Compare a matter wave 1yq)u9)- 0smrtuc h gz$\Psi$ to (a) a wave on a string, (b) an EM wave. Discuss similarities and differences.

Explain why Bohr's theory ofb40eb r hbdbh)v:x,8mwrjr3+ biq (9r the atom is not compatible with quantum mechanics, part(brb3x b bm:0b,hi 8dr4+ j9wvq)hrre icularly the uncertainty principle.

Explain why it is that the more massic 9np4;aax,(fas7y fqve an object is, the easier it becomes to predict its future posiyaqsp c(;7a4a9n,f xftion.

In view of the uncertaint7hk 64 wajb 2wm 33e,gohndn-qy principle, why does a baseball seem to have a well-defined position and speed whema4jh6hqn3egwd nb2k w-7,3 oreas an electron does not?

  Would it ever be possible to balance a very sharp needle precisely oyqid ;l )scf99:eui x5n its point? Explayi) ;qcdex9s fi59 l:uin.

  A cold thermometer is placed in a hot bowl of soup .4c,tnxwd(i ti cb:o4. Will the temperature reading of the thermometer be the same as the temperature of the hot socc ,(4tbwdoxin. ti 4:up before the measurement was made? Explain.

  Does the uncertainty principle set a limit to how well you can make any singl zko*a:p74f+ wvc*he qc9x kb)e measureme whcva+*49c:7q pfb)*z xkok ent of position?

If you knew the position of an object pr-x7 y5zlx9;;jzwl5;ls so g lrxv3er7ecisely, with no uncertainty, how well would you know itzs r;xj ;-5xvoz39se x yllg7w7;5lrl s momentum?

  When you check the pressure in a tire, doesn't some pni7 5va;zz kjs20l toz01kb0 air inevitably escape? Is it possible to avoid this escape of air altogether? 0v p0a12zjn0kkz7i;o s tl5zbWhat is the relation to the uncertainty principle?

  It has been said that the groundyt3tnxzn5- y 5out 48f-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 consistezyt8 ntn5u3 tx5y-f4ont with the uncertainty principle in its energy form? Explain.

Which model of the hydrogen atom, the Bohr model oreo ,y;yq7o4u9h7dbd x ug:2cu the quantum-mechanical model, predicts ye4q279go7dh ucyx udo,b;u : that the electron spends more time near the nucleus?

The size of atoms varies by only a factor of three or so, from largwk(kfusr t jeq;7sqse 2o9g n)00,c; yest to smallest, yet the number of electrons vage(tkf)9 qescynu,7;;w0kqo sj 0 2 rsries from one to over 100. Explain.

Excited hydrogen and e5h 7scqokp6ng)xd( a 2xcited helium atoms both radiate light as they jump down t 5xa(hqc dp)og2 7k6nso 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 t0(e0iaraav: gol7b;b qt 6wi 5here were no electron spin but otherwise quantum mechanics were valid00i;geb:t7raw6 a(oq5ba l iv? Consider the first 20 elements or so.

  Which of the following electron configqjtiv *o zg3kg h9.u0v3m9 h+vurations 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 urfnpj+pux;yix ud3uzrv p(9 v;07 o(x+anium atom (careful scrutiny across the periodic vnpd;+v;(xu yz jpor 3f i0 +97upuxx(table on the inside back cover will provide useful hints).

In what column of the periodic tabled ej1guh9jh*2ci9c 0o would you expect to find the atom with each of the followiu1*0gchihjc9 jeo d29ng 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 exhibit6ohtew: 7ut gw,;vf wr;u2lg52 uf.e72e zwlv similar properties?

Explain why potassium and sodium exhibit similar prkal p5wy83ig h.i07nw operties.

The ionization energy forf/-f1yq5j g k0 yrbh9o 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 a photon nature for lm: t07)k-nvgxa5n zwright? twm-zv0rn x7 5n)a g:k

Why do we not expect perfect agreement between measured values of X-9 lrxvb99 ;xftx4 t4msray line wavelenmxvs99;t 4t9 4lbf rxxgths and those calculated using Bohr theory?

How would you figure out which lines in an X-ra:egg+cx 3o4lmy k)9fay spectrum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transitions zo 8xkb 8(nkysyk,;y7deep within an atom to produce shorter wavelengths tha7 xk,8 bkyk;sozy y8(nn transitions by outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light *l /tzoh, ,s )vyxhho6differ from ordinary light? How is it the same?

Explain how a 0.0005-W laser beam, phot5z ++w+x mllyuographed at a distance, can seem much stronger than a 1000-W street lamp at the sayl+w5m lz +u+xme distance.

  Does the intensity of light from a laser fall off as z6 )711,ltd8l1qbh r6hohqeu8olizdhr : 4mg the inverse square of the distance? Explain 4 8 6lmqzl,)ogh b tehhdlod1iq18hrzr7u16:. ----待选择----yesno 

  The neutrons in a parallel beam, each having k8 8d3bfwo lz9ucx 4zf7inetic 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 spedyjujd c-l3 f /ig1*h0ed 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 xm2du )lcr/,jcan 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 of an atom fo*lbtn y h052q8eip0skr 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 muonk :4*z eqzf2xwzh3emye-,4qx $(m=105.7 \, \text{MeV/c}^2)$, 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 s2q o8halz/e91 nvh5a possible energy of a neutron contained in a typical nucleus of rhvzs 589ale qh n2o/a1adius $1.0 \times 10^{-15} \, \text{m}$. [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 prinsqby+4 *ea-uw ciple to show that if an electron were present in the nucl-*4+qwubasye eus $(r \approx 10^{-15} \, \text{m})$, 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 position o1*yg9zwo k) awf 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 3;zthtj h,v :lbe in the $n=6, l=3$ subshell?    elections

How many different states are possible for an electron whose principal quantum g23 cype7v (vn c:ar1in2c7piv( 1cn vgra3ey:umber is $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for kbryfk 5 -1h r/wi-blkrt g-570nkj)heach electron in the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each electron in the ground state 5n)6pdgynx 6v of nitrogen $(Z=7)$.

