Compare a matter wave e ao/gsd4tk/ .0nq. th$\Psi$ 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 metm k ,u3ybmixmsr8g1( jmu)/0chanics, particularly k (/ mxtmrm1su0b8y)ij,3gmu the uncertainty principle.

Explain why it is that the more massive an object is, the easier it becomes/tk7ijpu+phsn .,d h) to predict its futt /ndp,pkh.su7 hi)+jure position.

In view of the uncertainty principle, why does a baseball seem to haveui tng6 7l;;zv a well-defined position and speed whereas gz7;n iluvt6;an electron does not?

  Would it ever be possible to balance a very wkz/ jklf3(/dc a 4v5asharp needle precisely on its point? Explaifj a zvwda (3/k4k5/cln.

  A cold thermometer is p +7iyqafyt+ y*laced in a hot bowl of soup. Will the temperature reading of the thermometer be the same as the tempfy iaq7+ yy*+terature of the hot soup before the measurement was made? Explain.

  Does the uncertainty pbp8;b t72mj pmrinciple set a limit to how well you can make any single measurmj b;pm82t bp7ement of position?

If you knew the position of an object precisely, with no uncertaiw /3 z;3v fnncxw4ny+ux( fjx/nty, how well would youuf3yxc (n/4 znjw3n /wxv f+x; know its momentum?

  When you check the pressure in a tire, doesn't some air inevitably escape sd27b*e8wbwj. )sj . c) /.hljxursfx? Is it possible to avoid thiu. 7w)f8x2w s. h/ess bcjxj)rlbjd *.s escape of air altogether? What is the relation to the uncertainty principle?

  It has been said that the ground-state energy in the hydrogen hn+:jp 59vgz,p x gfljo9;5vbatom can be precisely known but the excited states have+o bzlgn ;h vv5:p ,j59xjg9fp 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 atomprv,u 8abc*+ x +gmydwhvoe /k+cr( 8/, the Bohr model or the quantum-mechanical model, predicts that the elh /cr8+(+* o cprv 8edygab,wmx+/uvkectron spends more time near the nucleus?

The size of atoms varies by only a factor of three or so, from largest to smalle, 6l pcvzu8l bxdca 4h9qd:z/;st, yet the number of electrons varies from one dcxv l89pb;lu4zh qzc:/ d 6,ato over 100. Explain.

Excited hydrogen and excited helium atoms both radiate light as they jump down db i;u bp6m4.pp9pt+j tpbu.t pp;j9b4p id 6m+o the $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic tabl+l 8vzayq:a/hhmfrfd a+s ftk3l;4( +e look if there were no electron spin but otherwise quantum meh(/dzrfa+ f+8s ma; 3avhtkfq+ ly: l4chanics were valid? Consider the first 20 elements or so.

  Which of the following electron c-up*- )*dostbac1wxo onfigurations 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 scrutipykxq )ycwnj 1e*;i d2u7iv* (ny across the periodic table on the inside back cover *j yk(e )idq ciuv7nw1x;* y2pwill provide useful hints).

In what column of the periodic table would you expect to find the atom +q/0wee.l7m (y tyzehwith each of the following configurati0yw+y(h/l.et em 7e zqons:
(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 proper6.h3g wrcl k/rties?

Explain why potassium and sodium exhiq1pm;t igtp9 2bit similar properties.

The ionization energy for neot0r0 s1 )jyzu rq.;,gqf9iefwn $(Z=10)$ is 21.6 eV, and that for sodium $(Z=11)$ is 5.1 eV. Explain the large difference.

Why does the cutoff wav; owbh-aggh*: elength in (Fig. Below) imply a photon nature for light? w;ogabhh-:* g

Why do we not expect perfect agreement between measured values of Xs:ctb k u89)se34gtlp -ray line wavelengths and those cal:ebssc) 3 4ttlu9k8 pgculated using Bohr theory?

