Compare a matter wave-+95wy:wzktba ffi (a/r) hf x $\Psi$ 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 quaqlirtcxn0) r*zm2n c9a96 z r5ntum mechanics, particularly the uncertainty rztx9ra5 *qi6 mcn0r clz2)9nprinciple.

Explain why it is that the more massive an objrtabob rk*7+0;.cyu qiaeu3 9ect is, the easier it becomes to predict its future poy o;ib0 t3*9.+kcur aqubear7 sition.

In view of the uncertainty principle, why does a baseball sev) u -dy0/ymbgd zsqo7q51n-zoq26egem to have a well-defined position and speed whereas an electron do1 bng0o/yyqgzs qvzd )mu257dq -e-6oes not?

  Would it ever be possible to balance a very sharp needle precisely on its point14iz 1c6a z8uoi(meh d)ahes.? Explain .4 i1ae )i dazhoh8zmu16sc(e.

  A cold thermometer is placed in umjo .p+ian1d/) rw/ba hot bowl of soup. Will the temperature reading of the thermometer . )+ridu/wjam1o /n pbbe the same as the temperature of the hot soup before the measurement was made? Explain.

  Does the uncertainty principle set a limit to how well ygkj .wd;nrw8+ ou can make any single measurgwdjw+ .r;8nkement of position?

If you knew the position of an object precisely, with no uncertal6eu8uy5r co) inty, how well would you know its momy uo5lc)er 6u8entum?

  When you check the pressure in a tire, doesn't some a61q wt5 1fbmdzr7az 2y 6bc,bjir inevitably escape? Is it possible to av7cz6drybtb1m ,w bj 1f62 z5qaoid this escape of air altogether? What is the relation to the uncertainty principle?

  It has been said that the ground-state energy in the hydrogen atom +rk/2 sflekb/5v vsl+nq0,9/b ps3be wj r w/tcan be precisely known but the excited stak9wj /vwlrkleb /qs ,+ sb /nts03 e+vp25b/frtes 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 hydrogen je:jxw,o.h3r 5s e8wv atom, the Bohr model or the quantum-mechanical model, predicts that the electron spends more time near te5w3r v: w,xjo.shje8he nucleus?

The size of atoms varies by only a factor of three or so, from large* r/j:xsb j5rkst to smallest, yet the number of electrons varies from one to orbj k:5* xjsr/ver 100. Explain.

Excited hydrogen and excited helium atoms both radiate light as they jump 0d r+cd ow)*uz91zjhx down to the hoj1u9 d zzw+d)rx0c *$n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table look if there were nqi7 6tilcaf; 9o electron spin but otherwise quantum mechanics were vaf aq6 t97i;ilclid? Consider the first 20 elements or so.

  Which of the following electron configurations aregbqo3 m.o s-r, 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 fo4(toehac p8l 1r a uranium atom (careful scrutiny ac p8ota1lc(4eh ross the periodic table on the inside back cover will provide useful hints).

In what column of the periodic table kc oibw7p v:rg4j )-2i:8 .2hdxwttldwould you expect to find the atom with each of the following confpcwo).r 7xt2 vj2hklt::iid -g4wb d8 igurations:
(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 siminz v twz8tt*-+lar properties?

Explain why potassium and sodium exhibit similae-pz.zab4 drnmd.+ y 8r properties.

The ionization energy i+zzgzqn*v -7for 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) p we9uzvmhgd07 00h4q 51qglj imply a photon nature for light? 500wuj7d0mlvzqqhp g4 h9e1 g

Why do we not expect perfect agreement between measured values of X-ray(-n sil4 ec.m+ dbai1t line wavelengthstbsi(n+1 -li4mae c.d and those calculated using Bohr theory?

How would you figure out which lines in an X-ray spec l sz-ks:9b3c5knjynxwum)*wt3d2 , ttrum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transitions u12+dra r yssog:l 4h(apnw:7deep within an atom to produce shorter wavelengths than trana1 saw l(sg 4dy:pnu:o7h+rr2sitions by outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ frxo ziwc0jf7a2t. 3oz- om ordinary light? How is it the same?

Explain how a 0.0005-W laser beam, photographed at a distance, can seem much wwq:,5p ilzgi+ut6:wel +f*n stronger than a 1000-W iqw:+5t*:gliu w+fpw6,e nl z street lamp at the same distance.

