Compare a matter wav9k dc )(+;kprccf5jly ),apb ue $\Psi$ to (a) a wave on a string, (b) an EM wave. Discuss similarities and differences.

Explain why Bohr's theory of th woz v8 ksg1/q/5wv:rnur8 ;fme atom is not compatible with quantum mechanics, particularly the uncertainty principle:r/ mwrv/ g85 zk1sfq vw;u8on.

Explain why it is that the more massive an object is, the easier it besyy+nlq4kj/xo d5nx/ ) :e gzru *m;3pcomes to predigk:mpq/e;jrn os x3n45l /d+*)xzu y yct its future position.

In view of the uncertainty principle, s1 s4r9 luf cc1eew s.ie4v31rwhy does a baseball seem to have a well-defined position a.e c3w11c s 9r4s4fer lv1siuend speed whereas an electron does not?

  Would it ever be possible to balance a very shakq 5i;agr2py :rp needle precisely on its point? Explain.y pa;g25:kr iq

  A cold thermometer is placed in a hot boa l:uohq;ilz j -i4+u8(ofpp. wl of soup. Will the temperature reading of the thermometer be the same as the temperature of the hot soup before the mehu lli+p ;8:aq pfu(ooi jz.-4asurement was made? Explain.

  Does the uncertainty principle set a limit to how well you .bzlq-al hi -8-a+ gnqcan make any single measurement of posnh+ l8.zga --b-aiqqlition?

If you knew the position of an object precisely, with no uncert- mv /a2xqcct/.7vzv xainty, how well would you knc.v/ ac7 xvqz2-/mv txow its momentum?

  When you check the pressure in a tire, doesn't some air inevitably esx)bnyk da.p4 +cape? Is it possible to avoid this escape of air altogether? What isx ya.pb)4 kdn+ the relation to the uncertainty principle?

  It has been said that th9- 2nxnq5brpn d pxn;07jwy7r e ground-state energy in the hydrogen atom can be precisely known but the exc 7xw0dn7b 5;9rpn-py qjrnxn2ited states have some uncertainty in their values (an "energy width"). Is this consistent with the uncertainty principle in its energy form? Explain.

Which model of the hydrogen atom, the Bohr modq qn.0z:vi4t mhu5rrj :(cdz(el or the quantum-mechanical model, predicts that the electron spen .m::qqr(ur (j40izvd 5hnct zds more time near the nucleus?

The size of atoms var; gwoy zn1o;3uwphf yg 15l4 gi+o8o-uies by only a factor of three or so, from largest to smallest, yet the number of electrons vaggou5 nooif1w; ol; zh4 +y-y8gpwu13ries from one to over 100. Explain.

Excited hydrogen and excited helium atoms both radiate light as they jump down t,k ys:(mfyr vcwfes,az7(/fk0aibk;pz3) 3oo th es3fk y(bmas ;03okckpz/w , f)7v:iyz r(,fae $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table look if there were no electron spin b3zqrud2 n b1+put otherwise q3bz+ prun2 dq1uantum mechanics were valid? Consider the first 20 elements or so.

  Which of the following eley(v n-u5cx )n6c6y3qtmf 4o fyctron configurations 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 configuration for a uranium atom (careful scrutinymun 58r-b 2bjv across the periodic table on the in5j-bub8rn2m v side back cover will provide useful hints).

In what column of the periodic table would you expect to find the atom w,pt2eg75ckv gmrk 1tq a+-lke55h0 g oith each of the following con72otkt1mr qk egvhkp5,l-+ 5a05ecg gfigurations:
(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 iodine exhibit similar properties?5j ,eddl1y mlqg1nxv3/22 ash

Explain why potassium and sodium exhibit simi53 pd8iavnh6 qgc*cw0 lar properties.

The ionization energy for neon n ayi g2s,p.e0$(Z=10)$ 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) imad ./5ys: aibuu1hk7 iply a photon nature for light? u:biuh/sa.k y7 1da5 i

Why do we not expect perfect agreement between measureh b. vwsxq.6ai9 b8.zfd values of X-ray line wavelengths and tbbvs8w.ah6 q9fzx.i. hose calculated using Bohr theory?

