Compare a matter wave tk(wxt 2+wz ,78evh1 j)g lqfk$\Psi$ to (a) a wave on a string, (b) an EM wave. Discuss similarities and differences.

Explain why Bohr's theory of the atom is not compatible with quantum mechz0k9f0vulji(( q s9br3j 0xgqanics, particularly the xu fv qs g(0kjb(qijrl09z903uncertainty principle.

Explain why it is that the more massive an object is, the easier it becomes tsano *)bao5,b o predict itsbaa nos5b)o* , future position.

In view of the uncertainty princ: d tuu2;4 cp7rt hribxl7,o)aiple, why does a baseball seem to have a well-defined position and speed:xir;2act op ud hr,7)t u74bl whereas an electron does not?

  Would it ever be possible to balance a very sharp needl-g lo 6w54b/eoy( btwse precisely on its point? Expyb- (wbtsgo4ow6 le /5lain.

  A cold thermometer i cgm 1.il2 ghupxn*b*4s placed in a hot bowl of soup. Will the temperature reading of.*gclpmgn*h2 xb 14 ui the thermometer be the same as the temperature of the hot soup before the measurement was made? Explain.

  Does the uncertainty principle set a limit to how well you j 5n dvvmozmun6z+pp40*u,um2+* syx can make any single mnupm,+u6vsd zp m5xv n*0+2*jzmy4 oueasurement of position?

If you knew the position of an object precisely, with no, re,z5bg7t gi8k7 wq4 bmc)re uncertainty, how well would youbmc r,re)b7 ge7 z84t5k,wq ig know its momentum?

  When you check the pressure in a tire, doesn't so:s3z./1i w vqfzx dr 5wlx2m,ome air inevitably escape? Is it powsf:i,dzm.wx3vqr ox21l z /5ssible to avoid this escape of air altogether? What is the relation to the uncertainty principle?

  It has been said that the ground-state energy in the hydrogen atom can be precis yirk64j k0j):ihs2re ely known but the excited states have some uncertainty in their values (an "ene sh jkki4r )0jr:6iy2ergy width"). Is this consistent with the uncertainty principle in its energy form? Explain.

Which model of the hydrogen atom, the Bo 1nc1tlm*1tcx/ d)vms a8l,gm9hsg) ahr model or the quantum-mechanical model, predicts th)l ,)sm m1 /8vgg1lctsac 91m*ntax hdat the electron spends more time near the nucleus?

The size of atoms varies by only a ff55/ +ndg ssumactor of three or so, from largest to smalles /mdu5sg+s 5fnt, yet the number of electrons varies from one to over 100. Explain.

Excited hydrogen and excited helium atoms both radiate light as they jump d xlcg rg k454;r8-kqylown to kgr5 8cl;qxkyr-l4g4the $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table look if qr- od+*nwdt d9+hly:.cuk-yoth,b*there were no electron spin but otherwise quantum mechanics were valid? Consider the ry* dh-y +uwntcddoltb+ : .q9o*-k, hfirst 20 elements or so.

  Which of the following electron configurations zn4 )b7ixxu5/yg -hyq are not allowed:
(a) $1s^2\,2s^2\,2p^4\,3s^2\,4p^2$;  ----待选择----allowednot allowed 
(b) $1s^2\,2s^2\,2p^3\,3s^1$;  ----待选择----allowednot allowed 
(c) $1s^2\,2s^2\,2p^6\,3s^2\,3p^3\,4s^2\,4d^3\,4f^1$? ----待选择----allowednot allowed 
If not allowed, explain why.

Give the complete electron configsn 0 w61 24jzikiiaeg+ipc6)yuration for a uranium atom (careful scrutiny across the periodic table on the inside back cover will provide useful hints)a6w2 pe+4ckg01i ijn)s iiy6z.

In what column of the periodic table would youoa .gpk2 fpb7,h;psij d5u0 ai+q 8u :c1yzj0f expect to find the atom with each of the following con2pbzjkpy diu:,sjfg0opac7+u;q.8 fi51 a 0hfigurations:
(a) $1s^22s^22p^63s^2$;
(b) $1s^22s^22p^63s^23p^6$;
(c) $1s^22s^22p^63s^23p^64s^1$;
(d) $1s^22s^22p^5$?

