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quantum mechanics

Table of Contents

1. Introduction

Quantum mechanics was discovered as a predictive framework in the early 1900's after a set of experiments (i.e. the photoelectric effect and the Stern Gerlach experiment) showed that particles were better described as waves, and that the probability of measurement of particle states were random. Indeed, other experiments that tested Bell's inequality have confirmed this assertion of quantum mechanics that the world is fundamentally random. Let's take a look at some of the postulates of quantum mechanics (for which the mathematics borrow from linear algebra quite a bit).

2. Postulates

Here, we describe all the postulates of quantum mechanics, with some motivation when needed.

2.1. Postulate 1

\(| \psi \rangle\) is the state vector in a Hilbert Space \(\mathcal{H}\) which describes the entire system.

2.2. Postulate 2

The norm \(\langle \psi | \psi \rangle\) of all state vectors are 1.

2.3. Postulate 3

If \(| \psi \rangle\) and \(| \phi\rangle\) represent two different quantum systems, the composite system can be described by \(| \psi \rangle \otimes | \phi \rangle\), where \(\otimes\) is the tensor product.

2.4. Postulate 4

Observable quantities are represented by Hermitian operators \(\hat{A}\) whose eigenvectors form a basis for \(\mathcal{H}\). Solutions to \(\hat{A}|\psi\rangle = a|\psi\rangle\) for eigenvalues \(a\) are the only possible observable values for a given eigenvector \(|\psi\rangle\).

2.5. Postulate 5

The time-evolution of the state vector \(|\psi\rangle\) can be given by the Schrodinger equation, a PDE:

\begin{align} \label{} \hat{H}|\psi\rangle = i\hbar\partial_{t}|\psi\rangle \end{align}

which is motivated by the De Broglie hypothesis:

\begin{align} \label{} p = hf \end{align}

where \(p\) is the momentum, and \(f\) is the frequency. De Broglie hypothesized that all matter behaves like a wave, after Einstein came up with the same relation for light:

\begin{align} \label{} p = \frac{E}{c} = hf \end{align}

where \(h\) is Planck's constant, which is specifically a relation

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