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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_2(x)} are both allowed states for a given system, then the following state is also allowed:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(x) = \alpha \psi_1(x) + \beta \psi_2(x) }

This leads to a notable consequence:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \Pr(x) & = | \alpha \psi_1(x) + \beta \psi_2(x) |^2 \\ & = (\alpha\psi_1 + \beta\psi_2)(\alpha\psi_1 + \beta\psi_2)^{*} \\ & = |\alpha|^2 |\psi_1|^2 + |\beta|^2\psi_2^2 + \alpha\beta^* \psi_1\psi_2^* + \alpha^*\beta\psi_1^*\psi_2 \\ & = \mathrm{Pr}_1(x) + \mathrm{Pr}_2(x) + \mathrm{interference} \\ \end{alignat} }

Notice that the final terms represent 'interference' between the two constituent states. This interference has no classical analogue; it is a quantum effect. Thus a superposition is not merely a 'joining' of the two states (e.g. "the particle can be in state 1 or state 2"), but a truly coherent interference between the two states. The superposition may be more generally written as:

Integral Notation	Dirac Notation
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int | \psi(x) |^2 \mathrm{d}x = 1}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \psi | \psi \rangle = 1}

The distribution of the particle described by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(x)} is given by:

Integral Notation	Dirac Notation
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(x) = \sum_n c_n \psi_n }	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle = c_1 | 1 \rangle + c_2 | 2 \rangle + c_3 | 3 \rangle + \cdots }

See Also

• Wikipedia: Quantum Mechanics