Compare a matter wave zs6g.1owfqm b*)a8l u$\Psi$ 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 mechanics, ichu b5i2) 6acmt5wx;:lkz+ c particularly the uncertaiw5i);h6l bikam :ccx 5ut 2z+cnty principle.

Explain why it is that the more massive an objezp vymf70o0w 6ct is, the easier it becomes to pwzy006 7fmvopredict its future position.

In view of the uncertao4k/sctrig ql8p at).1 izm9) inty principle, why does a baseball seem to have a well-defined position and speed whereik8cmo4qs z r )tg.1p9tali/)as an electron does not?

  Would it ever be possible,qo -kqf 94hfhv*4ilh to balance a very sharp needle precisely on its point? E4q h*,lf iqh-v94hkof xplain.

  A cold thermometer is placed in a hot bowl of soup. Wiomzkgy(vl56/g3co 8zzz1m t- ll the temperature reading of the t(l tmzzo 8o5gm3z g6 z/kv1yc-hermometer be the same as the temperature of the hot soup before the measurement was made? Explain.

  Does the uncertaintyw6kg2a fhh.a0 ec8w rc39z d 3ws8ny-r principle set a limit to how well you can make any single measurementhk2wre889zcc-6 0anhfd ga yw3 rsw3. of position?

If you knew the position of an object precisely, with no uncertaimupo4j+d/q gi+g wtc-70.hp xnty, how well would you know its momenji/+ 4+gdupxowpt - .cg0q7mhtum?

  When you check the presslo4n(e k 2ccw1b4.ub* nqf x1nure in a tire, doesn't some air inevitably escape? Is it possible to avoid thi *b4oc4n(1e un.klc fqbw1n2x s escape of air altogether? What is the relation to the uncertainty principle?

  It has been said that the ground-state energy in the hydroge djgz00ri prh4b3f)6 nysz szb,e887xn atom can be precisely known but the excysh84 dxzi3f)b, gz 60psb jz rrne780ited 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 hydrohe km+;g012w k. xc1wfyo za0hgen atom, the Bohr model or the quantum-mechanical model, predicts that the elec2we0 yhm;k1zh.a 1ko0 wg+cxftron spends more time near the nucleus?

The size of atoms varies by only a factor f7iode0(s38 xg-q qss/d+tv r of three or so, from largest to smallest, yet the number of electrons varies from one 8q3vt(+rf/s xgos7 -0qsed id to over 100. Explain.

Excited hydrogen and excited helium atoms both radiate light aswuv b+e 9/7ib8oz8uzh they jump down to 89w zbuh uoe/b+vi8 7zthe $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table look if there were no electron spin butfr+j3rf lxx7z 70d9xm otherwise quantum mechanics were valjrlfxzx7m0dx 3 9+f7r id? Consider the first 20 elements or so.

  Which of the following electron conogd e 3z fj3yql1m+x7/figurations 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:(rl 9 nnp/bgr (careful scrutiny across the periodic table on the inside back cover will provide useful/nn(r :9rgbpl hints).

In what column of the periodic table would you expect to find the atom withj3x,o/waz apeqqh ,/ss- w 8*n each of the following configuratjw-aw3,x*n 8/q ep,oqhs zas/ions:
(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 simila (2wecw12*di:y/ a- daahurojr properties?

Explain why potassium and sodium exhibit si ;fv3j+-14t yen; c*csjjawv fmilar properties.

The ionization energy for0;p6mmnbw 5m2nok8 mi 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 (Fig. Below) imply c*bdzxk e-bjlmoq3f h(8p(;b *tj6 -ba photon nature for light? 3-bh8eljkptc dxqj( b( ;bbm-z** f6o

Why do we not expect perfect du 706wy uq3ikt:ab (bagreement between measured values of X-ray line waveleqwa37 0yuid6ktu :(bb ngths and those calculated using Bohr theory?

