Difference between revisions of "Quantum Mechanics"
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A quantum system is completely specified by its '''Wave Function''': | A quantum system is completely specified by its '''Wave Function''': | ||
:<math> | :<math> | ||
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:<math> \psi(x) = \frac{1}{\sqrt{2 \pi}} \int \tilde{\psi}(k) e^{i k x } \mathrm{d}k </math> | :<math> \psi(x) = \frac{1}{\sqrt{2 \pi}} \int \tilde{\psi}(k) e^{i k x } \mathrm{d}k </math> | ||
:<math> \tilde{\psi}(k) = \frac{1}{\sqrt{2 \pi}} \int {\psi}(x) e^{-i k x } \mathrm{d}x </math> | :<math> \tilde{\psi}(k) = \frac{1}{\sqrt{2 \pi}} \int {\psi}(x) e^{-i k x } \mathrm{d}x </math> | ||
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| + | ==[[Wave packet]]== | ||
| + | TBD | ||
==See Also== | ==See Also== | ||
* [http://en.wikipedia.org/wiki/Quantum_mechanics Wikipedia: Quantum Mechanics] | * [http://en.wikipedia.org/wiki/Quantum_mechanics Wikipedia: Quantum Mechanics] | ||
Revision as of 15:56, 12 October 2014
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 |
|---|---|
| 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} |
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 \Pr(x) \mathrm{d}x = | \psi(x) |^2 } | 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 x | \psi \rangle |^2 } |
In the Copenhagen Interpretation, 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 \Pr(x)} is the probability of finding the particle at location 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 x} . In Universal Wave Function interpretations (e.g. MWI), 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 \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:
- 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) \longleftrightarrow \tilde{\psi} (k) }
These representations can be inter-related (c.f. Fourier transform):
- 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) = \frac{1}{\sqrt{2 \pi}} \int \tilde{\psi}(k) e^{i k x } \mathrm{d}k }
- 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 \tilde{\psi}(k) = \frac{1}{\sqrt{2 \pi}} \int {\psi}(x) e^{-i k x } \mathrm{d}x }
Wave packet
TBD