How would you figure out which lines 6cgr7le q.tb+58d9xgyn) wox in an X-ray spectrum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect elec+hf6p pg-qued aa+ f4(tron transitions deep within an atom to produce shorter wavelengths thefquh-d+ap6a + fgp(4 an transitions by outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light diffzu4.qsu(ms1 qogqg4va )p7 3q er from ordinary light? How is it the same?

Explain how a 0.0005-W laser beam, photographed at a distance, can seem much st.7erm:auj mw 4ronger than a 1000-Wa7je w.ru:4m m street lamp at the same distance.

  Does the intensity of light from a laser fall ofx vc k,ocd 96tmr:k;ui.p;ur;f as the inverse square of the distance?r6:kor p;9.ud;icmvc; x kt,u Explain. ----待选择----yesno 

  The neutrons in a parallel beam, each having kinetic 9vjkbkf jt 17k:phoq(9r *n(yenergy $0.40 \, \text{eV}$ 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 with a speed kgo0f ob0y6g( 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 c nzhp)5ypj,t ;0( i0txk 9icqjan be measured to an accuracy 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 iiw 9 c*s2s8xgze(s1x,pek;pan 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 muo afipev;0h)m0 n $(m=105.7 \, \text{MeV/c}^2)$, 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 energy of a neutron containedgu9an8uw :s(bj0at pe sy z:2/ in a typical nucleus of ra:b/uz:ap j(0 8gts9y2sewaun dius $1.0 \times 10^{-15} \, \text{m}$. [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 electron werx4gbuk+kbm- 4e present in the nucleuskbg u+-x4 mkb4 $(r \approx 10^{-15} \, \text{m})$, 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 positio -olf;+jgt b87ru8 aspn 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 thebe;h 5wcbx.x4fx *)xtl -r dl1;dqn :w $n=6, l=3$ subshell?    elections

How many different states are possible for an electron whose principal quantupzxd wy(( -+m3ga f5n6wyx,)+qjn j tum number is-dtw )++ n3( x5mwfyznq(ug,yxjjp6 a $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for , r s,da)cs4z sr256xcg:g otleach electron in the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each electr -iif )6c08x 1lwtbm8sxljk+ ton in the ground state of nitrogen $(Z=7)$.

  Suppose a certain hydrogen atoos jkj8 1mxfgs r 8 (.wohe)l;iq*++wnm 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 ths wu- 33,zlchhe angular 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 v peqi1j4png 0s;*weq)8ju-h j p1lt($m_l=-3$, 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 f;ho+- plm5zi18 electrons in a "g" subshell.    electrons

What is the full electron configuration in the ground state for elements withaf -mrjs2g-//e- *qjzgjb lse* 3z 4ki $Z$ equal to (a) 27, (b) 36, (c) 38? [Hint: see the periodic table inside the back cover.]

What is the full electron configuration for (a) selenium (Se,ce ( -sj4scjs), (b) gold (Au), (c) radium (Ra)? [Hint: see the perios esc(scj4- j,dic table inside the back cover.]

  A hydrogen atom is ih79(58b bwe:a i3 w,i v8jftwjklmde3 n the $6s$ 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 in lith nz16m* *x jwa;cgbv7rium using the Bohr theory. [Hinx; c aw j76r*bvg*1znmt: This electron has $n=2$ 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 for a :dl;hz lpgn55oi h)ia26;thy filled subshell.

  For each of the following atomic tra s3l:j9yj(lmk nsitions, state whether the transition is allowed or forbidden, an3s j9:k j(llmyd 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,xc7f m-12gn nf aml3y a $6d$ 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-wavelenkb3nh3( xfrn 2gth X-rays emitted by electrons striking the face onn (h3f2xbkr 3f a $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremsstrahlung X-rays emitted from an 3f5 rtnywmt/. X-ray tube havett/5y.3wmn r f $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wavelength-y)ut/rbbz+ 8-chamz $\lambda_0$ 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 wavelength emwgm r77nkghk(8/lg i9itted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength for aoydp :nydmgk(/e v0 jv*-d, .on $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the wavelengt n ls7tbp-vqpp241 v3ph 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 material38o:i dq kn2,i .tdvyizsq+v5 is bombarded with electrons. The waveleng+:5ynivqoivdd8.k 2i3,t q zsth of the $K_\alpha$ 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 retiz 4gg z1(fttt hpd0o 1zhx)*4fnas puts 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 uovrpp +y l-(40 . foy rpn5pyh3qh2g9a power of $0.50 \, \text{mW}$ 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/cu2 /wsdcuq ) a laser beam due to diffraction if the beam emerges/2) ucu wcq/sd through a $3.0$-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 wavelength of the He-Ne lac : i5tpp:otw8y3n1 kpko:rhg 0g 5j8rser?$\lambda $ =    nm