Why do chlorine and iodiqnx r.l73 nmi/ne exhibit similar properties?

Explain why potassium and sodium exhibit similar proph4*/7jbwl, gqfn *45p48jvz oo oc czterties.

The ionization energy for n d xq+7a zer;o*:xxe2reon $(Z=10)$ is 21.6 eV, and that for sodium $(Z=11)$ is 5.1 eV. Explain the large difference.

Why does the cutoff wavelength in (Fig. Below) imp9toyf2hkl hp(,3u(fm ly a photon nature for light? kh3typ lfo(2(f mu9 h,

Why do we not expect perfect agreement between measured values ofxguue+a6j)j7p/r*r: 8mv bpi X-ray line wavelengths and those calculated usp+)/ 6j vum r:jgbrx8 aui7*peing Bohr theory?

How would you figure out which lines ;9g udhzxa ib)0s 543om3s qvvin an X-ray spectrum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transitions deep within an atom to produce shorter wav( hei1 *0mwzy pc5jvd.z56uni 9.cjcg elengths than transitions byyj(iz5mecjz1ccni uw d*.p69v5g h0. outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from ordinary light? Hsbekh z- gs,8s,p/9yxow is it the same?

Explain how a 0.0005-W laser beam, phot7sbhj7o 8(ug n7a x)mlographed at a distance, can seem much stro) jho8an sb77um(gxl7nger than a 1000-W street lamp at the same distance.

  Does the intensity of light from a lasp9i8p 1edw 5sber fall off as the inverse square of the distance? Explain9pb wsi5e1p d8. ----待选择----yesno 

  The neutrons in a parallel beam, each having kinetic energy779wq o) wvn 0tcs-p.hjnytl : $0.40 \, \text{eV}$ are directed through two slits $0.50 \, \text{mm}$ apart. How far apart will the interference peaks be on a screen $1.0 \, \text{m}$ away? [Hint: First find the wavelength of the neutron.]
$\Delta y$ =    $ \times 10^{-7}\ \text{m}$

  Bullets of mass $3.0 \, \text{g}$ are fired in parallel paths with speeds of $220 \, \text{m/s}$ through a hole $3.0 \, \text{mm}$ in diameter. How far from the hole must you be to detect a $1.0 \, \text{cm}$-diameter spread in the beam of bullets? L =    $ \times 10^{28}\ \text{m}$

  A proton is traveling umm ; )fle y8y.talg87f,2 nyig6v2zx with a speed of $(6.560 \pm 0.012) \times 10^3 \, \text{m/s}$. With what maximum accuracy can its position be ascertained? [Hint: $\Delta p \approx m \Delta v$]
$\pm$    $ \times 10^{-11}\ \text{m}$

  If an electron’s position can be measured to an accuw030)gvq vhvt.7b t,lv8u ksrwf92 e nracy of $1.0 \times 10^{-8} \, \text{m}$, how accurately can its speed be known?
$\Delta v $=    $\times 10^{3}\ \text{m/s}$

  An electron remains in an 3v*bt3*ttmsij totn; f,jq,,excited state of an atom for typically $10^{-8} \, \text{s}$. What is the minimum uncertainty in the energy of the state (in eV)?
$\frac{\hbar}{\Delta t}$ =    $ \times 10^{-8}\ \text{eV}$

  The $Z^0$ boson, discovered in 1985, is the mediator of the weak nuclear force, and it typically decays very quickly. Its average rest energy is $91.19 \, \text{GeV}$, but its short lifetime shows up as an intrinsic width of $2.5 \, \text{GeV}$ (rest energy uncertainty). What is the lifetime of this particle?
$\frac{\hbar}{\Delta t}$=    $ \times 10^{-25}\ \text{s}$

  What is the uncertainty in the mass of a ;km. -uih+w*zuazu3k muon $(m=105.7 \, \text{MeV/c}^2)$, specified in $\text{eV/c}^2$, given its lifetime of $2.20 \times 10^{-6} \, \text{s}$? [Hint: $\Delta E \approx \hbar / \Delta t$ and $\Delta m = \Delta E / c^2$]
$\frac{\hbar}{\Delta t}$=    $ \times 10^{-10}\ \text{eV}$
${\Delta m}$ =    = $\times 10^{-10}\ \text{eV}$

