# Quantum_Mechanics

## How to Learn Quantum

A Good Book: The Theory of Quantum Information or can be found in my local copy

### Basics

Feynman Lectures on Physics, volume 3: Essential chapters: Chapter 1-12 (besides chapter 4 which is optional) I strongly recommend getting "Exercises for the Feynman Lectures on Physics" as well, and doing the problems. It's available online for fairly cheap (less than \$20 USD).

Introduction to Linear Algebra by Gilbert Strang:

• Chapter 1 Introduction to Vectors + Chapter 3.1 Spaces of Vectors (https://youtu.be/3ZfrJ0Sk5iY)

• Chapter 8 Linear Transformations (https://youtu.be/CBIO4xJ1Cok and https://youtu.be/ESKcF8XFzLM)

• Chapter 6 Eigenvalues and Eigenvectors

• Chapter 9 Complex Vectors and Matrices

• Solutions to the problem sets: here

Also refer to 3b1b's linear algebra video

Quantum Mechanics - The Theoretical Minimum by Leonard Susskind and Art Friedman: you can find lecture on Youtube, but it is difficult to absorb its material without problem set.

A Modern Approach to Quantum Mechanics by Townsend: This book goes a bit further than you might need, so below are the essential chapters, along with how these chapters match up with those in The Theoretical Minimum

• Chapter 1 (Chapter 1 TTM)

• Chapter 2&3 (Chapter 2&3 TTM)

• Chapter 4 (Chapter 4&5 TTM, but uncertainty stuff was partially covered in Chapter 3 of Townsend)

• Chapter 5 (Chapter 6&7 TTM)

• Chapter 6 (Chapter 7&8 TTM)

• Chapter 7 (Chapter 10 TTM)

Quantum Computing: The one classic text for this is by Nielsen and Chuang. However, if you just want an overview, you might be better off with lecture notes that are available online, for example Mermin's excellent notes: http://www.lassp.cornell.edu/mermin/qcomp/CS483.html

Quantum Complexity Theory: This is quantum computing for a computer science perspective. Scott Aaronson has a great book on it, based on these lecture notes: https://www.scottaaronson.com/democritus/

Quantum Foundations: Decoherence and the quantum to classical transition by Schlosshauer is a wonderful book that helps explain why the world seems classical when it's actually quantum. He makes an excellent case for the Many Worlds interpretation along the way. It does get quite technical though, and a lot of the value is toward the beginning. So I think this article by the author covers a lot of the most salient points: https://arxiv.org/pdf/quant-ph/0312059.pdf

• Sneaking a look at God's cards by Ghirardi: is one of my favourites and really put me onto foundations. It has an excellent discussion of the EPR paradox and hidden variables. Another good resource is Mermin's article on Bell's inequalities: https://cp3.irmp.ucl.ac.be/~maltoni/PHY1222/mermin_moon.pdf

• Emergent multiverse: is philosophy and physics legend David Wallace's defence of the Many Worlds interpretation of QM. A beautiful book, but it's dense in philosophy and physics so it's not a light read. Perhaps this will give the flavour of it: https://arxiv.org/pdf/quant-ph/0103092.pdf

Quantum Field Theory: An interpretive introduction to quantum field theory by Teller is nothing like any normal QFT textbook. Instead of all being mathematics, he spends a lot of time on what it means.

Quantum Chemistry & Computational Chemistry: I've recently been reading about this because this is expected to be an area that quantum computing will really help with. I don't know anywhere near enough to recommend a textbook, but here's a nice review article: https://arxiv.org/pdf/1812.09976.pdf

## Schrödinger Equation

Schrödinger Equation: where $\hslash = \frac{h}{2\pi}$ is Plank's (revised) constant.

i\hslash = \frac{\partial \Psi}{\partial t} = - \frac{\hslash^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + V\Psi

$\Psi(x, t)$ is a complex number because otherwise the equation is imaginary.

$\Psi(x, t)$ is a wave function that results a probability amplitude. Similar to Newton's second law, $\Psi(x, t)$ determines the "location" of object in future time given $\Psi(x, 0)$

The statistical interpretation: $|\Psi(x, t)|^2$ is the probability density of finding particle at point $x$ at time $t$.

\int_{-\infty}^\infty |\Psi(x, t)|^2 dx = 1
\int_a^b |\Psi(x, t)|^2 dx = \text{The probability of finding the particle between } a \text{ and } b \text{ at time } t

### Philosophical Consequence of Schrödinger Equation

Suppose I measure the position of the particle at point $x$, we ask where as the particle just before I made the measurement.

• Realist: "The particle was at $x$". This suggest quantum mechanics is an incomplete theory since particle has an exact position (nature is deterministic) but quantum mechanics is unable to tell as the exact position (because our ignorance). Hidden variable theory is needed to provide a complete description of the particle.

• Orthodox (Copenhagen Interpretation): "The particle wasn't really anywhere". The act of measurement disturb the particle and force it to produce an exact position.

• Agnostic (Wrong, eliminated by John Bell): Refuse to answer. Don't worry about things that can't be measured. Because you ask before, then we can't care about it. If you ask where the particle is at the measurement, then the particle is at $x$ by measurement.

• Many Worlds Interpretation: // TODO

Suppose I take another measurement immediately after the first measurement, would I get $x$ as the result?

• Yes: everybody agrees so. The measurement shrinks the wave function into a sharp peak around $x$. Wave function collapses.

### Solutions to Schrödinger Equation

Since Schrödinger Equation is a differential equation, if $\Psi$ is a solution, then $A\Psi$ is also a solution. So we impose the solution of Schrödinger Equation to sum to $1$. This process is called normalization.

\int_{-\infty}^\infty |\Psi(x, t)|^2 dx = 1

However, for non-normalizable (non-square-integrable) solutions (where $\Psi = 0 \lor \Psi = \pm \infty$), the solution can't represent particles (not physically realizable states) and must be rejected.

Now, Schrödinger Equation has property such that if we find constant $A$ that satisfy square-integrable along some specific time $t$, $A$ can also be used for other time $t$.

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