Use the uncertainty principle to e ;vr2qs9qqsb l(fs mq;4a8+qhd cp b;4stimate the position uncertainty for the 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 anywhere between this value and zero.] How does this result compare to r(fp9+sa2h;l4cdqqm; ; b8sv qbs4qqthe 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 a yzjb:j7;0 uo7ol+dsggl 8r 8jngular 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 gt 6ur.1f sgbxpbg)(6radius of an electron orbit, estimate the average distance from the nucleus.fg6sbxp)16 g urbg(t for 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 operate:e)suhjuf lm do xpmbr 300p:;jx n.(6s at $95 \, \text{kV}$ 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 thmd+1m)jw9j vci:9q hne outermost 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. Below) can b8p1dl gqvg8d5 ( 7tq)r*zlq xa-h70yb zf- hqte 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 vc ; t3izmm.hr9vwd), value 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 states possible for a given x9sv,hu+-lz 4wrd nkiyw(( w3value of zk,nw9(xr uwwhys-d + 34(v li$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons witq+z l4yi r4z pt*p7r9o5o4t/padi o3th energy 45 kV is shot through two narrow slits in a barrier. The slits are a dpii 4 do4t/q 9+op 75*lrytao3zpt r4zistance $2.0\times10^{-6}\ \text{m}$ 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 mor6eno9ful ock( 6 8+indel and by quantum mechanic9oio6nelnr 8f+u(6 cks. 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 speed oqww,4 gacrcj66 pf-*kf $(22 \pm 0.22)\ \text{m/s}$. With what maximum accuracy can its position be determined? $\Delta x =$    $\times 10^{-37}\ \text{m}$

  An atomic spectrum contains a line with a wa1tjpf 58a qzx/velength centered at 488 nm. Careful measurements show the line is really spread out between 487 and 489 nm. Estimate the lif 1a5/j zx8qftpetime of the excited state that produced this line.
$\Delta t = $    $\times 10^{-14}\ \text{s}$

  Protons are accelerated from rest across 550 V. Tqb/ :+bis3( lilkjo.ihey are then directed at two slits 0.70 mm apart. How far apart will theiblojs:b ki.+q li( /3 interference peaks be on a screen 28 m away? $\Delta y$ =    nm

An electron and a proton, each initially at rest, i1t c gl,67rn)kflbx c3xayi9e c 10p+are accelerated across the same voltage. Assuming that the uncertainty in their position is given by their de Broglie wavelength, find the ratio of the uncertainty ikygcc3 b6r9fx)t i1l ap7+0ec inx 1,ln their momentum.

  If the principal quantum number n we;ardkyi0q - m6re limited to the range from 1 to 6, how many elements would we findi-0k;m 6ydq ra in nature?   

  If your de Broglie wavelength were 0.50 m, how fast would8a:s;9 j0na . zn0auw avfd3ey1 lg7iz you be moving if your mass is 7z8w3a0nlzj na01d ase :yi9v ug ;af.75.0 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 o)ocj sf5vky4) kz:x;x f an unknown element shows a series of lines with one out of every four matching a f: s)o;)z yxvc4xjk k5line from the Lyman series of hydrogen. Assuming that the unknown element 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 arot(/n+ekv9d q pt(b3l e emitted from the surface of a certain metal when it is bombarded with high energy radiation. If this photon wavelength corresponds to th dtn9e/3+bq( opvt(l ke $$K_\alpha$$ line, what is the element? ----待选择----nickelcopperzinc 

Show that the diffractive spread of a laser beam,v- mfk sbgklrzw+ b4pe*1 j73 ptpy2:g6w(p5g $$\theta \approx \lambda/D$$ , 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.]