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

$${\skew 2\widehat{\skew{-1}\vec r}}= ({\widehat x}, {\widehat y}, {\widehat z})$$

and the lin­ear mo­men­tum op­er­a­tor

$${\skew 4\widehat{\skew{-.5}\vec p}}= ({\widehat p}_x,{\wide... ...c{\partial}{\partial y}, \frac{\partial}{\partial z} \right)$$

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 $x$-​po­si­tion ${\underline x}$, $y$-​po­si­tion ${\underline y}$, and $z$-​po­si­tion ${\underline z}$ will be de­noted by $R_{{\underline x}{\underline y}{\underline z}}(x,y,z)$. The eigen­value prob­lem is:

$$\begin{eqnarray*} && {\widehat x}R_{{\underline x}{\underline y}{\underline z}}... ...nderline z}R_{{\underline x}{\underline y}{\underline z}}(x,y,z) \end{eqnarray*}$$

(Note the need in this analy­sis to use $({\underline x},{\underline y},{\underline z})$ for the mea­sur­able par­ti­cle po­si­tion, since $(x,y,z)$ are al­ready used for the eigen­func­tion ar­gu­ments.)

To solve this eigen­value prob­lem, try again sep­a­ra­tion of vari­ables, where it is as­sumed that $R_{{\underline x}{\underline y}{\underline z}}(x,y,z)$ is of the form $X(x)Y(y)Z(z)$. Sub­sti­tu­tion gives the par­tial prob­lem for $X$ as

$$x X(x) = {\underline x}X(x)$$

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

To re­solve the am­bi­gu­ity, the func­tion $X(x)$ is taken to be the Dirac delta func­tion,

$$X(x)=\delta(x-{\underline x})$$

The delta func­tion is, loosely speak­ing, suf­fi­ciently strongly in­fi­nite at the sin­gle point $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ ${\underline x}$ 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.

Fig­ure 7.10: Ap­prox­i­mate Dirac delta func­tion $\delta_\varepsilon(x-\protect{\underline x})$ is shown left. The true delta func­tion $\delta(x-\protect{\underline x})$ is the limit when $\varepsilon$ be­comes zero, and is an in­fi­nitely high, in­fi­nitely thin spike, shown right. It is the eigen­func­tion cor­re­spond­ing to a po­si­tion $\protect{\underline x}$.

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 $f(x)$. Just take an in­ner prod­uct of the delta func­tion $\delta(x-{\underline x})$ with $f(x)$. It will pro­duce the value of $f(x)$ at the point ${\underline x}$, in other words, $f({\underline x})$:

$$\langle \delta(x-{\underline x})\vert f(x)\rangle = \int_{... ...nderline x}) f({\underline x}) { \rm d}x = f({\underline x})$$ (7.49)

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

The prob­lems for the po­si­tion eigen­func­tions $Y$ and $Z$ are the same as the one for $X$, and have a sim­i­lar so­lu­tion. The com­plete eigen­func­tion cor­re­spond­ing to a mea­sured po­si­tion $({\underline x},{\underline y},{\underline z})$ is there­fore:

$$\fbox{$\displaystyle R_{{\underline x}{\underline y}{\unde... ...quiv \delta^3({\skew0\vec r}-{\underline{\skew0\vec r}}) $} %$$ (7.50)

Here $\delta^3({\skew0\vec r}-{\underline{\skew0\vec r}})$ is the three-di­men­sion­al delta func­tion, a spike at po­si­tion ${\underline{\skew0\vec r}}$ whose vol­ume in­te­gral equals one.

Ac­cord­ing to the or­tho­dox in­ter­pre­ta­tion, the prob­a­bil­ity of find­ing the par­ti­cle at $({\underline x},{\underline y},{\underline z})$ for a given wave func­tion $\Psi$ should be the square mag­ni­tude of the co­ef­fi­cient $c_{{\underline x}{\underline y}{\underline z}}$ of the eigen­func­tion. This co­ef­fi­cient can be found as an in­ner prod­uct:

$$c_{{\underline x}{\underline y}{\underline z}}(t) = \langl... ...elta(y-{\underline y})\delta(z-{\underline z})\vert\Psi\rangle$$

It can be sim­pli­fied to

$$c_{{\underline x}{\underline y}{\underline z}}(t) = \Psi({\underline x},{\underline y},{\underline z};t)$$ (7.51)

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

How­ever, the ap­par­ent con­clu­sion that $\vert\Psi({\underline x},{\underline y},{\underline z};t)\vert^2$ gives the prob­a­bil­ity of find­ing the par­ti­cle at $({\underline x},{\underline y},{\underline z})$ 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; ${\skew0\vec r}$ is a con­tin­u­ously vary­ing vari­able, and the chances of find­ing the par­ti­cle at $({\underline x},{\underline y},{\underline z})$ 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

$$\vert\Psi(x,y,z;t)\vert^2 { \rm d}x {\rm d}y {\rm d}z$$

gives the prob­a­bil­ity of find­ing the par­ti­cle in an in­fin­i­tes­i­mal vol­ume ${\rm d}{x}{\rm d}{y}{\rm d}{z}$ around $(x,y,z)$. In other words, $\vert\Psi(x,y,z;t)\vert^2$ gives the prob­a­bil­ity of find­ing the par­ti­cle near lo­ca­tion $(x,y,z)$ 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 $\Psi(x,y,z;t)$ 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, $\Psi$ will again spread out as time in­creases.

Key Points

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Po­si­tion eigen­func­tions are delta func­tions.

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They are not prop­erly nor­mal­ized.

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The co­ef­fi­cient of the po­si­tion eigen­func­tion for a po­si­tion $(x,y,z)$ is the good old wave func­tion $\Psi(x,y,z;t)$.

