Compare a matter wave*gjv y 9 +lp*m;klwcq s13ja1f $\Psi$ 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 crmr mk(/1k4jm z;6tehompatible with quantum mechanics, particu1hk6 e4(mk j/r;m rztmlarly the uncertainty principle.

Explain why it is that the more massive an object is, the egzayvx .:,ij0 is+ht7asier it becomes to preits+.a0zi7x j: h,yvgdict its future position.

In view of the uncertainty principle, why does a baseball seem to have a wc6z8**h0cc2qg aeo k pell-defined position and speed whereas an elec*28qhp ezok*c6ac0cgtron does not?

  Would it ever be possible to balance a very sharp need wi ev5,xj bvfrt;b,17le precisely on its point? Explain.,,w b7vb5 jtre;xvi1f

  A cold thermometer is pq-kj1 t r+bs0vc-:.nnym.m+.jdh xx 5 )foxkplaced in a hot bowl of soup. Will the temperature reading of the thermometer be the same as the temperature of the hot soup before the measurx .n.hk-+5xv km1jq)mcybndr o +j.0 -:xtsfpement was made? Explain.

  Does the uncertainty princip5tu zc /5/j(d ygiie7 0lcrqn8le set a limit to how well you can make any single me 5c5clz d78ugyt /i/eiq0rnj(asurement of position?

If you knew the position of an object precisely, with no uncertainty, hps. t-/n kbz,mxar1o*ow well would you knoo* rk/s1- pntza,xb. mw its momentum?

  When you check the pressure in a tire, doesn't sj ayvkr +0; ow+i6e,qyome air inevitably escape? Is it possible to avoid this escape of air altogether? What is the relatkay+o je;ryqi6w+, 0 vion to the uncertainty principle?

  It has been said that the ground-state energy in thehp 9ppq:v usb01xagy6ub) 3-w hydrogen atom can be precisely known but the excited states have some uncertainty in their values (an "enpb) yp qgb1-hwu v9x a03:s6upergy width"). Is this consistent with the uncertainty principle in its energy form? Explain.

Which model of the hydrogen atom, the Bonpf- ulh1ey.0 hr model or the quantum-mechanical model, predicts that the electron spends more time ne l 1fy.neph-0uar the nucleus?

The size of atoms varies by only a factor of three or so, from largest to s1 wl*s9arfho1po o*u *mallest, yet the number of electrons varies from one to over ou11*9s* wr ho foa*pl100. Explain.

Excited hydrogen and excited helium atoms both radiatgi* k8wa:mp3oe light as they jump down to th w:k8ig3*po ame $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table look if there were no electron spin but othl7f,mog0)4wj kwt f s0erwise quantum melgf47j)m o ft k,0wsw0chanics were valid? Consider the first 20 elements or so.

  Which of the following electron configurations are no fk; s; evoaq0rr7f33jt 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 u*dzc *x.qvt3 nf*c ja9ranium atom (careful scrutiny across the3c*jnz * *q9ct.dfxv a periodic table on the inside back cover will provide useful hints).

In what column of the periodic table would you expeot9(g)w d wzcihtz 5ne2 ;(z-qct to find the atom with each of the following configuratiqhdi)9et nt -o((g5zczw;w z2ons:
(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 io1*kzp(ci 0wi fdine exhibit similar properties?

Explain why potassium and sodium exhibit ,r1;tkx 9 jdcalr+; qosimilar properties.

The ionization energy for neodgsg d;44c l+ie: r ue9i(dfj/6vpoc )n $(Z=10)$ 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. Belf 3nosxcxs** ib-oz; xg 9v 5,tenb.3mow) imply a photon nature for light? .ifxo sg;zx c33vte-5o **,9nmnbsbx

Why do we not expect perfect agreement betws b.g0zk;w1ymeen measured values of X-ray line wavelengths and ;s.wm0zgky1 bthose calculated using Bohr theory?

How would you figure out which linesxa-l-+,z/z q+nuoxq i in an X-ray spectrum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transitions deep with utr2hq5m)mq6in an atom to produce shorter wavelengt5qh6qmut 2m)r hs than transitions by outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light/eb,s8r)uu ek differ from ordinary light? How is it the same?

Explain how a 0.0005-W laser beam, photographed n6)scbub79v0-h + 75mjacdbyuhz b8t in3-yj at a distance, can seem much stronger than a 10jcy8 n sy0im 9ndz-u+thu ab7v5-6hbb)jb c3700-W street lamp at the same distance.

  Does the intensity of light from a laser fall off as the inverse squarewvx k-3 /lv*bb;la2zz of the distance? Ex3bvz-lwx ba 2vk*/ l;zplain. ----待选择----yesno 

  The neutrons in a parallel beam, each 4m*m wicn qc 0+a5 qbw*hd3z4phaving 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 (4b.q- bi p rcjnfzs;9a 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 measure 3f8pkf,g+ g,re+p6/xg9ydjr a:m vvsd 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 of an 3hls8 z: ydus1x((na batom 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 i2km fae r; 7b5q+qxt2tn the 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 possibeamj-t7w arz.n0rk6c90g -wele energy of a neutron contained in a typical nucleus of 7-.ee gmwazr 6j0-a0cw9t rnkradius $1.0 \times 10^{-15} \, \text{m}$. [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 prin c: u8qnm4vj0fn/bcjyr z26. uciple to show that if an electron were present in the nu:vrc/0 8c4 qfm2nunb.uz y jj6cleus $(r \approx 10^{-15} \, \text{m})$, 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 positionpaxq-pfa,nnc*lye 5 ci -; y ,*y8c;xn 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 b4la qvkth.n8/y-,n kw .ra-incn9l ;f e in the $n=6, l=3$ subshell?    elections

How many different states are possible for an electron whose principal quaop mok60equh0:y r(a :ntum num( 6h meookyu aqp:r:00ber is $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each electron k,re09by -)uzo lm (5lv6zmhwin the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each electron in the ground state okc-h(*ft7ycm6:ug p1q ;yl usf nitrogen $(Z=7)$.

