Compare a matter wave 2bvyhxa /lff; xc 19bp+a)cd2$\Psi$ 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 mechanicspj6*q d gl 3w/ u)ln :qt*pb*tmx-d,vp, particular /,tug*bn)x **lvw-q 3pljpmt qd6 p:dly the uncertainty principle.

Explain why it is that the more massive an objesyfp r:-1q0koct is, the easier it becomes to predict its future pos0-qofp:s rky 1ition.

In view of the uncertainty principle, why does a baseball seem to have a wermz+ ( z/ 0m/ub)cngihll-defined position and speed whereas anm /mhr(nb)gzu0zi +c/ electron does not?

  Would it ever be possible to balance a very sharp needle vbd uskn1n9r1 n2o*n qa,x77w precisely on its point? Expvn*k1dqanb2s7xnuw79 ,ro n1lain.

  A cold thermometer is placed in a hot bowl of soup. Will the temped ms qnzn w/nnrvfm9+ 8t7+7t.rature reading of the therm9dzsn.r87/t+ f7mqv n+nn twmometer be the same as the temperature of the hot soup before the measurement was made? Explain.

  Does the uncertainty principle s.n:0na 6 ockktet a limit to how well you can make any single measurement of posk k6tcn.noa0: ition?

If you knew the positiclvc1b-npyqm-8 09 tion of an object precisely, with no uncertainty, how well would qv1 0ibt9 ny8 --clmcpyou know its momentum?

  When you check the pressure in a tire, doesn u126flweaf1b't some air inevitably escape? Is it possible to avoid this escape of air altogether? What is the relation to f we61baf2 1ulthe uncertainty principle?

  It has been said that the groun,b w xzf8cv3)i xvts3 j;-zvx(d-state energy in the hydrogen atom can be precisely known but the excited states have some uncertainty in their values (an "energy wid,- vb 33vvx fzti8(j; x)wxzscth"). Is this consistent with the uncertainty principle in its energy form? Explain.

Which model of the hydrogen atom, the Bohr model or the quantum-mechanical modegb f2m(hko7h 3l, predicts that the electron spends more time near the nucleus?hmfgo7b (k23h

The size of atoms varies by only a factor of three or so, from largest to smaih/r0 y64l1hhhu6afg//p, n (rlyjzmllest, yet the number of electrons varies from one to over ,r(uhmy/y z/lnp6h 04gjh 1fri/ 6 lha100. Explain.

Excited hydrogen and6 yumj. hwy/z4 excited helium atoms both radiate light as they jump down tmhzyj6u wy./ 4o the $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table look if there were no electron spin but otherw,kb,2 gqx 2l en-vlfr(ise quantum mechanics were valid? Consider the first 20 elements or rlvbqkg2l, fexn ,2 (-so.

  Which of the followingl v,,0:il45g3af zhwc cn ;mji electron 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 i wr/x+ty.y 5tfor a uranium atom (careful scrutiny across the periodic table on the inside back cy y 5.xr+t/tiwover will provide useful hints).

In what column of the pem6vc5jwf du1yhr g)19riodic table would you expect to find the atom with each of t)j1 cy1hwm69vug5drf he following configurations:
(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 eit62ig/h 5ey yxhibit similar properties?

Explain why potassium and sodium exhibit simi3jlhtsy )t. 4zkyj1 )olar properties.

The ionization energy(:q (kf ;pc mv2at0y;ulh6hdm9k *j zo for neon $(Z=10)$ is 21.6 eV, and that for sodium $(Z=11)$ is 5.1 eV. Explain the large difference.

Why does the cutoff wavelength in (F mogpy7y b-n)+ig. Below) imply a photon nature for light? yn +7bo-ympg)

Why do we not expect pp uxfovs90)v5 0.ynbf dm5gi8erfect agreement between measured values of X-ray line wavelengths and those calculated usingf05.f s dionpgvv9 0m)buyx85 Bohr theory?

