Quantum Mechanics

Quantum mechanics is a theory that describes the interactions of all particles and systems. It underlies all physical phenomena, including scattering.

Wavefunction

A quantum system is completely specified by its Wave Function:

$\psi (x)$

The wavefunction is typically normalized:

Integral Notation	Dirac Notation
$\int |\psi (x)|^{2}\mathrm {d} x=1$	$\langle \psi |\psi \rangle =1$

The distribution of the particle described by $\psi (x)$ is given by:

Integral Notation	Dirac Notation
$\Pr(x)\mathrm {d} x=|\psi (x)|^{2}$	$|\langle x|\psi \rangle |^{2}$

In the Copenhagen Interpretation, $\Pr(x)$ is the probability of finding the particle at location $x$. In Universal Wave Function interpretations (e.g. MWI), $\Pr(x)$ can be thought of as the spatial distribution of the particle. The wavefunction contains all the information one can know about a system. It can thus be thought of as 'being' the particle/system in question. However, the wavefunction can be described in an infinite number of different ways. That is, there is not a unique basis for describing the wavefunction. So, for instance, one can describe the wavefunction using position-space or momentum-space:

$\psi (x)\longleftrightarrow {\tilde {\psi }}(k)$

These representations can be inter-related (c.f. Fourier transform):

$\psi (x)={\frac {1}{\sqrt {2\pi }}}\int {\tilde {\psi }}(k)e^{ikx}\mathrm {d} k$

${\tilde {\psi }}(k)={\frac {1}{\sqrt {2\pi }}}\int {\psi }(x)e^{-ikx}\mathrm {d} x$

Wave packet

TBD

Heisenberg Indeterminacy Relations

(Also known as Heisenberg Uncertainty Principle.)

$\Delta _{x}\Delta _{p}\geq {\frac {\hbar }{2}}$

$\Delta _{E}\Delta _{t}\geq {\frac {\hbar }{2}}$

Superposition

If $\psi _{1}(x)$ and $\psi _{2}(x)$ are both allowed states for a given system, then the following state is also allowed:

$\psi (x)=\alpha \psi _{1}(x)+\beta \psi _{2}(x)$

This leads to a notable consequence:

${\begin{alignedat}{2}\Pr(x)&=|\alpha \psi _{1}(x)+\beta \psi _{2}(x)|^{2}\\&=1\end{alignedat}}$

See Also

• Wikipedia: Quantum Mechanics