Sub­sec­tions

### 7.9 Po­si­tion and Lin­ear Mo­men­tum

The sub­se­quent sec­tions will be look­ing at the time evo­lu­tion of var­i­ous quan­tum sys­tems, as pre­dicted by the Schrö­din­ger equa­tion. How­ever, be­fore that can be done, first the eigen­func­tions of po­si­tion and lin­ear mo­men­tum must be found. That is some­thing that the book has been stu­diously avoid­ing so far. The prob­lem is that the po­si­tion and lin­ear mo­men­tum eigen­func­tions have awk­ward is­sues with nor­mal­iz­ing them.

These nor­mal­iza­tion prob­lems have con­se­quences for the co­ef­fi­cients of the eigen­func­tions. In the or­tho­dox in­ter­pre­ta­tion, the square mag­ni­tudes of the co­ef­fi­cients should give the prob­a­bil­i­ties of get­ting the cor­re­spond­ing val­ues of po­si­tion and lin­ear mo­men­tum. But this state­ment will have to be mod­i­fied a bit.

One good thing is that un­like the Hamil­ton­ian, which is spe­cific to a given sys­tem, the po­si­tion op­er­a­tor

and the lin­ear mo­men­tum op­er­a­tor

are the same for all sys­tems. So, you only need to find their eigen­func­tions once.

#### 7.9.1 The po­si­tion eigen­func­tion

The eigen­func­tion that cor­re­sponds to the par­ti­cle be­ing at a pre­cise -​po­si­tion , -​po­si­tion , and -​po­si­tion will be de­noted by . The eigen­value prob­lem is:

(Note the need in this analy­sis to use for the mea­sur­able par­ti­cle po­si­tion, since are al­ready used for the eigen­func­tion ar­gu­ments.)

To solve this eigen­value prob­lem, try again sep­a­ra­tion of vari­ables, where it is as­sumed that is of the form . Sub­sti­tu­tion gives the par­tial prob­lem for as

This equa­tion im­plies that at all points not equal to , will have to be zero, oth­er­wise there is no way that the two sides can be equal. So, func­tion can only be nonzero at the sin­gle point . At that one point, it can be any­thing, though.

To re­solve the am­bi­gu­ity, the func­tion is taken to be the Dirac delta func­tion,

The delta func­tion is, loosely speak­ing, suf­fi­ciently strongly in­fi­nite at the sin­gle point that its in­te­gral over that sin­gle point is one. More pre­cisely, the delta func­tion is de­fined as the lim­it­ing case of the func­tion shown in the left hand side of fig­ure 7.10.

The fact that the in­te­gral is one leads to a very use­ful math­e­mat­i­cal prop­erty of delta func­tions: they are able to pick out one spe­cific value of any ar­bi­trary given func­tion . Just take an in­ner prod­uct of the delta func­tion with . It will pro­duce the value of at the point , in other words, :

 (7.49)

(Since the delta func­tion is zero at all points ex­cept , it does not make a dif­fer­ence whether or is used in the in­te­gral.) This is some­times called the “fil­ter­ing prop­erty” of the delta func­tion.

The prob­lems for the po­si­tion eigen­func­tions and are the same as the one for , and have a sim­i­lar so­lu­tion. The com­plete eigen­func­tion cor­re­spond­ing to a mea­sured po­si­tion is there­fore:

 (7.50)

Here is the three-di­men­sion­al delta func­tion, a spike at po­si­tion whose vol­ume in­te­gral equals one.

Ac­cord­ing to the or­tho­dox in­ter­pre­ta­tion, the prob­a­bil­ity of find­ing the par­ti­cle at for a given wave func­tion should be the square mag­ni­tude of the co­ef­fi­cient of the eigen­func­tion. This co­ef­fi­cient can be found as an in­ner prod­uct:

It can be sim­pli­fied to
 (7.51)

be­cause of the prop­erty of the delta func­tions to pick out the cor­re­spond­ing func­tion value.

How­ever, the ap­par­ent con­clu­sion that gives the prob­a­bil­ity of find­ing the par­ti­cle at is wrong. The rea­son it fails is that eigen­func­tions should be nor­mal­ized; the in­te­gral of their square should be one. The in­te­gral of the square of a delta func­tion is in­fi­nite, not one. That is OK, how­ever; is a con­tin­u­ously vary­ing vari­able, and the chances of find­ing the par­ti­cle at to an in­fi­nite num­ber of dig­its ac­cu­rate would be zero. So, the prop­erly nor­mal­ized eigen­func­tions would have been use­less any­way.

In­stead, ac­cord­ing to Born's sta­tis­ti­cal in­ter­pre­ta­tion of chap­ter 3.1, the ex­pres­sion

gives the prob­a­bil­ity of find­ing the par­ti­cle in an in­fin­i­tes­i­mal vol­ume around . In other words, gives the prob­a­bil­ity of find­ing the par­ti­cle near lo­ca­tion per unit vol­ume. (The un­der­lines be­low the po­si­tion co­or­di­nates are no longer needed to avoid am­bi­gu­ity and have been dropped.)

