Compare a matter wavel71qnkwm q 2bgs 2feu8bjs (zm7296ot $\Psi$ 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 mechri+ib30tgrbjj 2rid 20j9ec9anics, pitej3i09 2jrjbbcdrgi2 +r0 9articularly the uncertainty principle.

Explain why it is that the more massive 3r7j:2bnz uv ran object is, the easier it becomes to predict its3rb 2 zn:r7uvj future position.

In view of the uncertainty principle, why does a baseball seem to havegu l6lpcs;8j : 6,a1nsz eq ,dp7zgek8 a well-defined position a 6;eukz,a c:ngp8 sq6lpg,jsl8 1edz7nd speed whereas an electron does not?

  Would it ever be possible to balance a very sharp needle precisely obxe3 +f hw9vg*n its point? Egb he3+wvf 9x*xplain.

  A cold thermometer is placed in a hot bowl of soup. Willsni jj. z2qm )yb gp4ov8c/-7f the temperature reading of the thermometer be th- giz 4sj).m8n/ 7ojyb2pcqv fe same as the temperature of the hot soup before the measurement was made? Explain.

  Does the uncertainty principle set a limit to how.*: faianpop8- vg;ak3,o.a(j q owrr well you can make any single measurement of positio.*oorgifa.j pp3 -(v ,;nq8awaa:kr on?

If you knew the position of an object precisely, wi4tf*r) nf0rvl g)noy*zx1g6y th no uncertainty, how well would youznn) fxg6 *vro104ry*t)lf gy know its momentum?

  When you check the pressure in a tire, dv,bp8jrs/ a s+oesn't some air inevitably escape? Is it possible to avoid this escape of air altogether? Whatv/j, 8bs +apsr is the relation to the uncertainty principle?

  It has been said that the ground-state energy in ant1 uxv; v4n+e1 bu xt42fu:sthe hydrogen atom can be precisely known but the excited states have some uncertainty in their values (an "energy width"). Is this consistent wi+ uvv x;4bfen1:au 42u1ntxt sth the uncertainty principle in its energy form? Explain.

Which model of the hydrogen atom, the Bohr model or the quantum o,hqpz1 ;c0qg ft aom6 ta:ls1a3*f*m-mechanical model, predicts a61z3gft*c0ma ofqsmop: ;qta h1*,l that the electron spends more time near the nucleus?

The size of atoms vaj r :e ) 5i.oc769hilvjvacn4tries by only a factor of three or so, from largest to smallest, yet the numbecv6a9 nrj7)i5.oi4 jcv :htler of electrons varies from one to over 100. Explain.

Excited hydrogen and excited helium atoms both radiate light as they jump downa 9.)c -hff)vjdg)2 zzpn4wx8fpxgd5 t4 )cp.d)2gxf j v 8pdz9f)nx-a5wgfzho 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 eld;q+/c s4szlfbagct+9on76 aectron spin but otherwise quantum mechanics were valid? Consider the first 20 elements or s9c+gcs6zoba7n ql/a;df 4t+ so.

  Which of the following electron confeyrbm,wvyf 4g 7wc7 r.q lo2;3igurations 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 ato /v6m o pfuxt t/)qdq7/iiq1/um (careful scrutiny across the periodic table on the inside bq/d)uq7tivpt 6i/f//xmo 1qu ack cover will provide useful hints).

In what column of the periodic table would you expect to find the 2y.c3iuy ;svqj(3hj batom with each of th3hv.23;ybjiy uqc(sj e 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 exhib* i;ls8rrrfo 1it similar properties?

Explain why potassium* m 72r,jbtovd6y ac*x and sodium exhibit similar properties.

The ionization energy for neoecn s pd 6d26f:+pj1fon $(Z=10)$ 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) i r* rz fhuv1/uz:jjg jg3:b:6pmply a photon nature for light? u/*pz6rbhu: : vfg:j jj1g3rz

Why do we not expect perfect agreementp9z.h mbxi4 ; ya*x;jg between measured values of X-ray line wavelengths and those calculated using Bohr thimhxa9ygj4*.bzx p; ;eory?