  Suppose a certain hyd3uolq poom1r0 ).re y4rogen 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 thell+ e ,gyf-do9 e xc(nq3;5mwtq;j4ny $n=4, l=3$ state of hydrogen.
L =    $\times 10^{-34}\ \text{kg} \cdot \text{m}^2/\text{s}$

  If a hydrogen atom hasjs.-w: ghgdb3swrg( 2 $m_l=-3$, 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 eleclapb7eg-q1 3 rtrons in a "g" subshell.    electrons

What is the full electron configuration in t:lj6 b)ilwu+l he 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) 0rm)nf pi .bl)5yu t1qgold (Au), (c) radium (Ra)? [Hint: see the ptq .l)inf0my1 5)ru bperiodic table inside the back cover.]

  A hydrogen atom is in *pav1+vzrq; dthe $6s$ 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/5e*4;y mfg f6onibdmm4lo2yj:fd e4 electron in lithium using the Bohr theory.m;y fedg4 de4l:m* ob4f 62yj5nim/fo [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 i).9jpmrlfu8z h s0 4gcs zero for a filled subshell.

  For each of the following atom ce omhg8,dtna(0 ci v..7e;) /fvzarkic transitions, state whether the transition is allowed or forbidden, and if forb.g7oedace h r,iaf8n/ )v . z0cv;tm(kidden, 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 a o)7fi xfw55lz)l g cz+$6d$ 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 electr4xxw00)jiyjc(*l xxifxm4;fj+ ( vsn ons striking the face of 4v*)f((xnxwyxxjs x;m 0ijl4i+j c0f a $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremsstrahlung X-rays emitte53htcq4agr+tl e 5lr+d from an X-ray tube havetl55h erqrg++l ct3a4 $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wavelen9:o5,g )v ncc*oeez:loe+mjg gth $\lambda_0$ 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 emi4lhxrv z ji+ w3s0prr6n83.wktted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength fody6a.nn*hqyj 0t x 1h2a 7igq5r an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the wave:x 3 q))mrr.vbygml48a.eyt x length 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 ne r,76ygrc17cvtqb p+jd 3k-an unknown material is bombarded with electrons. The wavelength ogr, ty- 3kb1pce+7vqnc6j 7 rdf 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 detaj;r pp*z,plxr37 7 tndched 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 physicka7x.e.37a/ oldg w (gd1zbe ds lab might have 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 spreafx+jwpr ) y3c*d of a laser beam due to diffraction if the beam emerges*p yw)fxr 3+cj 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 He-dzhb.;rco3x - Ne laser?$\lambda $ =    nm

Use the uncertainty principle to estimate the pojxed -3 d8 e*c:mjdzo) m/p/ozc+j.krsition uncertainty for the electron in the ground state of the hydrogen atom. [Hint: Determio ced8/ z:d3o.z)m/c d+*rpjxme-jj kne the 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 mon9fp 2x6vbc9t tzo1* chl8yz* bgu kw-;h7z1bmentum $L$ 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 4 we w-b 7unfb+0vlcf9-lf2 iffor the radius of an electron orbit, estimate the averagewl7 uveb 9i 4cff2l0fbwf --+n distance 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 operates a.r72v40vv f xh xz7sqvt $95 \, \text{kV}$ 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 boss*8q ac97zexron 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 that4i7ue) :.ppqjlxvp4)4 it9 jlj(jbrmm9 zid3 the 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 nus(ty5 bz os84cmber of 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 electron states pos0u u 83njb;0ayb tnb/rfd2b w:sible for a given value of u:08bybu0b r bnf/2ajd ;nt3w$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV is shot through two *nsh;d . mtq8: giyidxc4/du, narrow slits in a barrimus.gidi:qdd,y8/ xcnh 4*t ;er. The 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 hydrogen atom is givf v u 8xq+s z3x)9b9wy1acp0ohen both by the Bohr model and by quantum mechan wh pb103+f v8ox)aq9c9xszuyics. 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 *iw s yp.u3aiv(*an2pspeed 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. Carefu k 8 vq*imy)sp7ju7u-rl measurements show the line is really spread out betw8*mv7j s)-ryu ikp7queen 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 across 550 V. They are then directed at two dw hqr .r;ls995p0vxislits 0.70 mm apart.qr ;dwp lsv 9h9i.rx05 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 rest, are accelerated acrosshif ).;et:ex8pydb8g35ck h s the same voltage. Assuming that the uncertainty in their position is given by the8 t5k3hhfb s:y.8e;)cexpd gi ir de Broglie wavelength, find the ratio of the uncertainty in their momentum.

  If the principal quantum number n were lim gjv, mp9cd;j;ited to the range from 1 to 6, how many elements would we fidcvjj9m, ;; pgnd in nature?   

  If your de Broglie wavelength were( p*s)guwa- 2 npvfh1v 0.50 m, how fast would you be moving if your mass is 75.0 phngv2v 1*awp)u-( sfkg? $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 shows a series of li-pt 2nk 4uq+jfnes with one out of every four matching a line from the Lyman series of hydrogen. Assuming that the unknown element if k t+u4-n2pqjs an ion with Z protons and one electron, determine Z and the element in question. ----待选择----lithium berylliumboron 

  Photons of wavelength 0 6zrjh 7av**qx.154 nm are emitted from the surface of a certain metal when rqha*7*j xv6zit 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 spread9w aa:9hwzxlrdt3(i : m(7cf3 1 dxnhv 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.]