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This is a rather long winded questions so I hope that I can get my questions across as coherent as possible.

Background: I am a third Semester Nanoscience student and I am taking my first QM course this semester. So far, we have only learned some of the basics like "particle in a box" or the "quantum harmonic oscillator". In addition to all my other courses I (and some of my friends) need to collaborate and write a "paper" about quantum dots. I am writing the part that is supposed to describe the physics of Qdots and I am a bit lost.

What I have so far: I have been using two sources:

  1. Fundamental principles of quantum dots by Parak et al. LINK
  2. Chapter 9 from "Introductory Nanoscience" by Masaru Kuno LINK

What I have understood so far:

$$\boxed{\text{Seize of the system is comparable} \\\text{to the de-Broglie wavelength of }\\ \text{an electorn}} \longrightarrow \boxed{\text{System has to be described} \\ \text{using QM}} \\ \longrightarrow \boxed{\text{Solving the Schrödinger Eq. yields} \\ \text{the possible states the system can} \\ \text{be in}}$$

Questions: After this "basic" outline, both sources calculate the density of states for non-confined "free electron gas", 2D-electron gas, quantum wire and quantum dot systems. I sort of understand what density of states means (the number of states available at each energy) but I don't really see what that tells me about my confined system and how this actually leads to novel properties and applications like colloidal quantum dots, luminescence, conductivity etc. Does anyone have an idea how they are linked?

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The density of states becomes an important quantity in quantum mechanics, particularly in time dependent systems and understanding scattering. Since you're talking about quantum dots, let me focus on time dependent perturbations, in particular, light. Light is an electromagnetic field that oscillates in time. We can write an arbitrary E-field as a sum of planewaves with different amplitudes and phases. All of the electrons in your quantum dots are charged, and so they should want to wiggle when placed in an electric field. The ability of the electric field to excite an electron from one state to another is what leads to the process of light absorption and light emission.

But how likely is it for a quantum dot of a particular size to absorb or emit a photon?

To answer that question, we could try solving the Schrodinger equation. For a single electron interacting with light, we get $\hat{H}= -\hbar^2\frac{(\hat{p}+e\hat{A})^2}{2m_e} +\hat{V(\vec{r})}$ where A is the vector potential of the light field. That turns out to be very difficult to solve directly, but we can make approximations using perturbation theory. After a reasonable amount of math (I'd recommend Sakurai's book on Quantum Mechanics) under a heading called time dependent perturbation theory, what you'll find is that the probability to absorb the photon, or to emit it, will depend on the density of states at the final energy.

Or maybe your class isn't interested in using quantum dots for their optical properties, but maybe for their semiconducting properties. Well, if you want to inject charges into a quantum dot from an electrode, you'll have to have electrons either go over a contact barrier (thermionic emission) or tunnel through it. And the rate of tunneling is proportional to the density of states at a particular energy E. Similarly, if you wanted to think about the electronic resistance of a quantum dot (the part of the resistance which comes mainly from elastic collisions of electrons) that's also related to the density of states. Or maybe you're doing a photoemission experiment to explore the band structure of the dots? That's also related to the density of states.

Or maybe you wanted to know how much a quantum dot could be a superconductor? For that, you'd want to know what is the vibrational density of states that could couple two electrons together into a Cooper pair.

In short, there are lots of reasons and applications where the density of states of a system play a critical role in determining physical properties. The down side, is that in your first course in QM, you aren't aware of what they are. So if your instructor doesn't give you any motivation, it looks like a bunch of pointless math without relevance.

Hopefully this gives some ideas of where to go in your report, and I'm happy to talk further if you want more details. Cheers, and good luck!

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