# Matrices, Multiple Dimensions in Quantum Mechanics

There comes to be two main approaches to Quantum Mechanics. One approach is an equations approach which uses wavefunctions, operators and sometimes eigenstages. The other approach is a linear algebra approach that uses matrices, vectors and eigenvectors to describe quantum mechanics.

Consider an example of a quantum mechanical problem that uses linear algebra for the description of particle spin:

This allows for a more direct view of commutators as discussed in the previous article on quantum mechanics [link]. Matrices have an advantage of storing much more information elegantly and are convenient for commutations.

Matrices in fact can be written for x_hat, p_hat and other operators. Matrices are also useful for introducing more than one dimension. We can also make use of this method to give us a three-dimensional Schroedinger equation. First we will start by forming three dimensions of momentum p vectors.

# ECE530 Advanced Electronics and Optoelectronics 1/21/2020 Class Notes (1st lecture)

Textbook:        High speed electronics and optoelectronics: devices and circuits, by Sheila Prasad, Hemann Schumacher, and Anand Gopinath, Cambridge university press, 2009

Learning objective:    Principles of advanced electronics and optoelectronics are illustrated by showing their applications in advanced radar, wired/wireless communications, and electronic sensing. Key electronics/photonics devices including high speed transistors, diodes, lasers, high frequency modulators, photodetectors, amplifiers and passive circuitries are discussed.

Outcome:       Following the completion of this course students will be able to

1. Perform quantitative analysis of electronic and photonic systems using the basic principles covered in this course that include: wave propagation through dielectric media and optical waveguides, high frequency electronic circuits, generation and detection of light from semiconductor devices including semiconductor lasers, light emitting diodes and photodetectors and the modulation of light through the electro-optic
2. Articulate state-of-the-art electronics and photonics technology and future trends
3. Apply the theory of operation of electronic and photonic devices
4. Articulate the performance and design trade-offs amongst RF, Digital, and Photonic solutions in EW architecture

COURSE OUTLINE

• Review of semiconductor materials and physics
1. Semiconductor materials/crystal structure (1 week)
2. Carrier transport/recombination/generation (2 week)
3. Heterostructures (1 week)
• Electronic devices
1. High speed FET (2 week)
2. High speed HBT (1 week)
• Optoelectronics
1. Optical sources (2 week)
2. Photodetector (2 week)

Review of Quantum Mechanics

A course in devices would not be complete without device physics. The foundations of semiconductor devices are… Quantum Mechanics! A good resource for review in Quantum Mechanics, aside from the course textbook is the Quantum Physics course provided by MIT OpenCourseWare. This includes video lecutres, assignments, exams and more for three whole semesters’ worth of Quantum Mechanics. Quantum Physics is also important for studying the subject of Photonics and Quantum Electronics deeper and is necessary to become an expert in a related field. More review of Quantum Mechanics is to come soon.

Quantum Physics I
Video lectures: https://archive.org/details/MIT8.04S16/
Syllabus: https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2016/syllabus/

Quantum Physics II
https://ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/

Quantum Physics III
https://ocw.mit.edu/courses/physics/8-06-quantum-physics-iii-spring-2016/

Other courses available are found here: https://ocw.mit.edu/courses/find-by-topic/

This course features the use of Rsoft and T-CAD, Silvaco for device modeling, doping and bandgap engineering problems.

Semiconductors

Silicon wafers (~4″) go for about \$20. GaN wafers on the other hand go for about ~\$1k. The price differential may be one of the few things silicon has going for it. In other cases, consider that Silicon does not work at high speeds. For high speed semiconductor devices, III-V semiconductors are preferred and will work better. One other interesting downside of Silicon is that it is unable to emit light.

The Hybrid HBT is a better option for transistor technologies at higher frequencies. The HBT features higher electron mobility.

This course will also feature study of Ternary and Quarternary Compounds. Quarternary compounds are used for quantum well lasers. Ternary Compounds feature one variable x where quarternary compounds feature two variables x,y. Another important ternary compound not listed in InGaAs.

Material Growth

We will also discuss the process for fabrication and material growth. Material growth and fabrication is a process that generally requires a high level of technological know-how and thus only few countries manage to perform this operation. Currently, it is even possible to grow one atomic layer (Angstrom) at a time. Two methods of material growth are MOCVD (Nobel Prize awarded for this discovery) and MBE (should also get a Nobel Prize soon). MOCVD is particularly best for producing many at the same time, while MBE is better used for research production.

Bell Laboratories

As an aside, consider the company that had existed before breaking up, Bell Laboratories. At Bell Laboratories, given a monopoly it was able to fund scientists and researchers to conduct free-range scientific research and discovery. Today, researchers at Universities (primarily) need to provide evidence of advancement ever 6 months or so. A program such as Bell Laboratories allowed for aimless research to be conducted, which ended up being far more successful than could have been imagined.

Types of Solids

Consider the three types of solids, Crystalline, Polycrystalline and Amorphous. Crystalline structures feature atoms that are aligned periodically and produce a unique shape (example: Quartz). Polycrystalline solids include ceramic, saphire, even possibly other metals such as steel. Interesting point about saphire – saphire is used in PCVD as a film. Amorphous solids include liquidated solids, glass and other liquids.

Semiconductors feature a lattice structure as the two below:

Crystal Directions and Planes

The following are three types of crystal directions and planes:

It is of note that etching in crystals must be done in only certain directions according to the ‘grain’ of the crystal. In order to split a waver along a grain, make a notch at one point on the side and apply a pressure to the wafer.

Atomic Bonding
It is important to review the effect of covalent bonding between valence electrons.

Next class, the topic of wave equations will be covered.