How would you figure out whicw ;i4xi9:wtx3j j4qynh lines in an X-ray spectrum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron trxyx rw,mxgdt3 4b.bb-qj+.q 0 ansitions deep within an atom to produce shorter wavelengths than transitions by outer electrons?3q.x.qr b+0y4,bx btw d-mxj g

Compare spontaneous emission to stimulated emission.

How does laser light differ from ordinary ligy)a-s)vpk m3dvxg2 /(a yh/m; kb,qbnht? How is it the same?

Explain how a 0.0005-W laser beam, photographed at a distance, can seem mu- zd5 hfzbon9x-pj*)kch stronger than a 1000-h9kx-ob pdz -n5f)j z*W street lamp at the same distance.

  Does the intensity of light from a laser fall oqo9c.c m47rtr ff as the inverse square of the distan oc7t4qrc9m r.ce? Explain. ----待选择----yesno 

  The neutrons in a parallel beam, each havom hk hls5 l+a(;,wvb,ing 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 of gs)l7)x(lta3 qqtgx- $(6.560 \pm 0.012) \times 10^3 \, \text{m/s}$. 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 an accuracyj)/ x9avb, wpjpu)7 r(8fu5uoehvj7 l 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 ato:j.cidp skg/n) no8l8 m 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 8thdzazg +,0gmuon $(m=105.7 \, \text{MeV/c}^2)$, 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 typical nu- cm/xkae x*6gcleus ofma*-ec xkx/6 g 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 in th45q(viz7q ief hs9 g hpvda34k0 t*d(xe nucleu44t(a 0qdzkq vid5hs(ex9pg37 h*ivf s $(r \approx 10^{-15} \, \text{m})$, 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 positigog0a. mnf tgg /ca812on 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 fs ;ue1yb/l 3b$n=6, l=3$ subshell?    elections

How many different states are possible for an electron whose principal quantus4ut /biq+ r+2,vt 5a0njsfj i5 ;fa1zzyptn9m nspj; 9ruta + batn5fq +i,zjfy2n4s t0v1i5/zumber is $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each electron in tvosq)m)c5 r 1k, 1eoex7gwkl 8he ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each electron in the ground state of nitrogenx(b g6 p-r -bpv *mo6xya/ea+y $(Z=7)$.

  Suppose a certain hydroges vxjcz+ocf0j6fj 13 . fwzc4;n 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 an electron in the -el4 5 . ilpg0hwg7y0om; gzei$n=4, l=3$ state of hydrogen.
L =    $\times 10^{-34}\ \text{kg} \cdot \text{m}^2/\text{s}$

  If a hydrogen atom hasb*n2p5 r1sox0k ;sip:(ax eogzb+/xd $m_l=-3$, 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 electrons co4)o lm;p:n w 4 x2lwvfv /jl )wgvvo;c-dh61in a "g" subshell.    electrons

What is the full electron configuration in the ground state for elementscfb/w j)ro,m * 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) seleniy4cypqqu /zdr,p/* i.um (Se), (b) gold (Au), (c) radium (Ra)? [Hint: spiq4/,urqc ydz*p/ .y ee the periodic table inside the back cover.]

  A hydrogen atom is in 6rgvr9 q0tuh 31fzfg1the $6s$ 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 thir6u nor)n:y4ud 6/cuwk n5o/rfd electron in lithium using the Bohr theory. [Hint: To /c4y)k6n 5o :udr/ ufnwrun6his 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 subshell.tsn *-g7ne:7xg (u pqx