How would you figure out which lines in a31f11xr ss2h-nhvq2cp;v m qb n X-ray spectrum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transitions deep withi61oil3 ga 4b 1d6*9nuqwzkgrkn an atom to produce shorter wavelengths than transitions by outer electrl*nkgi azb9k rq413 wu g6od16ons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from ordjknqk5+anh2rn 80(aka x +/dh(vq 7fninary light? How is it the same?

Explain how a 0.0005-W laser beam, tl(u.h0+ thpfwjh)/0ix l ;kb l.aye0photographed at a distance, can seem much stronger than a 1000-W street lamp at the same dista0lbayti/jt+huhh (e.pw ;.x lk lf)0 0nce.

  Does the intensity of ligh( ks3:uqg u2( 3syxivbt from a laser fall off as the inverse square of the distance? vyg:3q(k2sb uu xsi(3 Explain. ----待选择----yesno 

  The neutrons in a parallel beam, eaf m95( dxzq;/++ wz)c7tm zbz xffvsd5n x7(mhch having kinetic 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 speed og +rjf5n+t+se1 umbbz em*7)th( zc t:f $(6.560 \pm 0.012) \times 10^3 \, \text{m/s}$. 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- qy-xxz4nv9/p qic e. can 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 an excited state :oawch (j46o fof 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 mahvd/a0+ i -uwsss 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 l odru2 nf ft8l ewg(61h*wd5xv12o8rin a typical nucleus of xr1d wg 6n*fh2uew8v82(fd llt 1or5 oradius $1.0 \times 10^{-15} \, \text{m}$. [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 were present /;pek;u.j n za4ob/aiin the nu .ja a;zpu;nio4/ bk/ecleus $(r \approx 10^{-15} \, \text{m})$, 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 posid8y033e lbxwlrfw9 0gtion 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 cs/t o6x7z ,e; +hbnlxsan be in the $n=6, l=3$ subshell?    elections

How many different states are possible for an electron whose principal quajocc 88y( ca*jntum numbecjjc8o(* 8yacr is $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each eleygauz*l /4k7;fqzf+ 5 .t qxb7ou w:zwctron in the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each electron in 3 efijicxxgk2a2qmg3t 7 *4f2the ground state of nitrogen $(Z=7)$.

  Suppose a certain hydrogen atom has / 2 xx zvve2c9*o2sbufeu3ar( $l=4$. 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 elecv o g(ruwgtwt:;k*:g.:tjs 3r tron 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 hhv.d-*7*yh9+d*cf sm pjc tvfas $m_l=-3$, 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 1p.cky,i/ot quj j-+ p38 electrons in a "g" subshell.    electrons

What is the full electron configurata ar*+c (e0onj2 ee3+zmdht0bion in 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 electron configuration for (a) selenium (Se), (b) goldgid+ g8 la *ja.yaey(; (Au), (c) radium (Ra)? [Hint: see the periodic table inside the back covereg(aldaj g iya.*8;y+ .]

  A hydrogen atom is in tx*t4s; gzt2r dhe $6s$ 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 enee-f( oqbbfs6pn .2 pz/rgy of the third electron in lithium using the Bohr theory. [Hint: This electron haf(s e/ b62pfbpn oq.-zs $n=2$ 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 sunmk:rncf6+ jd5) 8a 1(ynvsm zbshell.

  For each of the following atomic transitions, state whether,au8brj1ic e/hll c ,4 the transition is allowed or forbidden, and if forbidden, what rule u/ cir4b h, ja81,leclis 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 ;*g06ohlvc8 s5w0ua xyfyac .$6d$ 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 swhw y z uo1dqotna. 4*e+1/k7atriking the face ofyot wo1k+nwzu a1qh4a7* . d/e a $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength kk38 n)swi k+vbremsstrahlung X-rays emitted from an X-ray tube have i 38+sk)knkvw$\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wavelenguk0 ( t6 /d3nbw0 nhxqi-0gmisth $\lambda_0$ 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 wavel 69ggcmm(ss2t t-vs2o: i dw:oength emitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength 4;v by 9baq;elqm3lxic *-c *xfor an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the wavelen+r n* 0ousdfl n k/g(a75k 6r;lh1wlkngth 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 unknow 6mr7b0bxu8h zn material is bombarded with electrons. The wavelength 76buz8 h 0rxbmof 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 reti xpq6m7)qtd ubldj9qp w-4 +8zgim-6 xnas 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 a power iy8/1+swc gk,8ei *nby fgk+cof $0.50 \, \text{mW}$ 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 tu-7l t 2fk/7hoihye4 +et9ji wo diffraction if the beam emerges through a 4tht 7e+f7l2oujy9- e wi/hki $3.0$-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 wavelengt)3,:2f:rqkelbf mq mqh of the He-Ne laser?$\lambda $ =    nm

