3.2 The Heisen­berg Un­cer­tainty Prin­ci­ple

The Heisen­berg un­cer­tainty prin­ci­ple is a way of ex­press­ing the qual­i­ta­tive prop­er­ties of quan­tum me­chan­ics in an easy to vi­su­al­ize way.

Fig­ure 3.3: Il­lus­tra­tion of the Heisen­berg un­cer­tainty prin­ci­ple. A com­bi­na­tion plot of po­si­tion and lin­ear mo­men­tum com­po­nents in a sin­gle di­rec­tion is shown. Left: Fairly lo­cal­ized state with fairly low lin­ear mo­men­tum. Right: nar­row­ing down the po­si­tion makes the lin­ear mo­men­tum ex­plode.

Fig­ure 3.3 is a com­bi­na­tion plot of the po­si­tion $x$ of a par­ti­cle and the cor­re­spond­ing lin­ear mo­men­tum $p_x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $mv_x$, (with $m$ the mass and $v_x$ the ve­loc­ity in the $x$-​di­rec­tion). To the left in the fig­ure, both the po­si­tion and the lin­ear mo­men­tum have some un­cer­tainty.

The right of the fig­ure shows what hap­pens if you squeeze down on the par­ti­cle to try to re­strict it to one po­si­tion $x$: it stretches out in the mo­men­tum di­rec­tion.

Heisen­berg showed that ac­cord­ing to quan­tum me­chan­ics, the area of the blob can­not be con­tracted to a point. When you try to nar­row down the po­si­tion of a par­ti­cle, you get into trou­ble with mo­men­tum. Con­versely, if you try to pin down a pre­cise mo­men­tum, you lose all hold on the po­si­tion.

The area of the blob has a min­i­mum value be­low which you can­not go. This min­i­mum area is com­pa­ra­ble in size to the so-called Planck con­stant, roughly 10$\POW9,{-34}$ kg m$\POW9,{2}$/s. That is an ex­tremely small area for macro­scopic sys­tems, rel­a­tively speak­ing. But it is big enough to dom­i­nate the mo­tion of mi­cro­scopic sys­tems, like say elec­trons in atoms.

Key Points

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The Heisen­berg un­cer­tainty prin­ci­ple says that there is al­ways a min­i­mum com­bined un­cer­tainty in po­si­tion and lin­ear mo­men­tum.

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It im­plies that a par­ti­cle can­not have a math­e­mat­i­cally pre­cise po­si­tion, be­cause that would re­quire an in­fi­nite un­cer­tainty in lin­ear mo­men­tum.

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It also im­plies that a par­ti­cle can­not have a math­e­mat­i­cally pre­cise lin­ear mo­men­tum (ve­loc­ity), since that would im­ply an in­fi­nite un­cer­tainty in po­si­tion.