The particle in a box model is often used to make rough...

The particle in a box model is often used to make rough estimates of energy level spacings. Suppose that you have a proton confined to a one-dimensional box of length equal to a nuclear diameter (about $10^{-14} \mathrm{~m}$ ). (a) What is the energy difference between the first excited state and the ground state of this proton in the box? (b) If this energy is emitted as a photon as the excited proton falls back to the ground state, what is the wavelength and frequency of the electromagnetic wave emitted? In what part of the spectrum does it lie? (c) Sketch the wave function $\psi$ as a function of position for the proton in this box for the ground state and each of the first three excited states.

Key Concepts

Particle in a Box Model

This concept describes a quantum system in which a particle is confined within infinitely high potential barriers, leading to a quantized set of allowed energy values and corresponding wavefunctions. It serves as a foundational example in quantum mechanics to illustrate how spatial confinement results in discrete energy levels and a defined mathematical structure for the physical state of the particle.

Energy Quantization in Confinement

In confined quantum systems, such as the particle in a box, the energy states are quantized rather than continuous. The allowed energy levels depend on the quantum number n and are typically proportional to n². This quantization arises from the boundary conditions imposed on the wavefunction and highlights the deviation from classical systems where energy could vary continuously.

Quantum Transitions and Photon Emission

When a quantum system transitions between energy levels, the energy difference is either absorbed or released in the form of a photon. The energy of the emitted photon is directly equal to the difference between the initial and final energy states, and its wavelength and frequency are related to this energy by Planck’s relationship (E = hf) and the speed of light (? = c/f). This principle is central to understanding spectroscopic phenomena in quantum systems.

Wavefunction Characteristics in Quantum Mechanics

The wavefunction represents the state of a quantum particle and contains information about the probability distribution of its position. In confined systems like the particle in a box, the wavefunctions take on sinusoidal forms with nodes determined by the quantum number. Higher energy states exhibit more nodes and oscillations, reflecting the increased energy and complexity of the spatial distribution of the particle's state.

Transcript

00:01 In this exercise, we have to suppose that we have a protein inside a one -dimensional box of length 10 to the minus 14 meters.

00:09 Up here, i have written the equation for the energy of a particle confined to a one -dimensional box, that this is the energy for the nth level, en, and it's equal to n squared, h squared, divided by h times the massive the particle, times the length of the box squared.

00:31 And in question a, we have to find the difference between the energy of the first excited state, that is n equals 2, and the ground state that is n equals 1 for the proton.

00:44 So notice that e2 is 4 times h squared over 8ml squared, while e1 is h squared, divided by 8m l square.

01:01 So the difference, e2 minus e1, that i'm just going to define as d, delta e, and this is equal to 3h squared divided by 8ml square.

01:18 Now this is 3 times 6 .6 .3 times 10 to the minus 34, that's clung's constant, juul second squared divided by 8 times the mass 1 .67 times 10 to the minus 27 kilograms, that's the mass of the proton, times the length squared.

01:45 The length is 10 to the minus 27 kilograms, that's the mass of the proton, times the length squared.

01:49 The length is 10 to the minus 14 meters squared.

01:56 So the difference in energy delta e is 9 .5 .5 .5.

02:01 0 .87 times 10 to the minus 13 joules.

02:08 And if we divide this by 1 .6 times 10 to the minus 19 joules per electron volt, we get the delta e is equal to 6 .17 mega electron volts.

02:31 In question b, you have to suppose that a photon is emitted with the energy delta e, as the proton falls back to its ground state, so the energy of the photon egama equals delta e.

02:45 And based on this, we have to calculate the frequency of the photon and the wavelength.

02:52 So this is, we know that the energy of the photon is h times the wave, the frequency.

02:59 So the frequency is the energy of the photon, which in this case is just delta e divided by h.

03:06 Now, delta e is 6 .17 times 10 to the 6 electron volts, and we divide this by the planks constant, which is 4 .14 times 10 to the minus 15 electron volts second.

03:31 So the frequency is 1 .5 times 10 to the 21 hertz.

03:46 The wavelength omega is, it can be obtained actually by writing that the energy of a photon is hc over gamma.

03:58 I'm sorry, hc over lambda.

04:00 So lambda is hc divided by the energy of the photon, which is delta e.

04:07 Hc is 1 ,140 electron volts, and delta e is 6 .17 times 10 to the 6th electron volts.

04:18 Actually hc is 1240 electron volts nanometers...

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