  A free neutron $(m=1.67 \times 10^{-27} \, \text{kg})$ has a mean life of $900 \, \text{s}$. What is the uncertainty in its mass (in kg)?
${\Delta m}$=    $\times 10^{-54}\ \text{kg}$

  An electron and a $140$-g baseball are each traveling $150 \, \text{m/s}$ measured to an accuracy of $0.055\%$. Calculate and compare the uncertainty in position of each.
For the electron $\frac{\hbar}{m\Delta v}$=    $\times 10^{-3}\ \text{m}$
For the baseball $\frac{\hbar}{m\Delta v}$ =    $ \times 10^{-33}\ \text{m}$

  Estimate the lowest possible energ nyb2ls c*h*mke2kv+y0a0o 8py of a neutron contained in a typical nucleus of rad k *mk8vb0*h+02sl on ye2acpyius $1.0 \times 10^{-15} \, \text{m}$. [Hint: A particle can have an energy at least as large as its uncertainty.]
$\frac{\hbar}{\Delta x} $=    $\times 10^{-19}\ \text{kg} \cdot \text{m/s}$
E =    Mev

  Use the uncertainty principle to show that if an electron17 -xt5+ly cbikno/fu were present in the nuult71x ncio5-/yb +kf cleus $(r \approx 10^{-15} \, \text{m})$, its kinetic energy (use relativity) would be hundreds of MeV. (Since such electron energies are not observed, we conclude that electrons are not present in the nucleus.) [Hint: A particle can have an energy at least as large as its uncertainty.]
E $\approx$    Mev

  How accurately can the positi8 w*1xl4ac3ge2b+ oxpd swwp xua.20mon of a $3.00$-keV electron be measured assuming its energy is known to $1.00\%$?
$\frac{\hbar}{\Delta p} =$    $ \times 10^{-10}\ \text{m}$

  For $n=6$, what values can $l$ have?
$\ell$ =    or    or    or    or    or   

  For $n=5, l=3$, what are the possible values of $m_l$ and $m_s$?
$n_\ell$ = $\pm $    or $\pm $    or $\pm $    or $\pm $   
$m_s$ = $\pm $ 1/  

  How many electrons can be in theo+ u9qng:xta uw*dr9+ $n=6, l=3$ subshell?    elections

How many different states are possible for an electron whose principal quantumia03fseyw :m-x*a* p u nuf exys0:i-up a* 3*mawmber is $n=4$? Write down the quantum numbers for each state.

List the quantum numbm0p ,fyad rs*0ers for each electron in the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each electron in the ground state of nitro :kzo/tr-ua n,,cms2ygen $(Z=7)$.

  Suppose a certain hydrogen atobeibke 2o-pan4,h/5a *i trlq4z/b c )m has $l=4$. What are the possible values for $n, m_l$, and $m_s$?
n $\ge $   
$m_\ell$ =    $\pm$    $\pm$    $\pm$    $\pm$   
$m_s$ = $\pm$   

  Calculate the magnitude of the angula9f; 4lwvlj ow/r momentum of an electron in the $n=4, l=3$ state of hydrogen.
L =    $\times 10^{-34}\ \text{kg} \cdot \text{m}^2/\text{s}$

  If a hydrogen atom has y cmuty(a 7q.9z/ z)gp .ht1zf$m_l=-3$, what are the possible values of $n, l$, and $m_s$?
$\ell \ge $   
n$\ge \ell$ +    (minimum 4)
$m_s$ = $\pm $   

  Show that there can be 35h radxy6b4n 18 electrons in a "g" subshell.    electrons

What is the full electron configuration in the ground state for elements wibml1s a;8;ru qth $Z$ equal to (a) 27, (b) 36, (c) 38? [Hint: see the periodic table inside the back cover.]

What is the full electron configurati1.66z ggv),us m b3k hbzdek9bon for (a) selenium (Se), (b) gold (Au), (c) radium (Ra)? [Hint: see the period .mk961)gdkv 6zb3gzse ubbh,ic table inside the back cover.]