Use the uncertainty principle to estima9ihyblfm;y)scq- + 3 mte the position uncertainty for the electron in the ground state of the hydrogen atom. [Hint: Determine the momentumfmql)i; cb shy y39+-m 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 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 the5n6ldaw6z5 - rky-w; 1bsru th 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 formulam iz ,c5p.sif lg,ne/zp fmu*8mr 7s;5 for the radius of an electron orbit, estimate the average dista/cr*; pf,mpi. sl5 e8m,if mnu7sz5gznce from the nucleus 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 operatbok -:g* 0b81 bcd-gv -xvgwlxes 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 the9qqmx( g5r ci, 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. mm )r7p hxgq uqee7l,(-jng51Below) 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 diffddwft98xx +wjc c t;(8erent states for a given 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 /rrrp 6bbi 1 ny hhy2:bu5ni3ftq:/8pdifferent electron states possible for a given value of /bb8ny hy:rqnb:r6/r21 p t3 iuhf 5ip$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy;k p) fnyeavl7en/k ks6h j,++ 45 kV is shot through two narrow slits in a barrier. The kfn7e/h,ke +y a 6sk+nl;jvp)slits 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 hydro .tftpfw0kb(j7 jp/ s2gen atom is given both by the Bohr model and by quantum/0tp k.tjj(w7fspb2 f 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 snjj8.n v 61b itxwtxfw7zw(:,peed 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 wavelklhiyd27cr 6k x,u +. ;rca0goength centered at 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 thir yu0g ; i+2r6xalkc.7 odch,ks line.
$\Delta t = $    $\times 10^{-14}\ \text{s}$

  Protons are accelerated from rest gfz/g p+48(mtzi,t s bacross 550 V. They are then directed at two slitssi/mtgp zg,z 4f b8+(t 0.70 mm apart. How far apart will the interference peaks be on a screen 28 m away? $\Delta y$ =    nm

An electron and a proton, each initialeft;y. b1 dwh6) (v-cx q+motz *iyez3ly at rest, are accelerated across the same voltage. Assuming that qmct ivb d*xtw y6 ezh+)eoz3f1 ;.y(-the uncertainty in their position is given by their de Broglie wavelength, find the ratio of the uncertainty in their momentum.

  If the principal quantum nm s ihbp;+rhv2 kptl++q*.v-z umber n were limited to the range from 1 to 6, how many elements would vpvkq2sb.pi t+h h* m+l;+zr- we find in nature?   

  If your de Broglie wavelength were 0.50 m, how d2d2l k7 dr di4l7vwh 7c(ze0hfast would you be moving if your masszdvhl7id k l02r(h7w4d d7c e2 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 unkye d9d4)fg,bd) mea. kr 6u qzm4rc//ynown element shows a series of lines with one out of every four matching a line from the Lyman series of hydrogen. Assuming that the unknown element is an ion with Z protons and one electron, determine Z and // dk9)y.dqcz)mbf umee46gyar, dr4the element in question. ----待选择----lithium berylliumboron 

  Photons of wavelength 0.154 nm are emitted from r6s9ba npcq9 :the surface of a certain metal when it is rqpc6ab9n 9:s bombarded with high energy radiation. If this photon wavelength corresponds to the $$K_\alpha$$ line, what is the element? ----待选择----nickelcopperzinc 

Show that the diffractive spread of ajeg2d bw7*yr 7 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.]