  For each of the following atomic transitions, state whether the transitiox+y(g 4/in zbkn is allowed or forbidden, and if forbidden, what rule is b+(/z4kxin ygbeing 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 :ug0np8sh mo:l0 r )hha $6d$ 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-war g r;-fxe*g6svelength X-rays emitted by electrons striking the face ofx *r;frg6 -sge a $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength brlbchk9e8 gky*l otho)koh- g- -f 1a18emsstrahlung X-rays emitted from an X-ray tube haveg-of tl*khy lgh9o1 h- 18c8-oebk)ak $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wambo:( 5:fqoo dvelength $\lambda_0$ 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 emitted when odh /3ay/p)jfs53nd7j kw vp4a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength for g7hvw k-k60i ban $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the wavelength f6afsb:ia 6d9 mor 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 mater;/j3 um:ladevial is bombarded with electrons. The wavelength of the u: 3jdel/vma;$K_\alpha$ 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 retinas putspo83 p wvt42vfte5hp; 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 ha- 3d**:0u lvaixooda6nco l 8ave 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 n)k,;mm *q nq( hqnu88a(g xewdiffraction if the beam emergekuq8*na8m(en ( w) ngxq,q mh;s 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 bp mc7t(mss4owm6e;ki - n/e2He-Ne laser?$\lambda $ =    nm

Use the uncertainty principle to estimate the position uncertainty for the0-l5 v:co;h(9 wz: zbhsjonso electron in the ground state of the hydrogen atom. [Hint: Determine thnhjss vzhz :-w 9 oo;(:cl5b0oe 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 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 the angular mop 4mgd,5 ofgv5,xqn0zmentum $L$ 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 rade5d 1 *fpj qn(-er7coiius of an electron orbit, estimate the average distance from the nucle7 cej*d(r-oq1p e5 ifnus 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 operat ber lu1l.utc0h h r:*nyp: 0w-spk(*py.7apyes 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 outermost electron in c z:6vy7x./ + 5cilq svp* fq;ieoks;2o;rdxn 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.:dx.y-bh gnp 8 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 ofx11ah rbq sm +t88wwv39r gzkht8 xj04 different 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 different electron1jed,u xz7hk/6h el g3 states possible for a given value of u1hlg/e k,dhz j6ex37$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons wikz(4 /c)glfv6 6hzssuth energy 45 kV is shot through two narrow slits in a barrier. The slits are a di4h (g6s) kfuzcv l/6zsstance $2.0\times10^{-6}\ \text{m}$ 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 model ypv:*gx4y1 ipand by quantum mechanics. C:g4 pvyypx1 *iompare 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 ofny arwq1jwn vqv1:4p52hju+rhvf+65 $(22 \pm 0.22)\ \text{m/s}$. 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 cene7 gd2k3 s(p ytahmd4+1 gz:tqtered at 488 nm. Careful measureme d+3 7zt(ekqh4mp:atyds1g2g nts show the line is really spread out between 487 and 489 nm. Estimate the lifetime of the excited state that produced this line.
$\Delta t = $    $\times 10^{-14}\ \text{s}$

  Protons are accelerated from rest acroq 7 nuif; 5j6ifwvoj55ss 550 V. They are then directed at two slits 0.70 mm apaw7n6vo5jjq u i iff;55rt. How far apart will the interference peaks be on a screen 28 m away? $\Delta y$ =    nm

An electron and a proton, each initially at-f nx4-y hgct3dh(9rd1 u; 9:jkfvurt rest, are accelerated across the same voltage. A1d9 f v-ng;rur9-ucfykhdx jt(h3:t 4ssuming that 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 number n were limited to the range from pbvna 8n kn3+9s:ca8p 5j 4pkn j)ru4t1 to 6, how many elements would we find ps8b k v4nuktj8p3prj94:5ac)+ nann in nature?   

  If your de Broglie wavelength were 0.50 m, how fast would you 1/ 87ip hmqqg*fbv9kwbe moving if your mawvbigh8 qfq /9k17 m*pss 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 spectrumtx jk 2poqq0m+(1-chcw x2)xs of an unknown element shows a series of lines with one out of every four matching a line from the Lyman series of hydrogenqc+ )hsjk20qx 1o-( xxcp2 wmt. 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-b lvg77h;tbalelth),.imk : nm are emitted from the surface of a certain metal when it isg7h-:vkle ba il ,thtlb) ;m.7 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 a laser beetw kmc,;h )w.u58iu7pjw (fs36 gmh cam, $$\theta \approx \lambda/D$$ , 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.]