  A hydrogen atom is in t j lxbee;;mq5 iee;*m:he $6s$ state. Determine
(a) the principal quantum number, n =   
(b) the energy of the state, $E_6$ =    Mev
(c) the orbital angular momentum and its quantum number $l$ L =   
(d) the possible values for the magnetic quantum number.$m_\ell$ =   

  Estimate the binding energy of the third electron inv-- 9/ ejbgq /c;phbze lithium using the Bohr theory. [Hint: This electron haeg p/-qzc 9vb/h; bj-es $n=2$ and "sees" a net charge of approximately $+1e$.] The measured value is $5.36 \, \text{eV}$.    eV

Show that the total angular momentum is zero c 7ejd4xbwc 03for a filled subshell.

  For each of the following atomic transitions, state whether the tra7p l*;ne9pgw knsition is allowed or fw;p9eplng k*7 orbidden, and if forbidden, what rule is being violated:
(a) $3p \to 4p$  ----待选择----forbiddenallowed 
(b) $1s \to 2p$  ----待选择----forbiddenallowed 
(c) $3d \to 2d$;  ----待选择----forbiddenallowed 
(d) $4d \to 2p$;  ----待选择----forbiddenallowed 
(e) $4s \to 2p$. ----待选择----forbiddenallowed 

  An excited H atom is in a.m wclx/.q vj fn:qu9v (dk0k+ $6d$ state.
(a) Name all the states $(n, l)$ to which the atom is "allowed" to jump with the emission of a photon.
(b) How many different wavelengths are there (ignoring fine structure)?   

  What are the shortest-wavelength X-rays emitted by electrons/0tobdjcmt7.)oxr*ve) - cc a striking the face of0 tdmocrb )/aj).7cvc- *t xoe a $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremssk eizv jq;s3.6trahlung X-rays emitted from an X-ray tube have vsqj 6e;i3.zk $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wavelength2w0v4vqsm4 ua 3thg+r $\lambda_0$ is given by $\lambda_0 = \frac{1240 \, \text{nm} \cdot \text{V}}{V}$ where $V$ is the X-ray tube voltage in volts.

  Estimate the X-ray w/fpecjwxn) 2v 0 v16isavelength emitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength f13 t0xsukjbm o wumz0- p2 0cxsx .jaw9u)0:ohor an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate -;ostuppi gu* -g x , 56a+e*nfhaqbu0the wavelength for an $n=3$ to $n=1$ transition in molybdenum $(Z=42)$. The measured value is $0.063 \, \text{nm}$. Why do we not expect perfect agreement?
$\lambda $ =    nm
n =   

  A mixture of iron and an unknown material itboi;i-d1 a2x 3yaw0 us bombarded with electrons. The wavelength of a2d tuw 1xb-03;iiayothe $K_\alpha$ lines are $194 \, \text{pm}$ for iron and $229 \, \text{pm}$ for the unknown. What is the unknown material?
 ----待选择----copperironmagnesiumvanadiumchromiummanganese 

  A laser used to weld detached retinas pu:cgz dhjn 4xd 8*l7v jc5rk/ s9togm64ts out $28$-ms-long pulses of $640$-nm light which average $0.68$-W output during a pulse. How much energy can be deposited per pulse and how many photons does each pulse contain?
E =    J
N =    $\times 10^{16}\ \text{photons}$

  A low-power laser used in a physics lab might have a power o*y7lr.osvhhw.d7 x;dtgw . ,tf $0.50 \, \text{mW}$ and a beam diameter of $3.0 \, \text{mm}$. Calculate
(a) the average light intensity of the laser beam, I =    $W/m^2$
(b) compare it to the intensity of a lightbulb producing $40$-W light viewed from $2.0 \, \text{m}$.I =    $W/m^2$ The laser beam is more intense by a factor of   

  Estimate the angular spread of a laser beam due to diffraction if the beam*k 4w amrnq7nf4qh) ob:v0q1 x emerges through a bmn o17)4rafvqq* 0knq4xwh:$3.0$-mm-diameter mirror. Assume that $\lambda = 694 \, \text{nm}$. What would be the diameter of this beam if it struck
$\theta$ =    $ \times 10^{-4}\ \text{rad}$
(a) a satellite $300 \, \text{km}$ above the Earth, D =   $ \times 10^{2}\ \text{m}$
(b) the Moon?D =   $ \times 10^{5}\ \text{m}$

