An electron is confined to a one-dimensional box of length...

An electron is confined to a one-dimensional box of length $L$. When the electron makes a transition from its first excited state to the ground state, it emits a photon of energy 0.20 eV. (a) What is the ground-state energy (in electron-volts) of the electron in this box? (b) What are the energies (in electron-volts) of the photons that can be emitted when the electron starts in its third excited state and makes transitions downwards to the ground state (either directly or through intervening states)? (c) Sketch the wave function of the electron in the third excited state. (d) If the box were somehow made longer, how would the electron's new energy level spacings compare with its old ones? (Would they be greater, smaller, or the same? Or is more information needed to answer this question? Explain.)

Key Concepts

Particle in a One-Dimensional Infinite Potential Well

This concept describes a fundamental quantum mechanical model where a particle, such as an electron, is confined to move in a one-dimensional region with infinitely high potential barriers at the boundaries. The confinement creates discrete energy levels and quantized states that are central to understanding quantum behavior in bound systems.

Energy Quantization in Confined Systems

In a confined system like the 1D infinite potential well, the allowed energy levels are quantized. The energy of the nth state is proportional to n^2 and inversely proportional to the square of the length of the confinement. This relation arises from solving the Schrödinger equation under the boundary conditions imposed by the infinite potential walls.

Quantum Transitions and Photon Emission

When a quantum system transitions between energy levels, the difference between the initial and final energy states is emitted or absorbed as a photon. The energy of the photon directly corresponds to this energy difference, making it possible to deduce properties of the quantum states based on the photon energies observed.

Wave Function Characteristics in a Confined System

The wave functions for a particle in a 1D box are sinusoidal and form standing wave patterns inside the well. These functions, which are typically sine functions, describe the probability amplitude of finding the particle at a given position. Each quantized state has a distinct number of nodes, which corresponds to its quantum number.

Dependence of Energy Levels on Box Size

The energy levels of a particle in a 1D infinite potential well depend inversely on the square of the length of the box. A longer box results in smaller energy level spacings, meaning the quantized energy levels become closer together. This concept is crucial when considering how changes in the confinement dimensions affect the physical properties of the system.

Transcript

00:01 In this exercise, we have an electron confined to a one -dimensional box of length l, and the electron transitions from the first excited state that is n equals 2 to the ground state, and in the process it emits a photon that has an energy of 0 .2 electron volts.

00:19 This means that the difference in energy between the second excited, i'm sorry, the first excited state and the ground state is 0 .2 electron volts.

00:28 In question a, we have to find what is the energy of the ground state e1.

00:35 In order to solve this, notice that the difference delta e of energy between the first excited state and the ground state is e2 minus e1.

00:49 And as i have written here in red, highlighted in red, we have that the energy of the ground state is h squared divided by 8ml squared, while the energy of excited states, en, is equal to n squared times the energy of the ground state.

01:11 So for the first excited state, e2, we have that the energy is 4e1, which minus e1 is equal to 3e1.

01:24 So e1 is equal to delta e divided by 3.

01:28 So this is 0 .2 electron volts divided by 3, which is 0 .0667 electron volts.

01:39 This is the energy of the ground state.

01:44 In question b, we have to assume that the electron is in the third excited state, that is n equals 4, and then transitions to the ground state.

01:55 And we have to calculate all possible energies of the photons that can be emitted, during this transition.

02:02 So there are several ways for this transition to happen.

02:07 The electron can jump directly from the fourth to the first excited, i'm sorry to the third, from the third, the third to the ground state.

02:15 It can transition from the third excited state to the second, then to the first, then to the ground state.

02:22 It can also transition from the fourth to the third state, and then to the first, from the third excited state, then to the second, i'm sorry, then to the first, and then to the ground state.

02:39 So all possible transitions are from the fourth to the first, from the fourth to the second, and from the fourth to the third, but also from the third to the second, from the third to the first, and from the second to the first.

02:58 And we have to calculate the difference in energy for each one of these transitions.

03:05 And the difference in energy will be the energy of the emitted photon.

03:11 So let's start by calculating the difference in energy between the fourth and the first excited state.

03:20 And this is, i'm sorry, between the fourth state and the first state, that is the third excited state and the ground state.

03:27 So this is e4 minus e1.

03:31 This is 16e1 minus e1.

03:36 So this is 15e1.

03:41 And since we know that e1 is 0 .0667, 15 times 0 .667 is 1 electron volt.

03:52 Then we can calculate the difference in energy between the fourth and the second state.

03:57 So this is e4 minus e2.

04:03 This is 16.

04:04 E1 minus 4e1.

04:08 So this is 12e1, which is equal to 0 .8 electron volts.

04:16 Then the difference in energy between the fourth and the third states.

04:22 So this is a 4 minus e3, and this is 16e1 minus 9e3.

04:31 I'm sorry, e1.

04:32 So this is 7e1, which is equal to 0 .47 electron volts.

04:42 We still have three to go, so the difference in energy between the third, the second excited state and the ground state, is a 3 minus e1.

04:57 So this is 9e1 minus e1, which is 8e1, which is equal to 0 .50...

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