Compare a matter wave h rtbx9 ; 8ni+/wm- bgbxhyk)6$\Psi$ to (a) a wave on a string, (b) an EM wave. Discuss similarities and differences.

Explain why Bohr's thg4; hl*:xk.:m qcfp.kf c2u pgeory of the atom is not compatible with quantum mechanics, particularly the uncek4p*pf ul:hfkc q.mc;2gg: x.rtainty principle.

Explain why it is that the more massive an object is, the easier it become zcz(s58.j-zgblt vx f.) ck0ms to predict its future positio x(b ) czl.vjg.kmzft8-0czs5n.

In view of the uncertainty principle, why does a baseball seem to have a well-*lkvytcp d 9-oh 46rdme-k;;qdefiqlkyc -t6d;d rme*pv-ho4k 9 ;ned position and speed whereas an electron does not?

  Would it ever be possible to balance a very sharp needle precisely on its poinjp y.5k6 ubx4xp) pvhtor*1g4yn7s zpx;)n /zt? Expla;xn r/kopyv6j zxpu15 ts b7) 4*)pgn4 xz.phyin.

  A cold thermometer is placed in a hot bowl of soup. Will the zy92jar(.kjtf. tjcj 0ebh a)l*c*v( temperature reading of the thermometer be the same as the temperature of the hot soup before the measurement was made? Eahj0.bct) tfejz*k( rjacly(j.29* vxplain.

  Does the uncertainty principle set a limit to how well you can maked:pv* r z1 pn1xgwqwol(z4( 2l any single measurement wxdp(ro (lz1z*2 1n:4lgvp wqof position?

If you knew the position of an object precisely, with no uncertaintb 5 ht-iozylk5c( o)j2y, how well would you know its momen-(c)o5botkil25j y zhtum?

  When you check the pressure in a tire, doesn't some air inevitably escape? 7ybws6l0qz t ,Is it possible to avoid this escape of air altogether?sqz 6y,lt07 wb What is the relation to the uncertainty principle?

  It has been said thatwoml45eyyio,lh1-7e -z p 3eb the ground-state energy in the hydrogen atom can be precisely known but the excited states have some uncertainty in their values (an "energy width"). Is this consistent with the moew-h15 epe zlil,oyb y74-3 uncertainty principle in its energy form? Explain.

Which model of the hydrogen atom, the Bohr model or the quantum-mechanicq/:ssts /vekpa l gxl5d/027kal model, predicts that the electron spends more t vs0:aks/x/ kqe2pd g/ tsll75ime near the nucleus?

The size of atoms varies by only a factor of three or so, from largestuxlo0,2lyddz7ke *g.zvd x f y0/ 3-4n)tc gfu to smallest, yet the number of electrod 7)g/ y0-n0gzkx42*vuxfl,z yo3lt dud.ecfns varies from one to over 100. Explain.

Excited hydrogen and excited helium atoms bot3tjysifw9 * e2rth5q:h radiate light as they jump down to the r y95 sttf*wqj:e 3i2h$n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table look if t+z 9 ngeqs*7pthere were no electron spin but otherwise quantum mechanics were valid? Consider the first e*+s7n qp9tgz20 elements or so.

  Which of the following electron configurationsv9: x8y)usddf 8de/zm 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 uran2 y12d. w+lof1l*ud3qesighgium atom (careful scrutiny across the periodic table on theoq 2u dllsi1+y*hgf1w.ed 23g inside back cover will provide useful hints).

In what column of the periodic table would you expect to find the atomsas7n) tdw 1a- with each of the following confins t17a-)wdas gurations:
(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 exfz 0b ;sb-a9yue52dxn hibit similar properties?

Explain why potassium and sodium exhibit similar propej;g6bhov lg0l)v pa5 1rties.

The ionization energy for n2v3 470ab;yn,tebxh1aik r axeon $(Z=10)$ is 21.6 eV, and that for sodium $(Z=11)$ is 5.1 eV. Explain the large difference.

Why does the cutoff wavelengta g9zv64 ofd-dh in (Fig. Below) imply a photon nature for light? 9-d6 ozgfda4 v

Why do we not expect perfect agreement between measured values of X-ray lz- k gt3jtj k2 25h)ekbs4uux6ine waveleng25kjh u)-gt xk2tsjeu3zbk 46ths and those calculated using Bohr theory?