  What is the wavelengthl-b45ba) itf b of the He-Ne laser?$\lambda $ =    nm

Use the uncertainty principle to estimate the position uncertainty for thebm+mlxn6cf5)c vf2e vw*z* )i electron in the ground state of the hydrogen atom. [Hint: Determine the momentum using the Bohr model of Section 27-12 and assume the momentum can be anyw2fvb+))fx cvil*m z*cmw 6ne5 here between this value and zero.] How does this result compare to the Bohr radius?

  An electron in the $n=2$ state of hydrogen remains there on average about $10^{-8} \, \text{s}$ before jumping to the $n=1$ state.
(a) Estimate the uncertainty in the energy of the $n=2$ state. $\frac{\hbar}{\Delta t} =$    $ \times 10^{-8}\ \text{eV}$
(b) What fraction of the transition energy is this? $\frac{\Delta E}{E} =$    $ \times 10^{-9}$
(c) What is the wavelength, and width (in nm), of this line in the spectrum of hydrogen?
$\lambda$ =    nm
$\Delta \lambda =$    $ \times 10^{-7}\ \text{nm}$

  What are the largest and smallest possible values for the angulaydxwv 1s.78psc l/bi1 rh 6o+fr momentum $L$ of an electron in the $n=5$ shell?
$\text{The smallest value of } L \text{ is}$ =   
$\text{The largest value of } L \text{ is}$ =    $\times 10^{-34}\ \text{kg} \cdot \text{m}^2/\text{s}$

  Estimate
(a) the quantum number $l$ for the orbital angular momentum of the Earth about the Sun, $\ell$ =    $ \times 10^{74}$
(b) the number of possible orientations for the plane of Earth's orbit. N =    $ \times 10^{74}$

  A $12$-g bullet leaves a rifle at a speed of $180 \, \text{m/s}$.
(a) What is the wavelength of this bullet? $\lambda $ =    $ \times 10^{-34}\ \text{m}$
(b) If the position of the bullet is known to an accuracy of $0.60 \, \text{cm}$ (radius of the barrel), what is the minimum uncertainty in its momentum? $\frac{\hbar}{\Delta y}$ =    $\times 10^{-32}\ \text{kg} \cdot \text{m/s}$

  Using the Bohr formula for the radius of an electron orbit8xki6fjq o9pw +)k dffg57+la , estimate the average distance from the nucleus fd7 fwxal)i qj9fgkk+8 f o+5p6or an electron in the innermost $(n=1)$ orbit of a uranium atom $(Z=92)$. Approximately how much energy would be required to remove this innermost electron?
$r_{\text{uranium}} =$    $ \times 10^{-13}\ \text{m}$
$\text{the binding energy is }$    $ \ \text{keV}$

  An X-ray tube operates oo+vdy-w0v 5et lbbp 970*sojat $95 \, \text{kV}$ with a current of $25 \, \text{mA}$ and nearly all the electron energy goes into heat. If the specific heat of the $0.085$-kg plate is $0.11 \, \text{kcal/kg} \cdot ^\circ \text{C}$, what will be the temperature rise per minute if no cooling water is used?
$\frac{\Delta T}{t} =$    $ \times 10^{3}\ ^\circ\text{C}/\text{min}$

  The ionization (binding) energy of the outermo1q7 ddnj/1 p.f2pb35efdvvdx nhp2k/st electron in boron is $8.26 \, \text{eV}$.
(a) Use the Bohr model to estimate the "effective charge," $Z_{\text{eff}}$, seen by this electron. $Z_{\text{eff}} $=   
(b) Estimate the average orbital radius. z =    $ \times 10^{-10}\ \text{m}$

  Use the Bohr theory to show that the Moseley plot(Fig. Bdgd)s4ak 4:mg *i sx-nelow) can be written $\sqrt{\frac{1}{\lambda}} = a(Z - b)$, where $b \approx 1$, and evaluate $a$. a =    $\ \text{m}^{-\frac{1}{2}}$

(a) Show that the number of different states for a given value.7 +we 9qbpl(o*ywi zp of $l$ is equal to $2(2l + 1)$.
(b) What is this number for $l=0, 1, 2, 3, 4, 5,$ and $6$?