  Suppose a certain hydrogen atom has kc z/phux5/6e-+lmo q+gpsv(au ) *e v$l=4$. 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 angularmw*sv* /l4 mica/jai; 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 h;lxm 8w)h + l2yo1u/splzcmi*as $m_l=-3$, 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 electrok)ql8mbp58ho ns in a "g" subshell.    electrons

What is the full electron confv1 kck9(vq;nty/ +7w m s b:tw ebo,gh4j-fl;kiguration 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) gold 94izp;kr p(xg(f z*cb (Au), (c) radium (Ra)? [Hint: see xb z;g *p(9rf (zck4ipthe periodic table inside the back cover.]

  A hydrogen atom is in th c8jndfrus+,0ji+ pj4hf01abe $6s$ 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 lithium using9sy6 a 1a tthlfs9;,yx the Bohr theofyax9a,;h9ts 1t6sl yry. [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 momentli1:py nvs84 qum is zero for a filled subshell.

  For each of the following atomic transitionp3dqq; zah8a(arf )4l s, state whether the transition is allowed or forbidden, and if forbidden, what rule is bei8a) dfhrp;3zl qq(4aang 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 itf.4el7*,, qbkd9rp fc9x num n 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 X-rac, g) iev5id4fys emitted by electrons striking the face of a)ce45d i,gfv i $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelenaxwb *p -( osgu6b9u0x +*voav,lt3ldgth bremsstrahlung X-rays emitted from an X-ray tube have *- xbtbgud3 laao9u v p+*(6,osx 0vlw$\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wavelengt+ il5 j3lakpb4h $\lambda_0$ 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 m;ykkz8 7; g1xby9rl6ns ify/emitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength fo x t:k,55*v31t5o(i x*v lfe pvy.qagrfc7yfb r an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate thedy q7;b1b*t yp 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 materia b(wou(y2izvtrt+8slj si 3*0 3tbbx,l is bombarded with electrons. The wavelength of3vjxt3w oi,yb2tsz+8u0 s(b(i* br tl 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 re (8xzs odg/ s2/fbhth1p7oi 5gtinas 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 pyeyp.go/u/ 1hower 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 vd3an- -ogz/xdue to diffraction if the beam emerges thron /3- -gvzodxaugh 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 la(5 thk)*e evj)z1 jdokser?$\lambda $ =    nm

Use the uncertainty pripw , hrlb h0mt;4j;zzq7+gkj .nciple to estimate 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 bj, ;zh;ktl7hrz4mjp0qwb+ g. e 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 v2slfqsgl1i- -/ z1 zk*kl 0r7 krla/rialues 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 foh.uw ((e urfi(r the radius of an electron orbit, estimate the average distance from the nucleus fo r(.ei(hwuuf( r 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 59r7utz),rdag xd 5;5fmwx yg 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 czpgc*/ )or:c n6si3qof 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 thlc +y /1ozizc v0dl:em;h 7sc0e Moseley plot(Fig. Below) 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 va iley.,mqg 9i;lue 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 diffwk/ rm y 1g2/b56xi/d wknllb-l4cjt :erent electron states possible for a given value of iywg4b-jcbkrmd1k /:/ 5l2l/ l6nxt w$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV is shot throje-o5ytglc2jn2sd 2jj /ac27 ugh two narrow slits in a barrier. The slits are a dis /nj etsalo2y2cjdg-7j52 j 2ctance $2.0\times10^{-6}\ \text{m}$ 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 modelb46fr v tay7 dw1:a c2ahe2ee7 and by quantum mec yr27 dh41ef6cv:t2e abe7a awhanics. 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 witir tfbs :9(vf4h 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 contains a line with a wavelength centered at 488 nm. li:es52b0a tb feds0waf5 +mpz5 3ko, Careful measurements show the line is really spread out between 487 and 4kb505e2efw a5 : pod3mzlsista0+ bf,89 nm. Estimate the lifetime of the excited state that produced this line.
$\Delta t = $    $\times 10^{-14}\ \text{s}$

  Protons are accelerated from rest acrossjqs3vw x5-n(+c 16 bhs +wbzkn 550 V. They are then directed at two sln(bqw s5+ +b 6-zw3cxvk1shn jits 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, eai i)i2iz0x0x61ea blach initially at rest, are accelerated across the same voltage. Assuming that the uncertainty in their position is given by their de e2ax0x 0 aziib6i) l1iBroglie wavelength, find the ratio of the uncertainty in their momentum.

  If the principal quantum number n were limite ivpb9pwoya9 0zh-,h yas/4 p2d to the range from 1 to 6, how man0yho9azpps2v- 9,/ab hy4wp iy elements would we find in nature?   

  If your de Broglie wavelength were 0.50 m, how*guhpe +hg5p,b.gx+x fast would you be moving if your mas,ph. gbx* ge5hug+ p+xs 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 element jx vt.fqg* :h+ sms :p1z.ae0kshows 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 protonsh.gpxz:j.f* vsa e t:1 ks0q+m and one electron, determine Z and the element in question. ----待选择----lithium berylliumboron 

  Photons of wavelength 0.154 nm arba3llw,v:x ++fu a/m le emitted from the surface of a certain metal whe,3+ +alu/: lbfavl mxwn it is 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 sprem+ qtxk14ltld n4h xv;d.i99 fad 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.]