後續學習計劃安排+第一次筆記

和知乎上一些前輩們交流了一下，從大部分的前輩哪裡我確實學到了些東西，也多多少少意識到了一些不足：數學現在是制約我的一個方面，Peskin這本教材也有一定缺陷。

和幾位前輩特別交流了一下，綜合考慮了幾位前輩的意見，特別是和兩位前輩深入的交流之後，我會把學習的內容寫個簡單的note，發出來也好讓大家看看我是不是理解的有問題（其實我也蠻擔心的）。我可能會大部分情況下用英語寫，比較貼近我學習的閱讀情況。

我覺得我可能會1-2周放出一次筆記，初期可能會快一點，但是等真正學到新東西而不是做review note的時候我可能就會著重於學習了，並不會放出更多的note。再者說，接下來的一年我要完成好幾個重要的事情，精力分配也存在問題。

So I conclude my note-plan as follow: summarize what I have learned so far, while keep reading more literatures on some selected topics, which include but not limited to

• Mathematics

1. Lie group and Lie algebra, based on Howard Georgis Lie Algebras in Particle Physics.

2. More algebra (abstract algebra)
3. Spectrum analysis
4. (Topology)

• Physics

1. (Summary) Quantum Mechanics
2. (Summary) Statistical Mechanics
3. (Summary) QED
4. Review of General Physics (Optics & Atomic Physics)
5. Renormalization & renormalization group, in Ising model & field theory
6. More on field theory (Schwartzs book, Peskin + [Weinberg])

I would first more focus on physics notes. I will simultaneously work on summarization and progressing on new topics. The place I would like to release my note is, under serious consideration, here, Zhihu, for the following reasons:

• It supports LaTeX formula, even in a stupid manner (figure rather than mathjax).
• It has some basic layout which is sufficient for most situation.
• I can show-off easily.

I. Quantum Mechanics 1

I would like to start with quantum mechanics, which is somehow a symbol of physics-major student. It sounds like an inaccessible monster that rules the universe in a weird manner; however, from my point, the framework of quantum mechanics is nothing but linear algebra plus some fancy ideas. So my first notes would more likely to focus on the mathematical basis and its physics interpretation.

The first concept I would like to summarize and introduce here is a state, which could be described by density matrix. The reason people would more likely to use density matrix rather than state vector(bra, ket) is its invariant under U(1) phase transformation (global or local). And, for further exploration, when tracing out environment, density matrix is a natural generalization of description of a state. The requirement of a density matrix is its hermitian, and its trace is unity, which makes perfect sense with unity possibility; and, the trace of its square is no more than 1, which can be understood as the possibility of any single state is not negative. These are some straightforward idea.

The physical world is described by matter, which is discussed as state, together with interactions. Interactions, together with self energy (kinetic, self potential, etc) could be described by an operator named Hamiltonian. For a particle (within quantum mechanics framework, it will not be annihilated or created automatically), when we study its behavior, the Hamiltonian could be mostly described by the following form:, where the first term represents kinetic energy, is an operator of wave vector, and the second term is its potential; however, some potential dont directly enters the energy, but shift its translational invariance (say vector potential of magnetic field). In that case, the kinetic would also perceive a potential shift, such as where speed of light is omitted. This is a direct generalization of classical Hamiltonian treatment of mechanics, where the dynamics could be easily introduced. I will talk about this later.

So far, the operator is introduced without rigorous definition. Its nothing but a hermitian operator, and by a given basis, it could be expanded into a matrix form (either infinite or finite dimension is possible). In this case, the density matrix is also an operator. The operators evolution, or classically, an physical quantitys evolution, is determined by Hamiltonian. Hence, both the state (density matrix) and the operator (s exception value), could be determined when the initial state is given. Obviously, two operators could be not commutable, but the commutator could only be known by classical comparison, and by assumptions. A famous one is , which talks about coordinate and momentums commutator. Sakurais book gives a fabulous introduction to this part, and I highly recommend this. It applies an infinitesimal translation on a state, and derive the translation operators commutator with coordinate, and eventually comes to momentum.

However, quantum mechanics has a tricky but crucial concept: measurement. Let me ask a simple question: when we measure a quantity, say translational momentum, what we actually did? We have some machine, and it enforces some near-classical level influence on an incoming particle, and then receive a feedback. The feedback informs us the particles momentum, however, with great fluctuation.

Quantum mechanics says, when incident a particle, before measurement, it consists of many states to form a superposition. A superposition, naively, could be considered as a vector on a multidimensional space, for example, point in 2-d plane. But when the measurement happens, the states response to the influence contain multiple versions, and among them one would be chosen by the near-classical level machine. The probability is the same as the probability this response happens, or the weight this sub-state is in the state. For instance, the has same possibility to be in x direction or y direction if we measure the x/y binary situation; if we measure the , binary direction, the probability is 1 and 0, respectively. In words, the probability is the square of the state overlapping, or the trace of projection operator times density matrix. A projection operator is simply an operator that project any state/density matrix defined in the whole Hilbert-space to a subspace defined by the projection operator, and direct sum a zero matrix. For example, projection operator of is .

Technically, this is the measurement. Philosophically, it is less trivial then expected. It doesnt mean the mankinds currently deficiency to detect some details behind the state: its purely random. The difference is well explained by famous argument about Bohr versus Einstein. And recently released work also show the great loophole free test to the theory.

So, to summary, quantum mechanics is: deterministic, except for measurement process. Measurement is purely random, with a given possibility. For next note, I would use an example to demonstrate the basics of quantum mechanicss idea, and how it solve physics problems.

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