How would you figure out which lines in an X46oqj,q mx+ag*t d8 7 umtw8eu-ray spectrum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron tran.b m :jp+zc q/vru(8knsitions deep within an atom to produce shorter wavelengths than transitions by oute: b c+ z8vnk/uq(r.mpjr electrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from r)6ixi4p xxuq,:5 oewordinary light? How is it the same?

Explain how a 0.0005-W laser beam, photographed at a distance, can seem much-btsi6ho2 s3xwrl y1da-h -b9 stronger than a 1000-W street lamp at the samea6 odbyr-wb-h t-132h s9slxi distance.

  Does the intensity of light rozx*z,cf f-)i8n kq3from a laser fall off as the inverse square of the distanceofrfn z iq-)*kx3 c,z8? Explain. ----待选择----yesno 

  The neutrons in a parallel beam, each havikc7cvne vl oq8h3u5narn+(-8ng 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 wi jb2b4co:(,0qf ng cndboia3/ h41enxuj ho5/ th 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 measure 9vvx w7*bwdmy6mu(e;d 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 ofx5 stn vfw3m;t;*.mau 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 mazwod snxe/+9t ;v.:m vss 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 neutronxcmmb q/ 0**-fi ;xtef contained in a typical nucleus -mf/* qx0fct*e ;m bixof radius $1.0 \times 10^{-15} \, \text{m}$. [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 sp+ ky;il1d/ g1hm /byin the nuc;p gh+1mb/ kyis/1 ydlleus $(r \approx 10^{-15} \, \text{m})$, 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 pon-, x3(2qud bi skfnhe0+2j qesition 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 ismhm;bpy8r-l7*6,fad okd 5b n the $n=6, l=3$ subshell?    elections

How many different states are possibl) 6)-0ub 9vk( ,8llosr nljp5sdmlx jde for an electron whose principal quantum numb mjo0(lu5r,kj dd6nx - 9psls8lv)bl) er is $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each electron in the ground6 i ,j4-oe :*n3soyi3qapx mrk state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbeeh wl(*(pa/09o6sgxus1 gn xm rs for each electron in the ground state of nitrogen $(Z=7)$.

  Suppose a certain hydrogp 6:d . ap.xefut-ae 3/h/(zyjabr1qxen atom 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 2by(eb.69ahl9o npgp 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 8 2wbpmyc: m:p$m_l=-3$, 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 1n(w :y(dp4 -fxgsac k.*n ;dhv8 electrons in a "g" subshell.    electrons

What is the full electron configur,/bma(4qfwd6m n et3k7mj c0pation 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) selew39d (c 1dhmylnium (Se), (b) gold (Au), (c) radium (Ra)? [Hint: see the period3dlhdm1 y wc(9ic table inside the back cover.]

  A hydrogen atom is in thg4wm /v x/qc:re $6s$ 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 r5f+ wv+vub5yl/ j0kn in lithium using the Bohr theory. [Hint: This ryj nvb+0u5 5wkv/l+felectron has $n=2$ and "sees" a net charge of approximately $+1e$.] The measured value is $5.36 \, \text{eV}$.    eV

Show that the total angu n3:x 9buq(stplar momentum is zero for a filled subshell.

  For each of the follg58qh u7gjy2rdk0f*3d bwy/eowing atomic transitions, state whether the transition is allowed or forbidden, and if /f j5b2q3u*gh0k8yd r ydge7wforbidden, 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 1sajg :ak9lo,wxe 9h 3a $6d$ 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-wa,cc0k wn gcfzh .cfqfxce-/9qmiahmg ; -8 86,velength X-rays emitted by electrons striking the face og-.m;,n,hgqhifxfw 09 zmcakf 8-6c/8ccqe cf 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 rscx6l6j v4tuv(k4(w X-ray tube have v vlr46k(( 46jcuxwst $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wa m+-6q l,es/wo*kmxp.z+olp/ t: kufv1lr g) mvelength $\lambda_0$ 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 emzqu9 fz1rfg79 ;.pbo zitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength for7zdeq7 ldl2s+ an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the wd s:vqgjs - ajn63;w -.,xoytcavelength 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 unkno55.h hcrz, o4lepw-acwn material is bombarded with electrons. The wavelength of the.a-4,zcohl r 5wec5hp $K_\alpha$ 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 r1;ie jr0rzy(3k e*wt,afeceeq*7p k) etinas 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 ofc2*jfjtt 4n;o $0.50 \, \text{mW}$ 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 byfbtsmu ,h1g 3 36;vwteam due to diffraction if the beam emerges throu61f,hmy3vtsg t wb;3 ugh 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 lase w-cpmp0oxm)3whe19 pckc ex;g286 t. oh+ vicr?$\lambda $ =    nm

