Editing Uncertainty principle (section)

===Mandelstam–Tamm===
In 1945, [[Leonid Mandelstam]] and [[Igor Tamm]] derived a non-relativistic ''time–energy uncertainty relation'' as follows.<ref>L. I. Mandelstam, I. E. Tamm, [http://daarb.narod.ru/mandtamm/index-eng.html ''The uncertainty relation between energy and time in nonrelativistic quantum mechanics''] {{Webarchive|url=https://web.archive.org/web/20190607131054/http://daarb.narod.ru/mandtamm/index-eng.html |date=2019-06-07 }}, 1945.</ref><ref name="Busch2002"/> From Heisenberg mechanics, the generalized [[Ehrenfest theorem]] for an observable ''B'' without explicit time dependence, represented by a self-adjoint operator <math>\hat B</math> relates time dependence of the average value of <math>\hat B</math> to the average of its commutator with the Hamiltonian:

<math display=block> \frac{d\langle \hat{B} \rangle}{dt} = \frac{i}{\hbar}\langle [\hat{H},\hat{B}]\rangle. </math>

The value of <math>\langle [\hat{H},\hat{B}]\rangle</math> is then substituted in the [[#Robertson–Schrödinger_uncertainty_relations|Robertson uncertainty relation]] for the energy operator <math>\hat H</math> and <math>\hat B</math>:
<math display=block> \sigma_H\sigma_B \geq \left|\frac{1}{2i} \langle[ \hat{H}, \hat{B}] \rangle\right|, </math>
giving
<math display="block"> \sigma_H \frac{\sigma_B}{\left| \frac{d\langle \hat B \rangle}{dt}\right |} \ge \frac{\hbar}{2}</math>
(whenever the denominator is nonzero).
While this is a universal result, it depends upon the observable chosen and that the deviations <math>\sigma_H</math> and <math>\sigma_B</math> are computed for a particular state.
Identifying <math>\Delta E \equiv \sigma_E </math> and the characteristic time
<math display="block">\tau_B \equiv \frac{\sigma_B}{\left| \frac{d\langle \hat B \rangle}{dt}\right |}</math>
gives an energy–time relationship <math>\Delta E \tau_B \ge \frac{\hbar}{2}.</math>
Although <math>\tau_B</math> has the dimension of time, it is different from the time parameter ''t'' that enters the [[Schrödinger equation]]. This <math>\tau_B</math> can be interpreted as time for which the expectation value of the observable, <math>\langle \hat B \rangle,</math> changes by an amount equal to one standard deviation.<ref>{{Cite book |last=Naber |first=Gregory L. |url=https://books.google.com/books?id=kARGEAAAQBAJ |title=Quantum Mechanics: An Introduction to the Physical Background and Mathematical Structure |year=2021 |publisher=Walter de Gruyter GmbH & Co KG |isbn=978-3-11-075194-9 |pages=230 |language=en |access-date=2024-01-20 |archive-date=2024-02-23 |archive-url=https://web.archive.org/web/20240223155539/https://books.google.com/books?id=kARGEAAAQBAJ |url-status=live }}</ref>
Examples:
* The time a free quantum particle passes a point in space is more uncertain as the energy of the state is more precisely controlled: <math>\Delta T = \hbar/2\Delta E.</math> Since the time spread is related to the particle position spread and the energy spread is related to the momentum spread, this relation is directly related to position–momentum uncertainty.<ref name="GriffithsSchroeter2018" />{{rp|144}}
* A [[Delta particle]], a quasistable composite of quarks related to protons and neutrons, has a lifetime of 10<sup>−23</sup>&nbsp;s, so its measured [[Mass–energy equivalence| mass equivalent to energy]], 1232&nbsp;MeV/''c''<sup>2</sup>, varies by ±120&nbsp;MeV/''c''<sup>2</sup>; this variation is intrinsic and not caused by measurement errors.<ref name="GriffithsSchroeter2018" />{{rp|144}}
* Two energy states <math>\psi_{1,2}</math> with energies <math>E_{1,2},</math> superimposed to create a composite state
:<math display="block">\Psi(x,t) = a\psi_1(x) e^{-iE_1t/h} + b\psi_2(x) e^{-iE_2t/h}.</math>
:The probability amplitude of this state has a time-dependent interference term:
:<math display="block">|\Psi(x,t)|^2 = a^2|\psi_1(x)|^2 + b^2|\psi_2(x)|^2 + 2ab\cos(\frac{E_2 - E_1}{\hbar}t).</math>
:The oscillation period varies inversely with the energy difference: <math>\tau = 2\pi\hbar/(E_2 - E_1)</math>.<ref name="GriffithsSchroeter2018" />{{rp|144}}
Each example has a different meaning for the time uncertainty, according to the observable and state used.