### A.26 Fourier in­ver­sion the­o­rem and Par­se­val

This note dis­cusses Fourier se­ries, Fourier in­te­grals, and Par­se­val’s iden­tity.

Con­sider first one-di­men­sion­al Fourier se­ries. They ap­ply to func­tions that are pe­ri­odic with some given pe­riod :

Such func­tions can be writ­ten as a “Fourier se­ries:”
 (A.193)

Here the val­ues are those for which the ex­po­nen­tials are pe­ri­odic of pe­riod . Ac­cord­ing to the Euler for­mula (2.5), that means that must be a whole mul­ti­ple of , so
 (A.194)

Note that no­ta­tions for Fourier se­ries can vary from one au­thor to the next. The above form of the Fourier se­ries is the pref­ered one for quan­tum me­chan­ics. The rea­son is that the func­tions form an or­tho­nor­mal set:

 (A.195)

Quan­tum me­chan­ics just loves or­tho­nor­mal sets of func­tions. In par­tic­u­lar, note that the above func­tions are mo­men­tum eigen­func­tions. Just ap­ply the lin­ear mo­men­tum op­er­a­tor on them. That shows that their lin­ear mo­men­tum is given by the de Broglie re­la­tion . Here these mo­men­tum eigen­func­tions are prop­erly nor­mal­ized. They would not be us­ing dif­fer­ent con­ven­tions.

That any (rea­son­able) pe­ri­odic func­tion can be writ­ten as a Fourier se­ries was al­ready shown in {D.8}. That de­riva­tion took be the half-pe­riod. The for­mula for the co­ef­fi­cients can also be de­rived di­rectly: sim­ply mul­ti­ply the ex­pres­sion (A.193) for with for any ar­bi­trary value of and in­te­grate over . Be­cause of the or­tho­nor­mal­ity (A.195), the in­te­gra­tion pro­duces zero for all ex­cept if , and then it pro­duces as re­quired.

Note from (A.193) that if you known you can find all the . Con­versely, if you know all the , you can find at every po­si­tion . The for­mu­lae work both ways.

But the sym­me­try goes even deeper than that. Con­sider the in­ner prod­uct of a pair of func­tions and :

Us­ing the or­tho­nor­mal­ity prop­erty (A.195) that be­comes
 (A.196)

Now note that if you look at the co­ef­fi­cients and as the co­ef­fi­cients of in­fi­nite-​di­men­sion­al vec­tors, then the right hand side is just the in­ner prod­uct of these vec­tors. In short, Fourier se­ries pre­serve in­ner prod­ucts.

There­fore the equa­tion above may be writ­ten more con­cisely as

 (A.197)

This is the so-called “Par­se­val iden­tity.” Now trans­for­ma­tions that pre­serve in­ner prod­ucts are called “uni­tary” in math­e­mat­ics. So the Par­se­val iden­tity shows that the trans­for­ma­tion from pe­ri­odic func­tions to their Fourier co­ef­fi­cients is uni­tary.

That is quite im­por­tant for quan­tum me­chan­ics. For ex­am­ple, as­sume that is a wave func­tion of a par­ti­cle stuck on a ring of cir­cum­fer­ence . Wave func­tions should be nor­mal­ized, so:

Ac­cord­ing to the Born in­ter­pre­ta­tion, the left hand side says that the prob­a­bil­ity of find­ing the par­ti­cle is 1, cer­tainty, if you look at every po­si­tion on the ring. But ac­cord­ing to the or­tho­dox in­ter­pre­ta­tion of quan­tum me­chan­ics, in the right hand side gives the prob­a­bil­ity of find­ing the par­ti­cle with mo­men­tum . The fact that the to­tal sum is 1 means phys­i­cally that it is cer­tain that the par­ti­cle will be found with some mo­men­tum.

So far, only pe­ri­odic func­tions have been cov­ered. But func­tions in in­fi­nite space can be han­dled by tak­ing the pe­riod in­fi­nite. To do that, note from (A.194) that the val­ues of the Fourier se­ries are spaced apart over a dis­tance

In the limit , be­comes an in­fin­i­tes­i­mal in­cre­ment , and the sums be­come in­te­grals. Now in quan­tum me­chan­ics it is con­ve­nient to re­place the co­ef­fi­cients by a new func­tion that is de­fined so that

The rea­son that this is con­ve­nient is that gives the prob­a­bil­ity for wave num­ber . Then for a func­tion that is de­fined as above, gives the prob­a­bil­ity per unit -range.