How would you figure out which lines in an X-ra9mcu fn0enu0k6 y:n dtm:b-)zy spectrum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transi 3wugvn9,/,cb ns qfd3tions deep within an atom to produce shorter wavelengths than transitions by outer electrons? ,,s3 vc 9w/nubn3gqdf

Compare spontaneous emission to stimulated emission.

How does laser light differ from ordinary light? How is itx,21bmhv/+hs * jugmlnb7 q)4nb *fcr the same?

Explain how a 0.0005-W laser beam, photographed at a distance, can seem mucqo4zqermi3mi1- 3 y8m h stronger than a 1000-W street 3iqmze8om-y3mr41 qi lamp at the same distance.

  Does the intensity of light from a ljd1 u-6a s4regaser fall off as the inverse square of the distancjge 6ru d14-sae? Explain. ----待选择----yesno 

  The neutrons in a parallel beam, each havi + wlxs+ywmldyb9n7vr.s39 1x ng 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 entg )5 97uo+mr2 p/-bl rcxpwa 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 measured to an am qvn4 lx,xh rnn-*0-kccuracy 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 stater ,p/nth ptrjd5hclxx+yyg07t7r;3- of 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 , 33ysj9- kwfo-f cx/m5q agzpin 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 possible energy of a neutr,rx o4-o : to7u 10clqlgcif)von contained in a typical nucleus ofo -rvfogc,xl:1qi47u 0tlo c) 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 princip g86joo np+r.vle to show that if an electron were present in the nu8rp ojno. vg6+cleus $(r \approx 10^{-15} \, \text{m})$, 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 thfw0ea9 s4(k2y0hahbxe position 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 thwgum b;gh5o,.t;/:im lxmld *rke03 se $n=6, l=3$ subshell?    elections

How many different statpx0a r81bxf;g es are possible for an electron whose principal quantum number is gaxf0bx; r81p$n=4$? Write down the quantum numbers for each state.

List the quantum numbey 7p6gh7i2) nhuatt:,-gm v 1a4 guxzars for each electron in the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for epj07 u xv)mmh3ach electron in the ground state of nitrogen $(Z=7)$.

  Suppose a certain hydrogen ** hy qqf.nxfxl8ip)l7sm 6*xk8x6 tnatom 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 mol,i)3emmg4r qx+;cv bmentum 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 has 5inx 5n;8 prel$m_l=-3$, 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 in 9ggn9mx:0hw6ywmj( gqm+ahfq1 6 z6pa "g" subshell.    electrons

What is the full electron coq qr,,cg l7rtd.y4u xb8e d j*vw.v47gnfiguration 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):74z4u cddfz s8a 6ysr v:ea8.y.h usk selenium (Se), (b) gold (Au), (c) radium (Ra)? [Hint: see the periodic table inside the back cover.z s:ca 4d 6ayykf .4 zu7s8herd8v.:us]

  A hydrogen atom is inl.dn 7kcv*g0/u ty0fui/ zb(u the $6s$ 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 thir rcrw;0( 0o6udm9dwibd electron in lithium using the Bohr theory. [Hint: Thw rr;i 60b(cddow u90mis 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 momeny 8u cssy8 -ajd99avt1tum is zero for a filled subshell.

  For each of the following atomic transi qptm*ly2q* ;vvoi x,9tions, state whether the transition is allowed or forbidden, and 2moxlv q*ptyv;qi* ,9if forbidden, 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 +lb8bqp waa;m c1uw/(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-rays emitted by electrons stqz1ycux uft. d6+ne-.pe;ai6n4 a/ htriking the face ofp + aqntc.ux ene1-t.uh 6 z/ya;fd6i4 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 X-ra1d8xpj+ cyy ob. 2oo :u)ku)bsy tube havekpy8y xoo1:+) . cudo)s jubb2 $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wawwz+ dia(:c8il;gv .svelength $\lambda_0$ 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 em)e y 56eqz92tu(gbfzzitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelen oih0; a nb5qlv,1dzj+gth for an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the wavelenm;r. xbo e 4fimeyi(7(l 3xce(gth 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 material is bombarded with electrons. The wat2+mvs b ew( lr((olp2velength of thtwr(peblv sl+(2( 2 moe $K_\alpha$ 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 retinas0fu*3wr7ye.lkj ld +v3 +xdzs 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 labml + j8vuic(j ac.c-q2 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 spread of a laser beam due to diffraction if h5rgdl , xsb6y1zz/x3 the beam emerges thr5x6rgb1h 3yz xl,szd/ough 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 wavelengh;b f 4 sg3xgl1v9e5wjth of the He-Ne laser?$\lambda $ =    nm

