verified
Width of the box can be calculated by first finding the energy of the photon using its wavelength and then using the Quantum mechanical model for a particle in a box to relate this energy to the box width. The width is obtained that the transition from the first excited to ground state gives this energy.

Step by step solution

01

Understand the relationship between energy and wavelength of a photon

From the Quantum Mechanics, we know that the energy E of a photon is given by E = hc / 位, where h is Plank's constant (\(6.626 * 10^{-34} Js\)), c is the speed of light (\(3*10^8 ms^{-1}\)), and 位 is wavelength (in our case, 950 nm = \(950 * 10^{-9} m\)).
02

Calculate the energy of the photon

We substitute the values of h, c, and 位 into the equation to find the energy E.
03

Consider the energy levels in the quantum box

In a cubical box, the energy levels are given by E = h虏n虏/8mL虏, where n is the level (n=1 for ground state, n=2 for first excited state), m is the mass of electron, and L is the width of the box. In this case, we are interested in the transition from the first excited state (n=2) to the ground state (n=1). The energy difference 鈭咵 associated with this transition can be calculated using 鈭咵 = E鈧� - E鈧�.
04

Solve for width of the box

Since we know that 鈭咵 equals to the energy E of the emitted photon, we can equate the two, and solve for L.

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