# Tag Info

Quantum mechanics describes the microscopic properties of nature in a regime where classical mechanics no longer applies. It explains phenomena such as the wave-particle duality, quantization of energy and the uncertainty principle and is generally used in single body systems. Use the quantum-field-theory tag for the theory of many-body quantum-mechanical systems.

## When to Use the Tag

Use the tag when asking questions about small quantum mechanical systems, such as a single hydrogen atom, or general aspects of quantum mechanics, e.g. the uncertainty principle, the wave-particle duality or simple scattering. Use when asking questions about many-body quantum mechanical systems. There is most often no need to tag a question as both and .

## Introduction

Single-body quantum mechanics is usually done in a Hilbert space $\mathbf{H}$ of states, where the Bra-Ket notation is used. $|\Psi\rangle$ refers to an element of $\mathbf{H}$, $\langle \Psi |$ to an element of the dual, $\mathcal{H}^\star$. The projection of $|\Psi\rangle$ onto a particular basis gives a wave function in this basis, for example:

$$\langle \vec r | \Psi \rangle = \Psi(\vec r,t)\quad.$$

Operators, such as $\hat x$ and $\hat p$, are then defined to act on these states and project them onto a corresponding eigenstate. The most prominent example is the Hamilton operator $\mathcal{\hat H}$, which has the possible energy states of the system as eigenvalues. The eigenvalue equation of the Hamilton operator $\mathcal{\hat H}$ is the time-independent (sometimes referred to as TISE):

$$\mathcal{\hat H} |\Psi\rangle = E |\Psi \rangle\quad.$$

While the wave-function $\Psi(\vec r,t)$ and the states $|\Psi\rangle$ can, in general, be complex, operators that correspond to physical observables ($\hat x$, $\hat p$, $\mathcal{\hat H}$) must have real eigenvalues and real expectation values, implying that these operators are Hermitian in the meaning that, in a matrix representation of a Hermitian operator $\hat A$, we have

$$\left(\hat A^{\dagger} \right)_{ij} = \left( \hat A \right)^{\star}_{ji} = \left ( \hat A \right)_{ij}\quad.$$

The act of applying an Hermitian operator, such as $\hat x$, on a state $|\Psi\rangle$ is taken to be equal to a measurement. The eigenvalue of the resulting eigenstate of $\hat x$ is then the ‘measured’ value (in this case, the position of the particle). Most of these operators relate to classical mechanical operators by the correspondence principle.

## Equations of Motion

In the beginning, operators are usually taken to be constant, whereas the states $|\Psi\rangle$ evolve in time. This is known as the Schrödinger picture, and its time evolution is governed by the :

$$i \hbar \frac{\partial}{\partial t} |\Psi(t)\rangle = \mathcal{\hat H} | \Psi(t) \rangle \quad.$$

An equivalent view is the Heisenberg picture, where states $|\Psi\rangle$ are assumed to be constant and states evolve in time according to

$$\frac{\mathrm{d}}{\mathrm{d}t} \hat A(t) = \frac{i}{\hbar} [ \mathcal{\hat H} , \hat A(t) ] + \frac{\partial}{\partial t} A(t) \quad.$$

Yet another equivalent formulation is the Path Integral Formulation (or Feynman Formulation), which is as described in :

In the Path Integral formulation a functional called the phase is associated with each path:

$$\phi = A e^\frac{iS}{\hbar}$$

The Kernel or the Matrix Element is the path integral of this phase.

$$K(x ) =\int\phi\mbox{ } \mathcal{D}x$$

Where $x$ is actually the position of the particle in spacetime.

The wavefunction, finally is given by:

$$\Psi(x)=\int_{-\infty}^\infty \left(K(x,x_0)\Psi(x_0) \right) \mbox{d}x_0$$

The intuition behind this is summarised by the following diagram:

It is often surprising to many that the absolute value of the phase squared, $|\phi|^2$, is constant for all paths, at $A^2$. However, this actually makes sense, as the position of the particle is initially completely well-defined, so Heisenberg's Uncertainty Principle tells us that we In the Path Integral formulation, a functional, called the phase is associated with each path: would $$\phi = A e^\frac{iS}{\hbar}$$ have no idea about the momentum, and thus no idea about it's future position. However, the next moment, you know absolutely nothing about it's momentum, and so on. This process coarse-grains a particular path, the classical path, which means it is much more probable than the other paths. The mathematical description of this can be obtained by standard procedures (c.f. FeynmaPath integral mechanics (Due to Feynman) is a major formulation of Quantum Mechanics along with Matrix mechanics (Due to Heisenberg and Pauli), Wave Mechanics (Due to Schrödinger), and Variational Mechanics (Due to Dirac). n, Hibbs, Styer "Quantum Mechanics and Path Integrals", pg 77 - 79) and the final result is the (Time-Independent) Schrödinger's Equation. Path integral mechanics (Due to Feynman) is a major formulation of Quantum Mechanics along with Matrix mechanics (Due to Heisenberg and Pauli), Wave Mechanics (Due to Schrödinger), and Variational Mechanics (Due to Dirac).

It is often surprising to many that the absolute value of the phase squared, $|\phi|^2$, is constant for all paths, at $A^2$. However, this actually makes sense, as the position of the particle is initially completely well-defined, so Heisenberg's Uncertainty Principle tells us that we would have no idea about the momentum, and thus no idea about it's future position. However, the next moment, you know absolutely nothing about it's momentum, and so on. This process coarse-grains a particular path, the classical path, which means it is much more probable than the other paths.

The mathematical description of this can be obtained by standard procedures (c.f. Feynman, Hibbs, Styer "Quantum Mechanics and Path Integrals", pg 77 - 79) and the final result is the (Time-Independent) Schrödinger's Equation.

$$\left(-\frac{\hbar^2}{2m}\nabla^2-i\hbar\frac{\partial }{\partial t }+U\right) \Psi = 0$$

## Prerequisites to learn Quantum Mechanics:

Phys: Newtonian Mechanics; Classical Mechanics; Hamiltonian formalism.

Math: Linear algebra; Fourier Transformation; Partial Differential Equations (PDE); Operator Theory and Hilbert spaces; Lie algebras. Maybe also: Finite Groups, Discrete Groups, Lie Groups, and their representation theory; Young Tableau.