Use the uncertainty principle to estimate the pot4,nk 5*swk1)kcxr /bkn1sj e sition 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 a jr)t1xe*cns k5s1w bk k4/k,nnd 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-o1hx1 uq :kgz smallest possible 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, e aj9 pb:hr3d((.jijikstimate the average distance from the nucleus for an electron in the (bikrij(3 j. dj:9ahpinnermost $(n=1)$ 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 1nlx -o9vihv2 q/m1 skxy(/fr 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 the ou5yu*l *+ehs dfz*bx 5itermost 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 Mr/e/o3tiu ekj jf,e4 r5liq)6oseley 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 foa )9c lvzy,*uhr 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 different electron states p,k79 ssg7 lwqaossible for a given value of ss aw,l 79qkg7$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV is shot tog22q+h m2 tvdhrough two narrow slits in a barrier. The slits are a 2g+t 2 mvhd2oqdistance $2.0\times10^{-6}\ \text{m}$ 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 bys(x;xeo fw:0p*qjj1f 7p cz1e3bo 7un the Bohr model and by quantum mechanics. Compabw;(*o 0 7o f7xjxfcpe:uj 1en1sz3 qpre the results for $L = mvr$.
$L_{\text{Bohr}}$ =    $\hbar$
$L_{\text{QM}}$ =    $\hbar$ or $L_{\text{QM}}$ =    $\hbar$

  An 1100-kg car is travelilv)0rjn ldp5 0q 6u*trng 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 contains a linej/eqmj ayxn*xd ,. 4ip5w 9t4w with a wavelength centered at 488 nm. Careful measurements show the line is really spread out between 487 and 489 nm. Estimate the lifetime of the exciteiwqe yxtn pj/9ax dj,4w5m 4*.d state that produced this line.
$\Delta t = $    $\times 10^{-14}\ \text{s}$

  Protons are accelerated from rest across 550 V. They aregwg+ij qd t/lc j+79*x then directed at two slits 0.70 mm apart. How far apart will the interference peaks be on a scqxj9t++ 7*igjwl gd/creen 28 m away? $\Delta y$ =    nm

An electron and a proton, each initially at rest, are accelerat7lwx2 6a .gga9ijgcw; ed across the same voltage. Assuming that the uncertainty in their position is given by their de Broglie wavelength, find the ratio of the unc ;gx9 wj7ag6a.ilgwc2ertainty in their momentum.

  If the principal quantum number n were limited to the range from 1 to 6, ho,sgkt h:b-l.+xe i) akw many elements would we fh-s+tel ):.k,kxagi b ind in nature?   

  If your de Broglie wavelength were 0.50 m, how fast would you be movbzy.+aru)( yf,ey j 6fing if your r,aj+.e b)6yfufzyy(mass 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 u l518oxgpnbx t s;,2tpnknown 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 the element in questio;x12 p xo tsg85n,bptln. ----待选择----lithium berylliumboron 

  Photons of wavelength 0.154 nm are emitted fn1u(m-tnk ht( rom the surface of a certain metal when it is bombarded with high energ-1(nnmth (kuty 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 a lrjaz 0os ,yf1jx e7k 8g7q;n,caser 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.]