Sub­sec­tions

### D.2 Some Green’s func­tions

#### D.2.1 The Pois­son equa­tion

The so-called Pois­son equa­tion is

Here is sup­posed to be a given func­tion and an un­known func­tion that is to be found.

The so­lu­tion to the Pois­son equa­tion in in­fi­nite space may be found in terms of its so-called Green’s func­tion . In par­tic­u­lar:

Loosely speak­ing, the above in­te­gral so­lu­tion chops func­tion up into spikes . A spike at a po­si­tion then makes a con­tri­bu­tion to at . In­te­gra­tion over all such spikes gives the com­plete .

Note that of­ten, the Pois­son equa­tion is writ­ten with­out a mi­nus sign. Then there will be a mi­nus sign in .

The ob­jec­tive is now to de­rive the above Green’s func­tion. To do so, first an in­tu­itive de­riva­tion will be given and then a more rig­or­ous one. (See also chap­ter 13.3.4 for a more phys­i­cal de­riva­tion in terms of elec­tro­sta­t­ics.)

The in­tu­itive de­riva­tion de­fines as the so­lu­tion due to a unit spike, i.e. a delta func­tion, lo­cated at the ori­gin. That means that is the so­lu­tion to

Here is the three-di­men­sion­al delta func­tion, de­fined as an in­fi­nite spike at the ori­gin that in­te­grates to 1.

By it­self the above de­f­i­n­i­tion is of course mean­ing­less: in­fin­ity is not a valid num­ber. To give it mean­ing, it is nec­es­sary to de­fine an ap­prox­i­mate delta func­tion, one that is merely a large spike rather than an in­fi­nite one. This ap­prox­i­mate delta func­tion must still in­te­grate to 1 and will be re­quired to be zero be­yond some small dis­tance from the ori­gin:

In the above in­te­gral the re­gion of in­te­gra­tion should at least in­clude the small re­gion of ra­dius around the ori­gin. The ap­prox­i­mate delta func­tion will fur­ther be as­sumed to be non­neg­a­tive. It must have large val­ues in the small vicin­ity around the ori­gin where it is nonzero; oth­er­wise the in­te­gral over the small vicin­ity would be small in­stead of 1. But the key is that the val­ues are not in­fi­nite, just large. So nor­mal math­e­mat­ics can be used.

The cor­re­spond­ing ap­prox­i­mate Green’s func­tion of the Pois­son equa­tion sat­is­fies

In the limit , be­comes the Dirac delta func­tion and be­comes the ex­act Green’s func­tion .

To find the ap­prox­i­mate Green’s func­tion, it will be as­sumed that only de­pends on the dis­tance from the ori­gin. In other words, it is as­sumed to be spher­i­cally sym­met­ric. Then so is . (Note that this as­sump­tion is not strictly nec­es­sary. That can be seen from the gen­eral so­lu­tion for the Pois­son equa­tion given ear­lier. But it should at least be as­sumed that is non­neg­a­tive. If it could have ar­bi­trar­ily large neg­a­tive val­ues, then could be any­thing.)

Now in­te­grate both sides of the Pois­son equa­tion over a sphere of a cho­sen ra­dius :

As noted, the delta func­tion in­te­grates to 1 as long as the vicin­ity of the ori­gin is in­cluded. That means that the right hand side is 1 as long as . This will now be as­sumed. The left hand side can be writ­ten out. That gives

Ac­cord­ing to the [di­ver­gence] [Gauss] [Os­tro­grad­sky] the­o­rem, the left hand side can be writ­ten as a sur­face in­te­gral to give

Here stands for the sur­face of the sphere of ra­dius . The to­tal sur­face is . Also is the unit vec­tor or­thog­o­nal to the sur­face, in the out­ward di­rec­tion. That is the ra­dial di­rec­tion. The to­tal dif­fer­en­tial of cal­cu­lus then im­plies that is the ra­dial de­riv­a­tive . So,

Be­cause is re­quired to van­ish at large dis­tances, this in­te­grates to

The ex­act Green’s func­tion has equal to zero, so

Fi­nally the rig­or­ous de­riva­tion with­out us­ing poorly de­fined things like delta func­tions. In the sup­posed gen­eral so­lu­tion of the Pois­son equa­tion given ear­lier, change in­te­gra­tion vari­able to

It is to be shown that the func­tion de­fined this way sat­is­fies the Pois­son equa­tion . To do so, ap­ply twice:

Here means dif­fer­en­ti­a­tion with re­spect to the com­po­nents of in­stead of the com­po­nents of . Be­cause de­pends only on , you get the same an­swer whichever way you dif­fer­en­ti­ate.

It will be as­sumed that the func­tion is well be­haved, at least con­tin­u­ous, and be­comes zero rea­son­ably quickly at in­fin­ity. In that case, you can get a valid ap­prox­i­ma­tion to the in­te­gral above if you ex­clude very small and very large val­ues of :

In par­tic­u­lar, this ap­prox­i­ma­tion be­comes ex­act in the lim­its where the con­stants and . The in­te­gral can now be rewrit­ten as

as can be ver­i­fied by ex­plic­itly dif­fer­en­ti­at­ing out the three terms of the in­te­grand. Next note that the third term is zero, be­cause as seen above sat­is­fies the ho­mo­ge­neous Pois­son equa­tion away from the ori­gin. And the other two terms can be writ­ten out us­ing the [di­ver­gence] [Gauss] [Os­tro­grad­sky] the­o­rem much like be­fore. This pro­duces in­te­grals over both the bound­ing sphere of ra­dius , as well as over the bound­ing sphere of ra­dius . But the in­te­grals over the sphere of ra­dius will be van­ish­ingly small if be­comes zero suf­fi­ciently quickly at in­fin­ity. Sim­i­larly, the in­te­gral of the first term over the small sphere is van­ish­ingly small, be­cause is 1 on the small sphere but the sur­face of the small sphere is . How­ever, in the sec­ond term, the de­riv­a­tive of in the neg­a­tive ra­dial di­rec­tion is 1/, which mul­ti­plies to 1 against the sur­face of the small sphere. There­fore the sec­ond term pro­duces the av­er­age of over the small sphere, and that be­comes in the limit . So the Pois­son equa­tion ap­plies.

#### D.2.2 The screened Pois­son equa­tion

The so-called screened Pois­son equa­tion is

Here is sup­posed to be a given func­tion and an un­known func­tion that is to be found. Fur­ther is a given con­stant. If is zero, this is the Pois­son equa­tion. How­ever, nonzero cor­re­sponds to the in­ho­mo­ge­neous steady Klein-Gor­don equa­tion for a par­ti­cle with nonzero mass.

The analy­sis of the screened Pois­son equa­tion is al­most the same as for the Pois­son equa­tion given in the pre­vi­ous sub­sec­tion. There­fore only the dif­fer­ences will be noted here. The ap­prox­i­mate Green’s func­tion must sat­isfy, away from the ori­gin,

The so­lu­tion to this that van­ishes at in­fin­ity is of the form

where is some con­stant. To check this, plug it in, us­ing the ex­pres­sion (N.5) for in spher­i­cal co­or­di­nates. To iden­tify the con­stant , in­te­grate the full equa­tion

over a sphere of ra­dius around the ori­gin and ap­ply the di­ver­gence the­o­rem as in the pre­vi­ous sub­sec­tion. Tak­ing the limit then gives 1/, which gives the ex­act Green’s func­tion as

The rig­or­ous de­riva­tion is the same as be­fore save for an ad­di­tional term in the in­te­grand, which drops out against the one.