How would you figure out which lines in an X-ray spectrum cpg(zt*2( gt ngorrespond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transitions deep within an atom to produce sho l-u5vvf jg knki49.xcr; -km9rter wavv;kcx-r n f5k-giuv.jk4l m 99elengths than transitions by outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from ordinary ko43 ,2 ay9 wszb8+yz9*kxfb: rvujrtlight? How is it the same?

Explain how a 0.0005-W laser beztb ;8 kkw)ki6am, photographed at a distance, can seem much stronger than a 100zw ;ki)6kb tk80-W street lamp at the same distance.

  Does the intensity of li2ltbnn+l gping5-l .4 ght from a laser fall off as the inverse square of the distannlgti5 2pn+blng4- .lce? Explain. ----待选择----yesno 

  The neutrons in a parallel beam, each having 0hi 0 fny 0skkllqw586kinetic energy $0.40 \, \text{eV}$ 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 spx8( q.cv g7 fex,u+wmnd,9z,c1gfzh leed 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 3o/c o2gqkf sergd 0a;7;fqy5 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 an excited state of an atom for typicallcrbtd j6/(9iffxu)0e y $10^{-8} \, \text{s}$. 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 ty v1buw2r(6 oaa19*wgb iacj8he mass of a 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 energy of a neutron contained in a typu28c1 ql;clh/;sqi djn v c (fm.4ugl*ical nucleus of radiu ls ;d(v j. u n*;12ulcq84cg/qlmicfhs $1.0 \times 10^{-15} \, \text{m}$. [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/ hzexc p-a7.b if an electron were present in the nuc zehxp-b/a.7cleus $(r \approx 10^{-15} \, \text{m})$, 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 t wfpis5r -yb3h vq5m.+he position 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 the xb q(cwc0axj5 5;t9wq $n=6, l=3$ subshell?    elections

How many different states are possible for an electron whose principal quk9u5 a/-dxf7epw kso-antum number ispk5 -uoe-79/kdxw afs $n=4$? Write down the quantum numbers for each state.

List the quantum numbersaduvk k a h3ie82+6(ms for each electron in the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numtw 6i6 n oc3*mi7h(( oowf:hzebers for each electron in the ground state of nitrogen $(Z=7)$.

  Suppose a certain hydrogen atom cf/d . ;y2sdtqh-zf3qj tt+9r 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 angular momentum of an electrov)ru x5/njm07 )s,it iwxar5g n 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 h.y zuxi)c,y;9 f+kwvias $m_l=-3$, 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 18 electrozxx.8xmaj .ep170gkqk*h lg +ns in a "g" subshell.    electrons

What is the full electron configuration i8 -se wbb5/eden the ground state for elements with $Z$ equal to (a) 27, (b) 36, (c) 38? [Hint: see the periodic table inside the back cover.]

What is the full elecgyviq.2l5 m :k qokl.c3l1dw:tron configuration for (a) selenium (Se), (b) gold (Au), (c) radium (Ra)? [Hint: seev:l3k2g l:q qidyomc. .l 1wk5 the periodic table inside the back cover.]

  A hydrogen atom is in the ;/h9cveey*km /ax 3rj $6s$ 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 lit afaxyf /9kjf*m )ff-g4y l;xp6 bc-8xhium using the Bohr theoryyp -b xa)k8yx9fm fl-6c afgff4x/*;j. [Hint: 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 filled subshel7(dohl2n nm1e l.