If the above de­f­i­n­i­tion and are sub­sti­tuted into the Fourier se­ries ex­pres­sions (A.193), in the limit it gives the “Fourier in­te­gral” for­mu­lae:

 (A.198)

In books on math­e­mat­ics you will usu­ally find func­tion in­di­cated as , to clar­ify that it is a com­pletely dif­fer­ent func­tion than . Un­for­tu­nately, the hat is al­ready used for some­thing much more im­por­tant in quan­tum me­chan­ics. So in quan­tum me­chan­ics you will have to look at the ar­gu­ment, or , to know which func­tion it re­ally is.

Of course, in quan­tum me­chan­ics you are of­ten more in­ter­ested in the mo­men­tum than in the wave num­ber. So it is of­ten con­ve­nient to de­fine a new func­tion so that gives the prob­a­bil­ity per unit mo­men­tum range rather than unit wave num­ber range. Be­cause , the needed rescal­ing of is by a fac­tor . That gives

 (A.199)

Us­ing sim­i­lar sub­sti­tu­tions as for the Fourier se­ries, the Par­se­val iden­tity (A.197) be­comes

or in short
 (A.200)

This iden­tity is some­times called the “Plancherel the­o­rem,” af­ter a later math­e­mati­cian who gen­er­al­ized its ap­plic­a­bil­ity. The bot­tom line is that Fourier in­te­gral trans­forms too con­serve in­ner prod­ucts.

So far, this was all one-di­men­sion­al. How­ever, the ex­ten­sion to three di­men­sions is straight­for­ward. The first case to be con­sid­ered is that there is pe­ri­od­ic­ity in each Carte­sian di­rec­tion:

In quan­tum me­chan­ics, this would typ­i­cally cor­re­spond to the wave func­tion of a par­ti­cle stuck in a pe­ri­odic box of di­men­sions . When the par­ti­cle leaves such a box through one side, it reen­ters it at the same time through the op­po­site side.

There are now wave num­bers for each di­rec­tion,

where , , and are whole num­bers. For brevity, vec­tor no­ta­tions may be used:

Here is the “wave num­ber vec­tor.”

The Fourier se­ries for a three-di­men­sion­al pe­ri­odic func­tion is

 (A.201)

Here is short­hand for and is the vol­ume of the pe­ri­odic box.

The above ex­pres­sion for may be de­rived by ap­ply­ing the one-di­men­sion­al trans­form in each di­rec­tion in turn:

This is equiv­a­lent to what is given above, ex­cept for triv­ial changes in no­ta­tion. The ex­pres­sion for the Fourier co­ef­fi­cients can be de­rived anal­o­gous to the one-di­men­sion­al case, in­te­grat­ing now over the en­tire pe­ri­odic box.

The Par­se­val equal­ity still ap­plies

 (A.202)

where the left in­ner prod­uct in­te­gra­tion is over the pe­ri­odic box.

For in­fi­nite space

 (A.203)

 (A.204)

 (A.205)

These ex­pres­sions are all ob­tained com­pletely anal­o­gously to the one-di­men­sion­al case.

Of­ten, the func­tion is a vec­tor rather than a scalar. That does not make a real dif­fer­ence since each com­po­nent trans­forms the same way. Just put a vec­tor sym­bol over and in the above for­mu­lae. The in­ner prod­ucts are now de­fined as

For the picky, con­vert­ing Fourier se­ries into Fourier in­te­grals only works for well-be­haved func­tions. But to show that it also works for nasty wave func­tions, you can set up a lim­it­ing process in which you ap­prox­i­mate the nasty func­tions in­creas­ingly ac­cu­rately us­ing well-be­haved ones. Now if the well-be­haved func­tions are con­verg­ing, then their Fourier trans­forms are too. The in­ner prod­ucts of the dif­fer­ences in func­tions are the same ac­cord­ing to Par­se­val. And ac­cord­ing to the ab­stract Lebesgue vari­ant of the the­ory of in­te­gra­tion, that is enough to en­sure that the trans­form of the nasty func­tion ex­ists. This works as long as the nasty wave func­tion is square in­te­grable. And wave func­tions need to be in quan­tum me­chan­ics.

But be­ing square in­te­grable is not a strict re­quire­ment, as you may have been told else­where. A lot of func­tions that are not square in­te­grable have mean­ing­ful, in­vert­ible Fourier trans­forms. For ex­am­ple, func­tions whose square mag­ni­tude in­te­grals are in­fi­nite, but ab­solute value in­te­grals are fi­nite can still be mean­ing­fully trans­formed. That is more or less the clas­si­cal ver­sion of the in­ver­sion the­o­rem, in fact. (See D.C. Cham­p­eney, A Hand­book of Fourier The­o­rems, for more.)