Show that the number of different electron state5t(uf ju q+cv.s possible for a given value of cj(t+uf vqu.5$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV is shot through two narrow slits inz (mgfx ih3x (4c.a8jgs*jm(x a barrier. The slit8sgz(xahm( x4ji(cxm .f *gj3s are a distance $2.0\times10^{-6}\ \text{m}$ apart. If a screen is placed 35.0 cm behind the barrier, calculate the spacing between the "bright" fringes of the interference pattern produced on the screen. $\Delta y =$    $\ \mu\text{m}$

  The angular momentum in the hydrogen atom is given both by the Bohr model m r w4c uoni5;e(+ hc6zy-.maq4qbtl6 and by quan.y64h-atm(4lqo m+ecz6; bicw 5uqrntum mechanics. Compare the results for $L = mvr$.
$L_{\text{Bohr}}$ =    $\hbar$
$L_{\text{QM}}$ =    $\hbar$ or $L_{\text{QM}}$ =    $\hbar$

  An 1100-kg car is traveling with a 1(,itfz pm92uz3k occspeed of $(22 \pm 0.22)\ \text{m/s}$. With what maximum accuracy can its position be determined? $\Delta x =$    $\times 10^{-37}\ \text{m}$

  An atomic spectrum contains a line with a wavelength centered a o2tbe+ ;qv 5erxxh;-(u.lgl bt 488 nm. Careful measurements show the line is really spread out between 487 and 489 nm. Estimate the lifetime of the excited state that produced this (ubbqt-l;+ e o ;52erg.x xlhvline.
$\Delta t = $    $\times 10^{-14}\ \text{s}$

  Protons are accelerated dy;am* b vt1k.9lxtv/o(uv 3zfrom rest across 550 V. They are then directed at two slits 0.70 mm apart. How far apart will the inva xttkbv1 .do*v3y lu ;zm9(/terference peaks be on a screen 28 m away? $\Delta y$ =    nm

An electron and a proton, each initially at rest,9huhmy5u0j ,z are accelerated across the same voltage. Assuming that the uncertumhh 5zj0,y 9uainty in their position is given by their de Broglie wavelength, find the ratio of the uncertainty in their momentum.

  If the principal quantu sa8l7jfq3c/vm number n were limited to the range from 1 to 6, how many elements would we find in natuqa/8sf 7 3ljvcre?   

  If your de Broglie wavelength were 0.50 m, how fast fiy5t7k-c:t:k,cr +l7wp zwf would you be moving if your mass is +kpl,zcyt7w i-5k wff cr:t:775.0 kg? $v = $    $ \times 10^{-35}\ \text{m/s}$
Would you notice diffraction effects as you walk through a doorway?  ----待选择----yesno 
Approximately how long would it take you to walk through the doorway?$\Delta t \approx $    $ \times 10^{34}\ \text{s}$

  Suppose that the spectrum of an unknown element shows a series of lines with one12,hvh1wi s;ac7 ybdz out of every four matching a line from the Lyman series of hydrogen. Assuming that the unknown elem;7w2i,d1vsbazyh1 h cent is an ion with Z protons and one electron, determine Z and the element in question. ----待选择----lithium berylliumboron 

  Photons of wavelength 0.154 nm are emitlb*bj a2e r.1m; 3zuaated from the surface of a certain metal when it is bombarded with high energy radiation.a3 e z1lb*;ua mbr.2ja If this photon wavelength corresponds to the $$K_\alpha$$ line, what is the element? ----待选择----nickelcopperzinc 

Show that the diffractive spread of a l, v(nl . 69u(wqeqctvvaser beam, $$\theta \approx \lambda/D$$ , is precisely what you might expect from the uncertainty principle. [Hint: Since the beam's width is constrained by the dimension of the aperture D, the component of the light's momentum perpendicular to the laser axis is uncertain.]