Sub­sec­tions

### 4.5 The Com­mu­ta­tor

As the pre­vi­ous sec­tion dis­cussed, the stan­dard de­vi­a­tion is a mea­sure of the un­cer­tainty of a prop­erty of a quan­tum sys­tem. The larger the stan­dard de­vi­a­tion, the far­ther typ­i­cal mea­sure­ments stray from the ex­pected av­er­age value. Quan­tum me­chan­ics of­ten re­quires a min­i­mum amount of un­cer­tainty when more than one quan­tity is in­volved, like po­si­tion and lin­ear mo­men­tum in Heisen­berg's un­cer­tainty prin­ci­ple. In gen­eral, this amount of un­cer­tainty is re­lated to an im­por­tant math­e­mat­i­cal ob­ject called the com­mu­ta­tor, to be dis­cussed in this sec­tion.

#### 4.5.1 Com­mut­ing op­er­a­tors

First, note that there is no fun­da­men­tal rea­son why sev­eral quan­ti­ties can­not have a def­i­nite value at the same time. For ex­am­ple, if the elec­tron of the hy­dro­gen atom is in a eigen­state, its to­tal en­ergy, square an­gu­lar mo­men­tum, and -​com­po­nent of an­gu­lar mo­men­tum all have def­i­nite val­ues, with zero un­cer­tainty.

More gen­er­ally, two dif­fer­ent quan­ti­ties with op­er­a­tors and have def­i­nite val­ues if the wave func­tion is an eigen­func­tion of both and . So, the ques­tion whether two quan­ti­ties can be def­i­nite at the same time is re­ally whether their op­er­a­tors and have com­mon eigen­func­tions. And it turns out that the an­swer has to do with whether these op­er­a­tors “com­mute”, in other words, on whether their or­der can be re­versed as in .

In par­tic­u­lar, {D.18}:

Iff two Her­mit­ian op­er­a­tors com­mute, there is a com­plete set of eigen­func­tions that is com­mon to them both.
(For more than two op­er­a­tors, each op­er­a­tor has to com­mute with all oth­ers.)

For ex­am­ple, the op­er­a­tors and of the har­monic os­cil­la­tor of chap­ter 4.1.2 com­mute:

This is true since it makes no dif­fer­ence whether you dif­fer­en­ti­ate first with re­spect to and then with re­spect to or vice versa, and since the can be pulled in front of the -​dif­fer­en­ti­a­tions and the can be pushed in­side the -​dif­fer­en­ti­a­tions, and since mul­ti­pli­ca­tions can al­ways be done in any or­der.

The same way, com­mutes with and , and that means that com­mutes with them all, since is just their sum. So, these four op­er­a­tors should have a com­mon set of eigen­func­tions, and they do: it is the set of eigen­func­tions de­rived in chap­ter 4.1.2.

Sim­i­larly, for the hy­dro­gen atom, the to­tal en­ergy Hamil­ton­ian , the square an­gu­lar mo­men­tum op­er­a­tor and the -​com­po­nent of an­gu­lar mo­men­tum all com­mute, and they have the com­mon set of eigen­func­tions .

Note that such eigen­func­tions are not nec­es­sar­ily the only game in town. As a counter-ex­am­ple, for the hy­dro­gen atom , , and the -​com­po­nent of an­gu­lar mo­men­tum also all com­mute, and they too have a com­mon set of eigen­func­tions. But that will not be the , since and do not com­mute. (It will how­ever be the af­ter you ro­tate them all 90 de­grees around the -​axis.) It would cer­tainly be sim­pler math­e­mat­i­cally if each op­er­a­tor had just one unique set of eigen­func­tions, but na­ture does not co­op­er­ate.

Key Points
Op­er­a­tors com­mute if you can change their or­der, as in .

For com­mut­ing op­er­a­tors, a com­mon set of eigen­func­tions ex­ists.

For those eigen­func­tions, the phys­i­cal quan­ti­ties cor­re­spond­ing to the com­mut­ing op­er­a­tors all have def­i­nite val­ues at the same time.

4.5.1 Re­view Ques­tions
1.

The pointer state

is one of the eigen­states that , , and have in com­mon. Check that it is not an eigen­state that , , and have in com­mon.

#### 4.5.2 Non­com­mut­ing op­er­a­tors and their com­mu­ta­tor

Two quan­ti­ties with op­er­a­tors that do not com­mute can­not in gen­eral have def­i­nite val­ues at the same time. If one has a def­i­nite value, the other is in gen­eral un­cer­tain.

The qual­i­fi­ca­tion in gen­eral is needed be­cause there may be ex­cep­tions. The an­gu­lar mo­men­tum op­er­a­tors do not com­mute, but it is still pos­si­ble for the an­gu­lar mo­men­tum to be zero in all three di­rec­tions. But as soon as the an­gu­lar mo­men­tum in any di­rec­tion is nonzero, only one com­po­nent of an­gu­lar mo­men­tum can have a def­i­nite value.

