Quantum Dots and the "Particle in a 1-D Box" Model

[spaghettibretty]

Homework Statement

The "particle in a box" is the simplest quantum system. Despite this, it still reveals a lot of important new quantum features. Please use the "particle in a 1-D box" model to explain the reason why when the size of a quantum dot gets smaller, the wavelength of the emitted photon due to the n=1⇒n=2 transition becomes shorter (color shifts to blue).

Homework Equations

speed of light = wavelength * frequency
(it isn't listed in the question, but I use it for my answer)

The Attempt at a Solution

I'm not really sure how to tackle this question and give a sufficient answer. I tried reading around on the "particle in a 1-D box" model as well as quantum dots, but I'm not sure what I'm saying is sufficient.

My current answer is basically that the oscillation period of a particle in a 1-D box will decrease if the length of the box is reduced. A lower oscillation period equals a higher frequency, which, in turn, equals a lower wavelength. Therefore, if the length of the box decreases, the oscillation period decreases, and the wavelength decreases. This can be applied to a quantum dot. As the quantum dot's size is reduced, the frequency of its emitted photons will rise and cause the wavelength of those photons to decrease. The decreasing of the wavelength means that the perceived color of the photons will be more blue in comparison.

Is this answer enough? Are its components even correct? This material wasn't really covered by my professor and it's more of a self-research problem.

 

[BvU]

Oscillation period of a particle in a 1-D box ? It's abox, not a (harmonic) oscillator . The question is about photons that are emitted when transitions from higher energy levels to lower energy levels occur. (n=1 ⇒ n=2 is not one of these; an error in the text ?).

 

[spaghettibretty]

Oscillation period of a particle in a 1-D box ? It's abox, not a (harmonic) oscillator . The question is about photons that are emitted when transitions from higher energy levels to lower energy levels occur. (n=1 ⇒ n=2 is not one of these; an error in the text ?).

Oh, my mistake. It's suppose to be n=2⇒n=1. If the oscillation period thing is incorrect, that pretty much tears down my whole argument.
On another thought, I also know that if there are more nodes in the 1-D box, then there is a higher energy and higher energy equals higher frequency. However, I'm not sure how that ties in with the quantum dots.

 

[BvU]

It's not about the particles themselves. It's about the energy change of the (charge carrying) particles when they go from a higher energy state to a lower energy state. That energy difference is sent off in the form of a photon. So you want to find an expression for the energies of the possible states, and see if these depend on the size of the box. (A quantum dot is considered as a box here).

 

[paranormal]

Your answer is on the right track, but there are a few key points that you could expand on to make it more complete.

First, it would be helpful to explain why the "particle in a box" model is relevant to understanding quantum dots. A quantum dot can be thought of as a tiny box that confines electrons, similar to the 1-D box in the model. This confinement leads to quantized energy levels, just like in the model, and these energy levels determine the possible transitions and emitted photons.

Next, you could explain in more detail why the wavelength of the emitted photon decreases as the size of the quantum dot decreases. As you mentioned, this is due to the decrease in oscillation period. This can be further explained by the relationship between energy and wavelength. The energy of a photon is directly proportional to its frequency (and inversely proportional to its wavelength), so as the energy levels of the confined electrons increase, the frequency (and energy) of the emitted photons also increases. This results in a shorter wavelength and a shift towards the blue end of the spectrum.

Additionally, you could mention the concept of quantum confinement in relation to quantum dots. This refers to the confinement of electrons in a small space, which leads to changes in their behavior and properties. In the case of quantum dots, the quantum confinement leads to the quantization of energy levels and the resulting shift in emitted photon wavelengths.

Overall, your answer is a good start, but expanding on these points and providing more context and explanation will make it more complete and comprehensive.