Be­sides the nor­mal­iza­tion is­sue, an­other idea that needs to be some­what mod­i­fied is a strict col­lapse of the wave func­tion. Any po­si­tion mea­sure­ment that can be done will leave some un­cer­tainty about the pre­cise lo­ca­tion of the par­ti­cle: it will leave nonzero over a small range of po­si­tions, rather than just one po­si­tion. More­over, un­like en­ergy eigen­states, po­si­tion eigen­states are not sta­tion­ary: af­ter a po­si­tion mea­sure­ment, will again spread out as time in­creases.

Key Points
Po­si­tion eigen­func­tions are delta func­tions.

They are not prop­erly nor­mal­ized.

The co­ef­fi­cient of the po­si­tion eigen­func­tion for a po­si­tion is the good old wave func­tion .

Be­cause of the fact that the delta func­tions are not nor­mal­ized, the square mag­ni­tude of does not give the prob­a­bil­ity that the par­ti­cle is at po­si­tion .

In­stead the square mag­ni­tude of gives the prob­a­bil­ity that the par­ti­cle is near po­si­tion per unit vol­ume.

Po­si­tion eigen­func­tions are not sta­tion­ary, so lo­cal­ized par­ti­cle wave func­tions will spread out over time.

#### 7.9.2 The lin­ear mo­men­tum eigen­func­tion

Turn­ing now to lin­ear mo­men­tum, the eigen­func­tion that cor­re­sponds to a pre­cise lin­ear mo­men­tum will be in­di­cated as . If you again as­sume that this eigen­func­tion is of the form , the par­tial prob­lem for is found to be:

The so­lu­tion is a com­plex ex­po­nen­tial:

where is a con­stant.

Just like the po­si­tion eigen­func­tion ear­lier, the lin­ear mo­men­tum eigen­func­tion has a nor­mal­iza­tion prob­lem. In par­tic­u­lar, since it does not be­come small at large , the in­te­gral of its square is in­fi­nite, not one. The so­lu­tion is to ig­nore the prob­lem and to just take a nonzero value for ; the choice that works out best is to take:

(How­ever, other books, in par­tic­u­lar non­quan­tum ones, are likely to make a dif­fer­ent choice.)

The prob­lems for the and lin­ear mo­menta have sim­i­lar so­lu­tions, so the full eigen­func­tion for lin­ear mo­men­tum takes the form:

 (7.52)

The co­ef­fi­cient of the mo­men­tum eigen­func­tion is very im­por­tant in quan­tum analy­sis. It is in­di­cated by the spe­cial sym­bol and called the “mo­men­tum space wave func­tion.” Like all co­ef­fi­cients, it can be found by tak­ing an in­ner prod­uct of the eigen­func­tion with the wave func­tion:

 (7.53)

The mo­men­tum space wave func­tion does not quite give the prob­a­bil­ity for the mo­men­tum to be . In­stead it turns out that

gives the prob­a­bil­ity of find­ing the lin­ear mo­men­tum within a small mo­men­tum range around . In other words, gives the prob­a­bil­ity of find­ing the par­ti­cle with a mo­men­tum near per unit mo­men­tum space vol­ume. That is much like the square mag­ni­tude of the nor­mal wave func­tion gives the prob­a­bil­ity of find­ing the par­ti­cle near lo­ca­tion per unit phys­i­cal vol­ume. The mo­men­tum space wave func­tion is in the mo­men­tum space what the nor­mal wave func­tion is in the phys­i­cal space .

There is even an in­verse re­la­tion­ship to re­cover from , and it is easy to re­mem­ber:

 (7.54)

where the sub­script on the in­ner prod­uct in­di­cates that the in­te­gra­tion is over mo­men­tum space rather than phys­i­cal space.

If this in­ner prod­uct is writ­ten out, it reads:

 (7.55)

Math­e­mati­cians prove this for­mula un­der the name “Fourier In­ver­sion The­o­rem”, {A.26}. But it re­ally is just the same sort of idea as writ­ing as a sum of eigen­func­tions times their co­ef­fi­cients , as in . In this case, the co­ef­fi­cients are given by and the eigen­func­tions by the ex­po­nen­tial (7.52). The only real dif­fer­ence is that the sum has be­come an in­te­gral since has con­tin­u­ous val­ues, not dis­crete ones.

Key Points
The lin­ear mo­men­tum eigen­func­tions are com­plex ex­po­nen­tials of the form:

They are not prop­erly nor­mal­ized.

The co­ef­fi­cient of the lin­ear mo­men­tum eigen­func­tion for a mo­men­tum is in­di­cated by . It is called the mo­men­tum space wave func­tion.

Be­cause of the fact that the mo­men­tum eigen­func­tions are not nor­mal­ized, the square mag­ni­tude of does not give the prob­a­bil­ity that the par­ti­cle has mo­men­tum .

In­stead the square mag­ni­tude of gives the prob­a­bil­ity that the par­ti­cle has a mo­men­tum close to per unit mo­men­tum space vol­ume.

In writ­ing the com­plete wave func­tion in terms of the mo­men­tum eigen­func­tions, you must in­te­grate over the mo­men­tum in­stead of sum.

The trans­for­ma­tion be­tween the phys­i­cal space wave func­tion and the mo­men­tum space wave func­tion is called the Fourier trans­form. It is in­vert­ible.