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uncertainty relation

 Definition/Summary One of the most asked questions is concerning how to derive the Heisenberg Uncertainty Relation. Starting from almost basic concepts of Quantum Mechanics, a derivation is given here. Some details are left as minor exercises for the interested reader. The derivation on this page is based on the integral version of the Schwarz inequality, applied to wavefunctions. An alternative derivation, based on the expectation value version, and applied to bras and kets, is on the page "uncertainty principle".

 Equations $$\Delta x\Delta p \geq \hbar/2$$

 Scientists Werner Heisenberg (1901-1976)

 Recent forum threads on uncertainty relation

 Breakdown Physics > Quantum >> Non-Relativistic QM

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 Extended explanation For understanding this derivation, one might need to pick up results from libary items on Schwarz inequality and hermitian operators. The commutator of two operators $A$ and $B$ is defined as: $$\left[ A,B \right] = AB - BA.$$ Some commutator algebra: $$\left[ A,B + C \right] = \left[ A,B \right] + \left[ A,C \right] .$$ $$\left[ A,q \right] = 0 ,$$ if $q$ is a constant. The commutator for position and momentum in quantum mechanics (QM) is standard knowledge for a student of QM, thus the result is given here without proof: $$\left[ x,p \right] = \left[ x, -i\hbar \frac{d}{dx} \right] = i\hbar .$$ To prove this, act with this commutator on a test function. Now let us consider the variance in QM: $$(\Delta x ) ^2 = \int \psi^*(\Delta x ) ^2\psi dx$$ $$( \Delta x ) = x -$$ Now we have this very nice relation: $$\left[ \Delta x,\Delta p \right] = \left[ x,p\right] = i \hbar,$$ as an exercise, show this. Now substitute $a(x) = A\psi (x)$ and $b(x) = B\psi (x)$ in the Schwarz inequality: $$\int(A\psi)^*(A\psi)dx\int(B\psi)^*(B\psi)dx = \geq \left| \int (A\psi)^*(B\psi) dx \right|^2 = \left| \int \psi^*(A(B\psi)) dx \right|^2$$ Write: $$\int \psi^* AB \psi dx = \frac{1}{2}\int \psi^* (AB+BA) \psi dx + \frac{1}{2}\int \psi^* (AB-BA) \psi dx$$ We have: $$\geq \left| \frac{1}{2}\int \psi^* (AB+BA) \psi dx + \frac{1}{2}\int \psi^* (AB-BA) \psi dx \right|^2 \Rightarrow$$ Standard algebra: $|a+b|^2 \geq |a|^2 + |b|^2$ $$\geq \frac{1}{4}\left| \int \psi^* (AB+BA) \psi dx \right|^2+ \frac{1}{4}\left|\int \psi^* (AB-BA) \psi dx \right|^2$$ The first term on the right hand side is a number greater than zero (it is equal to the integral $I$ which is real, real number squared is a number greater than zero). So the lower limit is: $$\geq \frac{1}{4}\left|\int \psi^* \left[ A,B \right] \psi dx \right|^2$$ Make substituion $A \rightarrow \Delta x$ and $B \rightarrow \Delta p$ and use the fact that the expectation value of variance is equal to the variance, and that wavefunctions are normalized to unity: $$(\Delta x)^2(\Delta p)^2 \geq \frac{1}{4} \left|\int \psi^* \left[ \Delta x,\Delta p \right] \psi dx \right|^2$$ $$(\Delta x)^2(\Delta p)^2 \geq \frac{1}{4} (-i\hbar)(i\hbar)\cdot 1 = \hbar ^2/4$$ Thus: $$\Delta x\Delta p \geq \hbar/2$$ Consider Heisenberg Uncertainty Principle - Derivation (II) for same result obtained in bra-ket notation. We can also choose to perform this uncertainty with any two operators. It will become interesting if we consider two operators which does not commute. e.g let us consider the angular momentum operators, which have the following commutator: $$[ L_x, L_y] = i\hbar L_z$$

Commentary

 tiny-tim @ 05:07 AM Dec10-10 Fixed missing LaTeX. No other changes.

 tiny-tim @ 07:22 AM Feb10-09 Added to Definition a statement of the difference between this derivation and the one on the other page.

 malawi_glenn @ 05:39 AM Feb10-09 timmy: we also have two very good free material on QM in the tutorial session, I will add those soon!

 tiny-tim @ 02:56 AM Feb10-09 Removed "see also" reference to a textbook: it's no better than most other textbooks on the subject, and it's not free. For free online books, see for example http://freescience.info/books.php?id=2 Shortened title to "uncertainty relation".

 Hootenanny @ 02:30 AM Feb9-09 Added links to the The Schwarz Inequality and Hermiticity

 malawi_glenn @ 02:03 PM Feb7-09 Ok, I consider this one as finished. Can someone please fix URL links to library items on Schwarz inequality and hermitian operators :-)

 malawi_glenn @ 11:25 AM Feb7-09 Ok I'll start right away with that. I also think hats are old fashioned, but I will do it anyway ;-) Thanx for feedback timmy

 tiny-tim @ 09:18 AM Feb7-09 Yes, "Schwarz inequality" and "hermitian" would each get plenty of autolinking, which together with the "see also" interlinkings would considerably increase the number of views. (and if you think, after comparison, that it's clearer without hats, then i say go hatless … hats are so old-fashioned )

 malawi_glenn @ 08:50 AM Feb7-09 ok, I can put hat's to the operators, and also split into smaller library posts. One might be the Schwarz inequality and another one could be about hermitian operators. Then I could add more examples on uncertainty relations, such as angular momenta etc. And I could move some results here on expectation values to the library item which now are called "expected value". What do you think about that?

 tiny-tim @ 04:46 AM Feb7-09 i agree … i think the Schwarz inequality makes this explanation too long (and most members will be happy to take it for granted anyway) i suggest the title be as short as possible, so as to maximise the autolinking … some people write "Heisenberg", but some write "Heisenberg's", so probably best to leave Mr H out of the title completely (a forum search for "Heisenberg's uncertainty relation" got 18 hits, "Heisenberg uncertainty relation" got 35 hits, "uncertainty relation" 227 hits, and "uncertainty principle" 500 hits in 22 months) changed title from "Heisenberg Uncertainty Relation (HUP)" to "uncertainty relation" (leaving room for a separate entry on "uncertainty principle") ~EDIT: the Edit log just got completely wiped … what's happening?

 cristo @ 04:30 AM Feb7-09 Isn't it standard to put hats on operators? I also think the Schwarz inequality should have its own article, since it is not solely used in quantum mechanics.