Sub­sec­tions

### D.14 The spher­i­cal har­mon­ics

This note de­rives and lists prop­er­ties of the spher­i­cal har­mon­ics.

#### D.14.1 De­riva­tion from the eigen­value prob­lem

This analy­sis will de­rive the spher­i­cal har­mon­ics from the eigen­value prob­lem of square an­gu­lar mo­men­tum of chap­ter 4.2.3. It will use sim­i­lar tech­niques as for the har­monic os­cil­la­tor so­lu­tion, {D.12}.

The im­posed ad­di­tional re­quire­ment that the spher­i­cal har­mon­ics are eigen­func­tions of means that they are of the form where func­tion is still to be de­ter­mined. (There is also an ar­bi­trary de­pen­dence on the ra­dius , but it does not have any­thing to do with an­gu­lar mo­men­tum, hence is ig­nored when peo­ple de­fine the spher­i­cal har­mon­ics.) Sub­sti­tu­tion into with as in (4.22) yields an ODE (or­di­nary dif­fer­en­tial equa­tion) for :

It is con­ve­nient de­fine a scaled square an­gu­lar mo­men­tum by so that you can di­vide away the from the ODE.

More im­por­tantly, rec­og­nize that the so­lu­tions will likely be in terms of cosines and sines of , be­cause they should be pe­ri­odic if changes by . If you want to use power-se­ries so­lu­tion pro­ce­dures again, these tran­scen­den­tal func­tions are bad news, so switch to a new vari­able . At the very least, that will re­duce things to al­ge­braic func­tions, since is in terms of equal to . Con­vert­ing the ODE to the new vari­able , you get

As you may guess from look­ing at this ODE, the so­lu­tions are likely to be prob­lem­atic near , (phys­i­cally, near the -​axis where is zero.) If you ex­am­ine the so­lu­tion near those points by defin­ing a lo­cal co­or­di­nate as in , and then de­duce the lead­ing term in the power se­ries so­lu­tions with re­spect to , you find that it is ei­ther or , (in the spe­cial case that 0, that sec­ond so­lu­tion turns out to be .) Ei­ther way, the sec­ond pos­si­bil­ity is not ac­cept­able, since it phys­i­cally would have in­fi­nite de­riv­a­tives at the -​axis and a re­sult­ing ex­pec­ta­tion value of square mo­men­tum, as de­fined in chap­ter 4.4.3, that is in­fi­nite. You need to have that be­haves as at each end, so in terms of it must have a fac­tor near 1 and near 1. The two fac­tors mul­ti­ply to and so can be writ­ten as where must have fi­nite val­ues at 1 and 1.

If you sub­sti­tute into the ODE for , you get an ODE for :

Plug in a power se­ries, , to get, af­ter clean up,

Us­ing sim­i­lar ar­gu­ments as for the har­monic os­cil­la­tor, you see that the start­ing power will be zero or one, lead­ing to ba­sic so­lu­tions that are again odd or even. And just like for the har­monic os­cil­la­tor, you must again have that the power se­ries ter­mi­nates; even in the least case that 0, the se­ries for at 1 is like that of and will not con­verge to the fi­nite value stip­u­lated. (For rigor, use Gauss’s test.)

To get the se­ries to ter­mi­nate at some fi­nal power , you must have ac­cord­ing to the above equa­tion that , and if you de­cide to call the az­imuthal quan­tum num­ber , you have where since and , like any power , is greater or equal to zero.

The rest is just a mat­ter of ta­ble books, be­cause with , the ODE for is just the -​th de­riv­a­tive of the dif­fer­en­tial equa­tion for the Le­gendre poly­no­mial, [41, 28.1], so the must be just the -​th de­riv­a­tive of those poly­no­mi­als. In fact, you can now rec­og­nize that the ODE for the is just Le­gendre's as­so­ci­ated dif­fer­en­tial equa­tion [41, 28.49], and that the so­lu­tions that you need are the as­so­ci­ated Le­gendre func­tions of the first kind [41, 28.50].

To nor­mal­ize the eigen­func­tions on the sur­face area of the unit sphere, find the cor­re­spond­ing in­te­gral in a ta­ble book, like [41, 28.63]. As men­tioned at the start of this long and still very con­densed story, to in­clude neg­a­tive val­ues of , just re­place by . There is one ad­di­tional is­sue, though, the sign pat­tern. In or­der to sim­plify some more ad­vanced analy­sis, physi­cists like the sign pat­tern to vary with ac­cord­ing to the so-called lad­der op­er­a­tors. That re­quires, {D.64}, that start­ing from 0, the spher­i­cal har­mon­ics for 0 have the al­ter­nat­ing sign pat­tern of the lad­der-up op­er­a­tor, and those for 0 the un­vary­ing sign of the lad­der-down op­er­a­tor. Physi­cists will still al­low you to se­lect your own sign for the 0 state, bless them.

The fi­nal so­lu­tion is

 (D.5)

where the prop­er­ties of the as­so­ci­ated Le­gendre func­tions of the first kind can be found in ta­ble books like [41, pp. 162-166]. This uses the fol­low­ing de­f­i­n­i­tion of the as­so­ci­ated Le­gendre poly­no­mi­als:

where is the nor­mal Le­gendre poly­no­mial. Need­less to say, some other au­thors use dif­fer­ent de­f­i­n­i­tions, po­ten­tially putting in a fac­tor .