A mea­sure for the amount to which two op­er­a­tors and do not com­mute is the dif­fer­ence be­tween and ; this dif­fer­ence is called their “com­mu­ta­tor” :

 (4.45)

A nonzero com­mu­ta­tor de­mands a min­i­mum amount of un­cer­tainty in the cor­re­spond­ing quan­ti­ties and . It can be shown, {D.19}, that the un­cer­tain­ties, or stan­dard de­vi­a­tions, in and in are at least so large that:

 (4.46)

This equa­tion is called the “gen­er­al­ized un­cer­tainty re­la­tion­ship”.

Key Points
The com­mu­ta­tor of two op­er­a­tors and equals and is writ­ten as .

The prod­uct of the un­cer­tain­ties in two quan­ti­ties is at least one half the mag­ni­tude of the ex­pec­ta­tion value of their com­mu­ta­tor.

#### 4.5.3 The Heisen­berg un­cer­tainty re­la­tion­ship

This sec­tion will work out the un­cer­tainty re­la­tion­ship (4.46) of the pre­vi­ous sub­sec­tion for the po­si­tion and lin­ear mo­men­tum in an ar­bi­trary di­rec­tion. The re­sult will be a pre­cise math­e­mat­i­cal state­ment of the Heisen­berg un­cer­tainty prin­ci­ple.

To be spe­cific, the ar­bi­trary di­rec­tion will be taken as the -​axis, so the po­si­tion op­er­a­tor will be , and the lin­ear mo­men­tum op­er­a­tor . These two op­er­a­tors do not com­mute, is sim­ply not the same as : means mul­ti­ply func­tion by to get the prod­uct func­tion and then ap­ply on that prod­uct, while means ap­ply on and then mul­ti­ply the re­sult­ing func­tion by . The dif­fer­ence is found from writ­ing it out:

the sec­ond equal­ity re­sult­ing from dif­fer­en­ti­at­ing out the prod­uct.

Com­par­ing start and end shows that the dif­fer­ence be­tween and is not zero, but . By de­f­i­n­i­tion, this dif­fer­ence is their com­mu­ta­tor:

 (4.47)

This im­por­tant re­sult is called the “canon­i­cal com­mu­ta­tion re­la­tion.” The com­mu­ta­tor of po­si­tion and lin­ear mo­men­tum in the same di­rec­tion is the nonzero con­stant .

Be­cause the com­mu­ta­tor is nonzero, there must be nonzero un­cer­tainty in­volved. In­deed, the gen­er­al­ized un­cer­tainty re­la­tion­ship of the pre­vi­ous sub­sec­tion be­comes in this case:

 (4.48)

This is the un­cer­tainty re­la­tion­ship as first for­mu­lated by Heisen­berg.

It im­plies that when the un­cer­tainty in po­si­tion is nar­rowed down to zero, the un­cer­tainty in mo­men­tum must be­come in­fi­nite to keep their prod­uct nonzero, and vice versa. More gen­er­ally, you can nar­row down the po­si­tion of a par­ti­cle and you can nar­row down its mo­men­tum. But you can never re­duce the prod­uct of the un­cer­tain­ties and be­low , what­ever you do.

It should be noted that the un­cer­tainty re­la­tion­ship is of­ten writ­ten as or even as where and are taken to be vaguely de­scribed un­cer­tain­ties in mo­men­tum and po­si­tion, rather than rig­or­ously de­fined stan­dard de­vi­a­tions. And peo­ple write a cor­re­spond­ing un­cer­tainty re­la­tion­ship for time, , be­cause rel­a­tiv­ity sug­gests that time should be treated just like space. But note that un­like the lin­ear mo­men­tum op­er­a­tor, the Hamil­ton­ian is not at all uni­ver­sal. So, you might guess that the de­f­i­n­i­tion of the un­cer­tainty in time would not be uni­ver­sal ei­ther, and you would be right, chap­ter 7.2.2.

Key Points
The canon­i­cal com­mu­ta­tor equals .

If ei­ther the un­cer­tainty in po­si­tion in a given di­rec­tion or the un­cer­tainty in lin­ear mo­men­tum in that di­rec­tion is nar­rowed down to zero, the other un­cer­tainty blows up.

The prod­uct of the two un­cer­tain­ties is at least the con­stant .

4.5.3 Re­view Ques­tions
1.

This sounds se­ri­ous! If I am dri­ving my car, the po­lice re­quires me to know my speed (lin­ear mo­men­tum). Also, I would like to know where I am. But nei­ther is pos­si­ble ac­cord­ing to quan­tum me­chan­ics.