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Be­cause of the fact that the delta func­tions are not nor­mal­ized, the square mag­ni­tude of $\Psi(x,y,z;t)$ does not give the prob­a­bil­ity that the par­ti­cle is at po­si­tion $(x,y,z)$.

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In­stead the square mag­ni­tude of $\Psi(x,y,z;t)$ gives the prob­a­bil­ity that the par­ti­cle is near po­si­tion $(x,y,z)$ per unit vol­ume.

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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 $(p_x,p_y,p_z)$ will be in­di­cated as $P_{p_xp_yp_z}(x,y,z)$. If you again as­sume that this eigen­func­tion is of the form $X(x)Y(y)Z(z)$, the par­tial prob­lem for $X$ is found to be:

$$\frac{\hbar}{{\rm i}} \frac{\partial X(x)}{\partial x} = p_x X(x)$$

The so­lu­tion is a com­plex ex­po­nen­tial:

$$X(x)= Ae^{{\rm i}p_x x/\hbar}$$

where $A$ 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 $\vert x\vert$, the in­te­gral of its square is in­fi­nite, not one. The so­lu­tion is to ig­nore the prob­lem and to just take a nonzero value for $A$; the choice that works out best is to take:

$$A=\frac{1}{\sqrt{2\pi\hbar}}$$

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

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

$$\fbox{$\displaystyle P_{p_xp_yp_z}(x,y,z) = \frac{1}{\sqrt{2\pi\hbar}^3} e^{{\rm i}(p_x x + p_y y + p_z z)/\hbar} $} %$$ (7.52)

The co­ef­fi­cient $c_{p_xp_yp_z}(t)$ of the mo­men­tum eigen­func­tion is very im­por­tant in quan­tum analy­sis. It is in­di­cated by the spe­cial sym­bol $\Phi(p_x,p_y,p_z;t)$ 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:

$$\fbox{$\displaystyle \Phi(p_x,p_y,p_z;t) = \frac{1}{\sqrt{... ...e^{{\rm i}(p_x x + p_y y + p_z z)/\hbar} \vert \Psi\rangle $}$$ (7.53)

The mo­men­tum space wave func­tion does not quite give the prob­a­bil­ity for the mo­men­tum to be $(p_x,p_y,p_z)$. In­stead it turns out that

$$\vert\Phi(p_x,p_y,p_z;t)\vert^2 {\rm d}p_x {\rm d}p_y {\rm d}p_z$$

gives the prob­a­bil­ity of find­ing the lin­ear mo­men­tum within a small mo­men­tum range ${\rm d}{p}_x{\rm d}{p}_y{\rm d}{p}_z$ around $(p_x,p_y,p_z)$. In other words, $\vert\Phi(p_x,p_y,p_z;t)\vert^2$ gives the prob­a­bil­ity of find­ing the par­ti­cle with a mo­men­tum near $(p_x,p_y,p_z)$ per unit mo­men­tum space vol­ume. That is much like the square mag­ni­tude $\vert\Psi(x,y,z;t)\vert^2$ of the nor­mal wave func­tion gives the prob­a­bil­ity of find­ing the par­ti­cle near lo­ca­tion $(x,y,z)$ per unit phys­i­cal vol­ume. The mo­men­tum space wave func­tion $\Phi$ is in the mo­men­tum space $(p_x,p_y,p_z)$ what the nor­mal wave func­tion $\Psi$ is in the phys­i­cal space $(x,y,z)$.

There is even an in­verse re­la­tion­ship to re­cover $\Psi$ from $\Phi$, and it is easy to re­mem­ber:

$$\fbox{$\displaystyle \Psi(x,y,z;t) = \frac{1}{\sqrt{2\pi\h... ...+ p_y y + p_z z)/\hbar} \vert \Phi\rangle_{{\skew0\vec p}} $}$$ (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:

$$\fbox{$\displaystyle \Psi(x,y,z;t) = \frac{1}{\sqrt{2\pi\h... ..._y y + p_z z)/\hbar} { \rm d}p_x {\rm d}p_y {\rm d}p_z $} %$$ (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 $\Psi$ as a sum of eigen­func­tions $\psi_n$ times their co­ef­fi­cients $c_n$, as in $\Psi$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\sum_nc_n\psi_n$. In this case, the co­ef­fi­cients are given by $\Phi$ and the eigen­func­tions by the ex­po­nen­tial (7.52). The only real dif­fer­ence is that the sum has be­come an in­te­gral since ${\skew0\vec p}$ has con­tin­u­ous val­ues, not dis­crete ones.

Key Points

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The lin­ear mo­men­tum eigen­func­tions are com­plex ex­po­nen­tials of the form:

$$\frac{1}{\sqrt{2\pi\hbar}^3} e^{{\rm i}(p_x x + p_y y + p_z z)/\hbar}$$

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They are not prop­erly nor­mal­ized.

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The co­ef­fi­cient of the lin­ear mo­men­tum eigen­func­tion for a mo­men­tum $(p_x,p_y,p_z)$ is in­di­cated by $\Phi(p_x,p_y,p_z;t)$. It is called the mo­men­tum space wave func­tion.

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Be­cause of the fact that the mo­men­tum eigen­func­tions are not nor­mal­ized, the square mag­ni­tude of $\Phi(p_x,p_y,p_z;t)$ does not give the prob­a­bil­ity that the par­ti­cle has mo­men­tum $(p_x,p_y,p_z)$.

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In­stead the square mag­ni­tude of $\Phi(p_x,p_y,p_z;t)$ gives the prob­a­bil­ity that the par­ti­cle has a mo­men­tum close to $(p_x,p_y,p_z)$ per unit mo­men­tum space vol­ume.

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

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