  For each of the following atomic transitions, state whether the transition is an) w*6u *-z)afiliwwell* 6ewnfi)wau- )i w*zlowed or forbidden, 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 i*le p m7b+8qlnn 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-wavelength mf,83tlxp re)X-rays emitted by electrons striking the face of38pefl, xtmr) a $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremsstrahlung X-rays em, j l*k+lui6yeitted from an X-ray tube have l y6k*eil+uj ,$\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wx,8g:h a43-mcqa nttbn6u;o-+vj*i ekp i /fiavelength $\lambda_0$ 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 sa7z ;o:u a2t(xf+qy67d ktihavelength emitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength for av mb6u;cl3l n ;2hkhbmo2j68ln $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate t mm e8+soffz8vk8qr+ 7he 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 7s:ll b77 9ugk 1cziegis bombarded with electrons. The wavelength of th e:zicsu7gkl 71gb7 l9e $K_\alpha$ 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 weldnd)wp4):n waq, c:i sj detached retinas 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 havexh5a41;b3wmplnrm 53o 7/mg z ldo1b2zrna q3 a 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 a laser beam due to diffraction if the 0iix.r(3pms2)i o1iu 1 rgezjbeam emerges tp1 mgu ze.i irj 2(r)is0ox3i1hrough 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 theah v8dw yn*)o hrwy) /12;fbra He-Ne laser?$\lambda $ =    nm

Use the uncertainty principle to estimate the positiolevw, xkm4 93ln 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 tlv4m w9e 3,lkxhis 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 possksc w2 x3no .jedfi/k0:jdn:3ible values for the angular 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 orbit, estimat8k3 ,h6; uzkb tg.tceke the average distance from the nucleus for an electron ibc.u k;h zgt8ek6,t 3kn 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 aq187e . 3ebntbo3p n. qfw130thukpqkt $95 \, \text{kV}$ 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) ene+7qr :lm20wg : jgwtd0w( b5uaech9sprgy of the 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) r,1ef ny f+8v ep;cre6can 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 of,z0p+i.yfaif $l$ 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 fo woupk 04z7ihkl p+6p5unfxds14hkm.a2g8f/r a given value of kh8h+ 1g 02pxff7ou kzi5/kupwm s4 n6pdla4. $n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV is shot through two narrow slits in a ba:/- 91xxggri vcvbc0errier. The slits are axr:b vx0/ig c9 ev-cg1 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 hyyyy 8bjbf):curzabxa7 2 ,9 8ldrogen atom is given both by the Bohr model and by quantum mechanb87xr :aauybyc89)b2 lfjz,yics. Compare the results for $L = mvr$.
$L_{\text{Bohr}}$ =    $\hbar$
$L_{\text{QM}}$ =    $\hbar$ or $L_{\text{QM}}$ =    $\hbar$

  An 1100-kg car is travelinpk ld- )h ln;:3bjm*yzg with a speed 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 ci2mt5;b+h nfcv)xnmc(fy1w ; ontains a line with a wavelength centered at 488 nm. Careful measurements show the line is really spread out between 487 y+m tf; n(h2nvc mwf )cx5i1b;and 489 nm. Estimate the lifetime of the excited state that produced this line.
$\Delta t = $    $\times 10^{-14}\ \text{s}$

  Protons are accelerated fr ;pv,6+a0vd.vbpyk dwrq7z8 mom rest across 550 V. They are then directed at two slits 0.70 mm apart. How f8zw 0;,rypkd6vvd. am 7vb +qpar apart will the interference peaks be on a screen 28 m away? $\Delta y$ =    nm

An electron and a proton, each initially at rest, ar6y- 6dtal 2vyle accelerated across the same voltage. Assuming that the uncertainty in their position is given by their de Broglie wavelength, find the ratio of the und6ya2l y-6v tlcertainty in their momentum.

  If the principal quantum number n were limited to the range from 1 to 6by :ur- 6uma ht*7a9cx, how many elements would w9x: mtcy ura7hb-6 u*ae find in nature?   

  If your de Broglie wavelength were 0.5kq u4zm, gs* sq(y o6rk)3t)gw0 m, how fast would you be moving if your ma )qmqs *w,oygukz)4rtg6s3k (ss is 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 of an unknown elemes6bw4r3;0lwh)ha 7eips jg5 jnt shows a series of lines with one out of every four matching bj 03wj;w 5ia 7hgss4plr)6h ea line 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 are emitted from the surfaze ax:w4d g7(9nlb f j57qwng1ce of a certain metal when it is bombarded with high energ( wgd41n7 lq5 jxafbw97ne zg:y radiation. If this photon wavelength corresponds to the $$K_\alpha$$ line, what is the element? ----待选择----nickelcopperzinc 

Show that the diffractive sprecp;hz(xx4vb:du( ok , ad of a laser 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.]