Use the uncertainty principle to estimate the position uncertainty for thbv6i4w5ag/c7 )ltp ume electron in the ground state of cltwu i5/pgm6vb) 74athe hydrogen atom. [Hint: Determine 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 momene +lttutbh jj: u4( 0xb+kv.j) 7s:cqe1o8syutum $L$ 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 forxi 2jn pu*y8*i the radius of an electron orbit, estimate the average distance from the nucleus for anu*8 ixpij*yn2 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 operatestvsfr(scl0h 6w*jd qww5-b ,* 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 413t3p pjav pjboron 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. Below) can be writtb16ta k:;u(f wnuzvuml/p2c, en $\sqrt{\frac{1}{\lambda}} = a(Z - b)$, where $b \approx 1$, and evaluate $a$. a =    $\ \text{m}^{-\frac{1}{2}}$

(a) Show that the numk vmek1,edoly nn (otx;w7 513ber 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 differenzlqxrl8y -jjlvg *d l* y7dlm6y*67g) t electron states possible for a given value of j dl)l*g6r-g m86x7vdyyyl7 qj zll* *$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV is shot through two na.qzh l(:(hz ) kg7jvok a;i vb66ptdc pe(5h5grrow slits in a barrier. The slits are a distanc5 .z(ctve(bgoq ag)phi6l:6jdz k7h pvh( 5k; e $2.0\times10^{-6}\ \text{m}$ 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 t6 vcox,z.xw8t he Bohr model and by quantum mechanics. Compare the res8 6w tczx.vxo,ults for $L = mvr$.
$L_{\text{Bohr}}$ =    $\hbar$
$L_{\text{QM}}$ =    $\hbar$ or $L_{\text{QM}}$ =    $\hbar$

  An 1100-kg car is trao mtn 72bg-8dhveling 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 line with a wavelength centered at 488 nm. Carefuxof-)42 jpc6 onref1c l measurements show the line is really spread out between 487 and 489 nm. Estimate the lifetime oc2 nf1pxojr)o64c-e ff the excited state that produced this line.
$\Delta t = $    $\times 10^{-14}\ \text{s}$

  Protons are accelerated from rest a3/e5zzi(iw l,jekpc9yh4+ e scross 550 V. They are then directed at two slits 0.70 mm apart. How far apart will the interference peaks be on a l 4s+ejwc3/y pz 9ezhe,i(5ik screen 28 m away? $\Delta y$ =    nm

An electron and a proton, each initially at rest, are accelerated across the 05x k0 0j nd5.jaudeuosame voltage. Assum0je.n ako05 duj0ux5ding 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 razbw wk/a- 0tny nirz9*8z9/bqnge from 1 to 6, how many elemenbyq znw/znr z 9b*-9k/0awit8ts would we find in nature?   

  If your de Broglie wavelength were 0.50 m, hm b61wco+ 3;egt2 dzmsow fast would you be moving if your mass is 75; s3e m1t26gmzdb+owc.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 unkn ),3klf7 ywnzbown element shows a series of lines with one out of every four matching a line from the Lyman series of hydrob),lk z ywn73fgen. 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 nm are)ct+f uamb9v * emitted from the surface of a certain metal when it is bombarded with high energy radiation. If this phoutf)+cab9* mvton wavelength corresponds to the $$K_\alpha$$ line, what is the element? ----待选择----nickelcopperzinc 

Show that the diffractive spreab v+s(9 uv: 6fj.ejygxd 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.]