Exploring Quantum Physics: A High School Senior's Journey

[smize]

I am a high school senior who is new to these forums. So far, I have found them intriguing, but I do have a question that I have asked many of my teachers, but sadly, had no real response.

What are the basics of quantum physics? Basic concepts, theories, equations for example. I haven't been exposed to the field, but I wish to learn it. I have no idea where to start and would like guidance on that. Thank-you!

 

[Dickfore]

1. The wave-particle duality;

2. Probabilistic interpretation of the wave function;

3. Heisenberg's Uncertainty Principle;

4. Quantization of dynamic quantities.

 

[smize]

Thank-you! Of those, I'm somewhat familiar with the wave-particle duality and Heisenberg's Uncertainty Principle, but do you think you could give a generals summary of the other two?

 

[_DJ_british_?]

A friend of mine once said : "Introductory QM is just Fourier analysis vomiting on a page." It's funny so I thought I'd share... In other words : study some Fourier stuff!

 

[Dickfore]

I ordered the points in somewhat logical manner. There is a lot of mathematical baggage that you need to master before those concept can reveal their true content. For example, you must know:

1. Complex numbers;

2. Differential equations;

3. Statistics;

4. Linear algebra

to understand the concepts involved. Are you familiar with them?

 

[smize]

Statistics I am currently taking (but am familiar with already), complex numbers I understand from Algebra, Differential equations I have worked with a little (I am currently taking AP Calculus BC which is the equivalent of Calculus 2, but I have also borrowed my teacher's Differential equations book to study), and I have also borrowed my teacher's book on linear algebra and know enough of the basics to keep up with the concepts.

 

[vanhees71]

You should cancel "wave-particle duality" from the list. This is an old concept which has been outdated with the advent of "modern quantum theory", which has been discovered by Heisenberg in 1925.

The important concepts are states and observables. You should start with one-particle non-relativistic quantum theory, where the states can bei represented by (up to some subtleties, you should not worry about in the very beginning) complex valued functions of the position of the particle that are square integrable, i.e., you have a function $\psi(\vec{x})$ which is falling off at infinity such that you can set, by choice of the overall magnitude of this "wave function"

$$\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} |\psi(\vec{x})|^2=1.$$

The physical interpretation of this wave function is that $|\psi(\vec{x})|^2 \mathrm{d}^3 \vec{x}$ is the probability to find the particle in a small volume $\mathrm{d}^3 \vec{x}$ around the position $\vec{x}$, provided the particle has been prepared in a state, discribed by this wave function $\psi$.

This concept sounds simpler than it is, because it causes debates about the meaning of quantum theory since it's discovery nearly 90 years ago! Don't bother yourself with these more philosophical questions, before you haven't get some understanding about the theory itself. To work with the minimal interpretation is enough, where one just states that the wave function has this probabilistic meaning and nothing else.

The linear algebra comes in through the fact that you can multiply wave functions with complex numbers and add them, i.e., you can build arbitrary linear combinations from any set of square integrable functions, and you always get another square-integrable function. All these can represent states.

Further, since the wave functions are square integrable, also the scalar product,

$$\langle \psi_1 | \psi_2 \rangle=\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} \psi_1^*(\vec{x}) \psi_2(\vec{x})$$

exists for any two square-integrable functions $\psi_1$ and $\psi_2$. With this scalar product the vector space of square-integrable functions becomes the Hilbert space of square integrable functions, called $L^2$.

The next step is to understand, how to describe observables. This is not so easily explained. What comes out after some hand-waving arguments is the concept to describe observables as so-called self-adjoint linear operators. E.g., the position of the particle is described by the mutliplication of the wave function with $\vec{x}$, i.e.,

$$\hat{\vec{x}} \psi(\vec{x}):=\vec{x} \psi(\vec{x}).$$

Momentum is described by taking the derivatives with respect to $\vec{x}$, i.e.,

$$\hat{\vec{p}} \psi(\vec{x})=\frac{\hbar}{\mathrm{i}} \vec{\nabla} \psi(\vec{x}).$$

An operator $\hat{A}$ is self-adjoint, if for any pair of wave functions $\psi_1$ and $\psi_2$, you have

$$\langle \psi_1 | \hat{A} \psi_2 \rangle=\langle \hat{A} \psi_1|\psi_2 \rangle.$$

As an exercise you should check that both $\hat{\vec{x}}$ and $\hat{\vec{p}}$, defined as said above, are self-adjoint operators.

I hope this gives some hints about, which math you need to learn to understand quantum theory. It's not easy but great fun. You should take a good introductory physics book on the subject and try to learn from it. Very good books are the Feynman Lectures on Physics (three volumes, covering mechanics, electromagnetism, and quantum mechanics). As I said, it's not an easy subject, but very rewarding!

 

[Dickfore]

You should cancel "wave-particle duality" from the list. This is an old concept which has been outdated with the advent of "modern quantum theory", which has been discovered by Heisenberg in 1925.

Actually, the concept is not that outdated as it may seem. For example, what is a photon according to the Copenhagen Interpretation?

 

[homeomorphic]

You can approach quantum mechanics in various different ways. If you want to understand it on a deep level, it can get very involved. It takes 3-4 years of studying math and physics. And it takes several more years in grad school to really master it.

Typically, you would study more advanced mechanics and possibly electromagnetism before tackling quantum mechanics.

You can learn a lot about quantum mechanics by thinking very deeply about a few basic experiments and thought experiments, most notably the double-slit experiment. That's sort of the place to start.

You can try this video for the double-slit experiment



Just ignore the part at the end about observers collapsing the wave function by the act of observing. It's a very nice clip, but it's taken from the mostly retarded movie, "what the bleep do we know", which is a pseudo-scientific piece of excrement, overall. But it did happen to contain a couple nice clips like this one.

 

[smize]

My biggest issue with equations is when it doesn't define what all the variables are. How often is this an occurence? And where can I find out all the variables of the different equations?

 

[jtbell]

This is less of a problem if you use real textbooks and work through them systematically, instead of trying to piece things together from a bunch of Web sites.

 

[smize]

This is less of a problem if you use real textbooks and work through them systematically, instead of trying to piece things together from a bunch of Web sites.

Yeah, which is why I hate high school. The highest level classes are too easy, and when I try learning more, I don't have the proper resources to learn the material correctly.

 

[mal4mac]

Why not buy a real textbook and try reading it.

You may get stuck at some point, then you can ask here for a specific recommendation for another book that will get you over the point you are stuck on.

There are, of course, dozens of quantum textbooks at several different levels, so it's difficult to make one specific recommendation. One, at a 'beginning to intermediate' level, is:

Principles of Quantum Mechanics by R. Shankar

 

[yenchin]

The easiest (and rather thin) intro book to QM I have seen is Quantum Mechanics: An Accessible Introduction. It even teaches what is a complex number so the math is pretty much self-contained.

 

[micromass]

Why not read the Feynman lectures?? It takes a while until you get to quantum mechanics (the third volume), but the other books are very interesting (and challenging) to read!

 

[Dickfore]

you can also watch them online. Project Tuva from Microsoft has them uploaded.

 

[smize]

Why not buy a real textbook and try reading it.

My largest issue is money. My family is just above the income level for government aid, but we are just below the income we need. But, I will attempt to get a hold of the book you recommended; I typically try to get them from my library, but usually they have nothing on the subjects I'm reading at the level I want, so I have to use interlibrary exchange.