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== Odvození ==
Princip neurčitosti má přímočaré matematické odvození. Klíčovým krokem je uplatnění [[Cauchyho-Schwarzova nerovnost|Cauchy-Schwarzovy nerovnosti]], jednoho z nejužitečnějších teorémů lineární algebry. Relace neurčitosti pak odpovídají vlastnostem [[Fourierova transformace|Fourierovy transformace]], kdy jisté spektrální šířce odpovídá minimální délka v původním prostoru (např. čase).
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For two arbitrary [[Self-adjoint operator|Hermitian operators]] ''A'': ''H'' → ''H'' and ''B'': ''H'' → ''H'', and any element ''x'' of ''H'', then
:<math> \langle B A x | x \rangle = \langle A x | B x \rangle = \langle B x | A x \rangle^{*}</math>
In an [[inner product space]] the Cauchy-Schwarz inequality holds.
:<math>\left|\langle B x | A x \rangle\right |^2 \leq \|A x \|^2 \|B x \|^2 </math>
Rearranging this formula leads to:
:<math>
\begin{align}
\|A x \|^2 \|B x \|^2 \geq \left|\langle B x | A x \rangle\right |^2 &\geq \left|\mathrm{Im}\{\langle B x | A x \rangle\}\right |^2 \\
&= \frac{1}{4} \left|2 \, \mathrm{Im}\{\langle B x | A x \rangle\}\right |^2 \\
&= \frac{1}{4} \left| \langle B x | A x \rangle - \langle B x | A x \rangle^{*} \right |^2 \\
&= \frac{1}{4} \left| \langle B x | A x \rangle - \langle A x | B x \rangle \right |^2 \\
&= \frac{1}{4} \left| \langle A B x | x \rangle - \langle B A x | x \rangle \right |^2 \\
&= \frac{1}{4} |\langle (AB - BA)x | x \rangle|^2
\end{align}
</math>
Consequently, the following general form of the uncertainty principle, first pointed out in [[1930]] by [[Howard Percy Robertson]] and (independently) by [[Erwin Schrödinger]], holds:
:<math>\frac{1}{4} |\langle [A,B]x | x \rangle|^2\leq \| A x \|^2 \| B x \|^2,</math>
where the operator [''A'',''B''] = ''AB'' - ''BA'' denotes the [[Commutator#Ring_theory|commutator]] of ''A'' and ''B''. This inequality is called the [[Robertson-Schrödinger relation]].
To make the physical meaning of this inequality more directly apparent, it is often written in the equivalent form:
:<math>
\Delta_{\psi} A \, \Delta_{\psi} B \ge \frac{1}{2} \left|\left\langle\left[{A},{B}\right]\right\rangle_\psi\right|
</math>
where
:<math>\left\langle X \right\rangle_\psi = \left\langle \psi | X \psi \right\rangle</math>
is the operator [[mean]] of observable ''X'' in the system state ψ and
:<math>\Delta_{\psi} X = \sqrt{\langle {X}^2\rangle_\psi - \langle {X}\rangle_\psi ^2}</math>
is the operator [[standard deviation]] of observable ''X'' in the system state ψ. This formulation can be derived from the above formulation by plugging in <math>A - \lang A\rang_\psi</math> for ''A'' and <math>B - \lang B\rang_\psi</math> for ''B'', and using the fact that
:<math>[A,B]=[A - \lang A\rang, B - \lang B\rang].</math>
This formulation acquires its physical interpretation, indicated by the suggestive terminology "mean" and "standard deviation", due to the axioms of [[quantum statistical mechanics]]. Particular uncertainty relations, such as position-momentum, can usually be derived by a straightforward application of this inequality.
==Energy, time and further generalizations==
General arguments, connected with the theory of relativity, point out that seemingly a relation like the following should exist:
:<math> \Delta E \Delta t \ge \frac{\hbar}{2} </math>.
But its correct mathematical formulation [http://daarb.narod.ru/mandtamm-eng.html was given] only in 1945 by [[Leonid Mandelshtam|L. I. Mandelshtam]] and [[Igor Tamm|I. E. Tamm]]. This is the only known formulation of the time-energy uncertainty relation rigorously derived from the mathematical structure of quantum mechanics. For a quantum system in a non-stationary state <math>|\psi\rangle</math> and an observable <math>B</math> represented by a self-adjoint operator <math>\hat B</math>, the formula takes the following form:
:<math> \Delta_{\psi} E \frac{\Delta_{\psi} B}{\left | \frac{\mathrm{d}\langle \hat B \rangle}{\mathrm{d}t}\right |} \ge \frac{\hbar}{2} </math>,
where <math>\Delta_{\psi} E</math> is the standard deviation of the energy operator in the state <math>|\psi\rangle </math>,
<math>\Delta_{\psi} B</math> stands for the standard deviation of the operator <math>\hat B</math> and
<math> \langle \hat B \rangle </math> is the expectation value of <math>\hat B</math> in that state.
Although, the second factor in the left-hand side has dimension of time, it is different from the time parameter that
enters [[Schrödinger equation]]. It is a lifetime of the
state <math>|\psi\rangle</math> with respect to the observable <math>B</math>. In other words, this is the time
after which the expectation value <math>\langle\hat B\rangle</math> changes appreciably.
The relation has an important implication for spectroscopy. As excited states have a short lifetime their energy uncertainty is not negligible. For this reason sharp lines cannot be obtained even under ideal conditions. This relation helps also to give an idea of the "chaotic" behavior of the space-time, wherein very small time steps authorize huge energy variations.
Historically, the time-energy uncertainty relation was motivated by many physicists, like [[Niels Bohr]], especially in the vaguer form <math>\Delta E \Delta t \approx h</math>, simply by relating ''E'' and ''t'' to ''x'' and ''p'' in the original relation.
In particular, one false formulation is still present. It says that the energy of a quantum system measured over the time
interval <math>\Delta t</math> has to be inaccurate, with the inaccuracy <math>\Delta E</math> given by the inequality
<math>\Delta E \Delta t \ge \frac {\hbar}{2}</math>. This formulation has been invalidated by [[Yakir Aharonov|Y. Aharonov]] and [[David Bohm|D. Bohm]] in 1961. One can actually determine accurate energy of a quantum system in an arbitrarily short
interval of time.
Moreover, as [http://xxx.lanl.gov/abs/quant-ph/0512223 recent research] indicates, for quantum systems with discrete
energy spectra the product <math>\Delta E\Delta t</math> is bounded from above by a statistical noise that vanishes if sufficiently many identical copies of the system are used. This vanishing upper bound removes the possibility of
lower bounds including many false formulations of the time-energy uncertainty.
In 1926 Dirac offered an alternative rigourous derivation of the uncertainty relation
:<math> \Delta E \Delta t \ge \frac{\hbar}{2} </math>
as coming from a relativistic quantum theory of "events".-->
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