#### 4.5.4 Com­mu­ta­tor ref­er­ence

It is a fact of life in quan­tum me­chan­ics that com­mu­ta­tors pop up all over the place. Not just in un­cer­tainty re­la­tions, but also in the time evo­lu­tion of ex­pec­ta­tion val­ues, in an­gu­lar mo­men­tum, and in quan­tum field the­ory, the ad­vanced the­ory of quan­tum me­chan­ics used in solids and rel­a­tivis­tic ap­pli­ca­tions. This sec­tion can make your life eas­ier deal­ing with them. Browse through it to see what is there. Then come back when you need it.

Re­call the de­f­i­n­i­tion of the com­mu­ta­tor of any two op­er­a­tors and :

 (4.49)

By this very de­f­i­n­i­tion , the com­mu­ta­tor is zero for any two op­er­a­tors and that com­mute, (whose or­der can be in­ter­changed):
 (4.50)

If op­er­a­tors all com­mute, all their prod­ucts com­mute too:
 (4.51)

Every­thing com­mutes with it­self, of course:

 (4.52)

and every­thing com­mutes with a nu­mer­i­cal con­stant; if is an op­er­a­tor and is some num­ber, then:
 (4.53)

The com­mu­ta­tor is an­ti­sym­met­ric; or in sim­pler words, if you in­ter­change the sides; it will change the sign, {D.20}:

 (4.54)

For the rest how­ever, lin­ear com­bi­na­tions mul­ti­ply out just like you would ex­pect:
 (4.55)

(in which it is as­sumed that , , , and are op­er­a­tors, and , , , and nu­mer­i­cal con­stants.)

To deal with com­mu­ta­tors that in­volve prod­ucts of op­er­a­tors, the rule to re­mem­ber is: “the first fac­tor comes out at the front of the com­mu­ta­tor, the sec­ond at the back”. More pre­cisely:

 (4.56)

So, if or com­mutes with the other side of the op­er­a­tor, it can sim­ply be taken out at at its side; (the sec­ond com­mu­ta­tor will be zero.) For ex­am­ple,

if and com­mute.

Now from the gen­eral to the spe­cific. Be­cause chang­ing sides in a com­mu­ta­tor merely changes its sign, from here on only one of the two pos­si­bil­i­ties will be shown. First the po­si­tion op­er­a­tors all mu­tu­ally com­mute:

 (4.57)

as do po­si­tion-de­pen­dent op­er­a­tors such as a po­ten­tial en­ergy :
 (4.58)

This il­lus­trates that if a set of op­er­a­tors all com­mute, then all com­bi­na­tions of those op­er­a­tors com­mute too.

The lin­ear mo­men­tum op­er­a­tors all mu­tu­ally com­mute:

 (4.59)

How­ever, po­si­tion op­er­a­tors and lin­ear mo­men­tum op­er­a­tors in the same di­rec­tion do not com­mute; in­stead:
 (4.60)

As seen in the pre­vi­ous sub­sec­tion, this lack of com­mu­ta­tion causes the Heisen­berg un­cer­tainty prin­ci­ple. Po­si­tion and lin­ear mo­men­tum op­er­a­tors in dif­fer­ent di­rec­tions do com­mute:
 (4.61)

A gen­er­al­iza­tion that is fre­quently very help­ful is:

 (4.62)

where is any func­tion of , , and .

Un­like lin­ear mo­men­tum op­er­a­tors, an­gu­lar mo­men­tum op­er­a­tors do not mu­tu­ally com­mute. The com­mu­ta­tors are given by the so-called “ fun­da­men­tal com­mu­ta­tion re­la­tions:”

 (4.63)

Note the or­der of the in­dices that pro­duces pos­i­tive signs; a re­versed or­der adds a mi­nus sign. For ex­am­ple be­cause fol­low­ing is in re­versed or­der.

The an­gu­lar mo­men­tum com­po­nents do all com­mute with the square an­gu­lar mo­men­tum op­er­a­tor:

 (4.64)

Just the op­po­site of the sit­u­a­tion for lin­ear mo­men­tum, po­si­tion and an­gu­lar mo­men­tum op­er­a­tors in the same di­rec­tion com­mute,

 (4.65)

but those in dif­fer­ent di­rec­tions do not:
 (4.66)

Square po­si­tion com­mutes with all com­po­nents of an­gu­lar mo­men­tum,
 (4.67)

The com­mu­ta­tor be­tween po­si­tion and square an­gu­lar mo­men­tum is, us­ing vec­tor no­ta­tion for con­cise­ness,
 (4.68)

The com­mu­ta­tors be­tween lin­ear and an­gu­lar mo­men­tum are very sim­i­lar to the ones be­tween po­si­tion and an­gu­lar mo­men­tum:

 (4.69)

 (4.70)

 (4.71)

 (4.72)

The fol­low­ing com­mu­ta­tors are also use­ful:

 (4.73)

Com­mu­ta­tors in­volv­ing spin are dis­cussed in a later chap­ter, 5.5.3.

Key Points
Rules for eval­u­at­ing com­mu­ta­tors were given.

Re­turn to this sub­sec­tion if you need to fig­ure out some com­mu­ta­tor or the other.