#### D.14.2 Par­ity

One spe­cial prop­erty of the spher­i­cal har­mon­ics is of­ten of in­ter­est: their “par­ity.” The par­ity of a wave func­tion is 1, or even, if the wave func­tion stays the same if you re­place by . The par­ity is 1, or odd, if the wave func­tion stays the same save for a sign change when you re­place by . It turns out that the par­ity of the spher­i­cal har­mon­ics is ; so it is 1, odd, if the az­imuthal quan­tum num­ber is odd, and 1, even, if is even.

To see why, note that re­plac­ing by means in spher­i­cal co­or­di­nates that changes into and into . Ac­cord­ing to trig, the first changes into . That leaves un­changed for even , since is then a sym­met­ric func­tion, but it changes the sign of for odd . So the sign change is . The value of has no ef­fect, since while the fac­tor in the spher­i­cal har­mon­ics pro­duces a fac­tor un­der the change in , also puts de­riv­a­tives on , and each de­riv­a­tive pro­duces a com­pen­sat­ing change of sign in .

#### D.14.3 So­lu­tions of the Laplace equa­tion

The Laplace equa­tion is

So­lu­tions to this equa­tion are called “har­monic func­tions.” In spher­i­cal co­or­di­nates, the Laplace equa­tion has so­lu­tions of the form

This is a com­plete set of so­lu­tions for the Laplace equa­tion in­side a sphere. Any so­lu­tion of the Laplace equa­tion in­side a sphere is a lin­ear com­bi­na­tion of these so­lu­tions.

As you can see in ta­ble 4.3, each so­lu­tion above is a power se­ries in terms of Carte­sian co­or­di­nates.

For the Laplace equa­tion out­side a sphere, re­place by 1 in the so­lu­tions above. Note that these so­lu­tions are not ac­cept­able in­side the sphere be­cause they blow up at the ori­gin.

To check that these are in­deed so­lu­tions of the Laplace equa­tion, plug them in, us­ing the Lapla­cian in spher­i­cal co­or­di­nates given in (N.5). Note here that the an­gu­lar de­riv­a­tives can be sim­pli­fied us­ing the eigen­value prob­lem of square an­gu­lar mo­men­tum, chap­ter 4.2.3.

#### D.14.4 Or­thog­o­nal in­te­grals

The spher­i­cal har­mon­ics are or­tho­nor­mal on the unit sphere:

 (D.6)

Here is de­fined to be 0 if and are dif­fer­ent, and 1 if they are equal, and sim­i­lar for . In other words, the in­te­gral above is 1 if and , and 0 in every other case. This ex­presses phys­i­cally that the spher­i­cal har­mon­ics, as eigen­func­tions of the Her­mit­ian and square an­gu­lar mo­men­tum op­er­a­tors, are or­tho­nor­mal. Math­e­mat­i­cally, it al­lows you to in­te­grate each spher­i­cal har­monic sep­a­rately and quickly when you are find­ing for a wave func­tion ex­pressed in terms of spher­i­cal har­mon­ics.

Fur­ther

 (D.7)

This ex­pres­sion sim­pli­fies your life when you are find­ing the for a wave func­tion ex­pressed in terms of spher­i­cal har­mon­ics.

See the no­ta­tions for more on spher­i­cal co­or­di­nates and .

To ver­ify the above ex­pres­sion, in­te­grate the first term in the in­te­gral by parts with re­spect to and the sec­ond term with re­spect to to get

and then ap­ply the eigen­value prob­lem of chap­ter 4.2.3.

#### D.14.5 An­other way to find the spher­i­cal har­mon­ics

There is a more in­tu­itive way to de­rive the spher­i­cal har­mon­ics: they de­fine the power se­ries so­lu­tions to the Laplace equa­tion. In par­tic­u­lar, each is a dif­fer­ent power se­ries so­lu­tion of the Laplace equa­tion 0 in Carte­sian co­or­di­nates. Each takes the form

where the co­ef­fi­cients are such as to make the Lapla­cian zero.

Even more specif­i­cally, the spher­i­cal har­mon­ics are of the form

where the co­or­di­nates and serve to sim­plify the Lapla­cian. That these are the ba­sic power se­ries so­lu­tions of the Laplace equa­tion is read­ily checked.

To get from those power se­ries so­lu­tions back to the equa­tion for the spher­i­cal har­mon­ics, one has to do an in­verse sep­a­ra­tion of vari­ables ar­gu­ment for the so­lu­tion of the Laplace equa­tion in a sphere in spher­i­cal co­or­di­nates (com­pare also the de­riva­tion of the hy­dro­gen atom.) Also, one would have to ac­cept on faith that the so­lu­tion of the Laplace equa­tion is just a power se­ries, as it is in 2D, with no ad­di­tional non­power terms, to set­tle com­plete­ness. In other words, you must as­sume that the so­lu­tion is an­a­lytic.

#### D.14.6 Still an­other way to find them

The sim­plest way of get­ting the spher­i­cal har­mon­ics is prob­a­bly the one given later in de­riva­tion {D.64}.