  Does the intensity of light from a laser fall off as the:vn u8 st.10f/ l;ulj/rfad nr inverse square of the distance? Expl0/ /u unrr.8fj; at1 nvdsf:llain. ----待选择----yesno 

  The neutrons in a parallel beam, each having kinetic energycjuhy f.3)n k( $0.40 \, \text{eV}$ 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 apgd: ; yxv94yb 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 pv. s/ks75 tmkbe 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 foroy /e0 cwnzl ghc4qe) :f,4;mg 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 muo/6eh0yt h kdb.:i89(tu pce cf7khz(un $(m=105.7 \, \text{MeV/c}^2)$, 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 poss1rxgnvzuhfyz-pr9:-j* 5 n* wible energy of a neutron contained in a typical nucleus oy:n-xfh-zg 5 1urj zwr*n*p9 vf 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 tobm*ll3tyl 4nar 8a(7h show that if an electron were present in the nucleus *7(tabhy l l4am8l3n r$(r \approx 10^{-15} \, \text{m})$, 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 posl0evu3y1/5r e jzn(qm b+s f1pition 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 i oly;y040;ewt gex* nbbuyucu l: ;27en the $n=6, l=3$ subshell?    elections

How many different states are l57d6x3ua;5:.dtld r9 ngidfivsha:possible for an electron whose principal quantum numberria:.d:h 39 t5a; fxgn 5i6su l7ddvdl is $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each electron in the ground stw ;b.(jsegp5o ( . h (erlkdevpv.6u9wate of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each electron in the ground state ofonmlc ya+x1;qo xnol 2g:: h)( nitrogen $(Z=7)$.

  Suppose a certain hydrogen atom haq-v; 5unvm21tp td f7:9tud ais $l=4$. 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 eb7qu(n1 j kfbn9.cthiu -kd;1lectron 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 haskmdm;n*f ; (ua $m_l=-3$, 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 a "g" suxujg9,i7ka )tb+;b37id 8rf.neucltbshell.    electrons

What is the full electron configurat ta*jbmyos56f86zlvs9z hsb f +o9+(yion 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 configubuix+;vkc 08 cp:- d7hez6 hktration for (a) selenium (Se), (b) gold (Au), (c) radium (Ra)? [Hint: see the periodic table inxc+ ;e7 0bdkcpv hikz -6th:u8side the back cover.]

  A hydrogen atom is in thepk3 ouwlm7vi6ge4s0n2yo. v0y xi,7 x $6s$ 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 thnk9z+h w)ey5b e third electron in lithium using the Bohr theory. [Hint: This eleb)kw5ey z+h9 nctron 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 moueep *:. vau7/ aj7fulmentum is zero for a filled subshell.

  For each of the following atomic transitiz6p2t0hs5cve ons, state whether the transition is allowed or forbidden, and if fh0 6tsc ezv2p5orbidden, 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-clu0208zu . g 3y zqbtjuw)fj 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 strik d7w*aj w:zcs*gl(uo) ing the face oa u7w(dcsg*zl) *jo: wf a $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremsstrahlung X-rays e/o w()yv3dgk7yw so )imitted from an X-ray tube have dw v)o(3ywk 7/)gyosi$\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wgd52qwswszzfwk+v 1- k /jn3r8f mt( 9avelength $\lambda_0$ 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 waveai moh91q akcfpi6 /wdlqyt+r5/3p08length emitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength for +h,; okiiez e.an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to esti jo3qu:cf*(ucmate the 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 material is bombarded with ele*py xpg::gp, ip-*vadypo6 92 wga2fyod 12g octrons. The wavelengi*2 ayvx9pp6o2:pg*p -p:g ygowoyd 1 f2d ,agth of 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 retinr 2ap+y2++v kxzserf3 sn: sz*as 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 9i3yf8 a1vkhu 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 -4t:p, 87w yk sk080qnv,gg cai uuo8:xkcbecdue to diffraction if the beam emerges t0: ick: 7cvx8ncag4os ,uy-u,bgk tkp08 qw8ehrough 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 qa 2 ao85bsc;51 ezenc8csd,b He-Ne laser?$\lambda $ =    nm

Use the uncertainty priz97b1u w zfpzz19 9y 9b ogxax++/qfaynciple to estimate the position uncertainty for the electron in the ground state91f oxq79xzp9 yba1fa+ y+9z w/ugz bz 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 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 s+ma *z* bp)za(:+hrs9j)swsb hex2z,v/ bj u 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 foip pjmx q/q m,n9h3/8pr the radius of an electron orbit, estimate the average distance from the nucleus for an electron in the innerpnmqp qp//39ijh8x m, most $(n=1)$ 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 operate 4su8 bs b,ii*sc*.wzms 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 boron *ioux; e 9 kk s4 vo/tv9gjmbm+x)/dl.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 written .svy nu 4zpo+jj ;);yg$\sqrt{\frac{1}{\lambda}} = a(Z - b)$, where $b \approx 1$, and evaluate $a$. a =    $\ \text{m}^{-\frac{1}{2}}$

(a) Show that the number of different states for a-e f,;hcgn(4)lwyc 2q d6gea3rs v(pc 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 differeno(h ap p2hfqq6,i1,;i 9j vzqcn-w h4jt electron states possible for a given value of w ; z-cq vqjj,ph,q6ha2np1 o4hii9f($n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV is shot through9we xp5m+n8ijzcyj 72 two narrow slits in a barrier. The w95yejj 7p2c8znmx+i 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 p+pig2ag 6iqvv, /f3h(l+ml m is given both by the Bohr model and by quantum mech 2gpvmp/ ,ilvq 6lhm(gf+ia +3anics. 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 n(p * jm-62ttcp r.pl*+en zjza 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 n nb -x(a-ki2yzm. Careful measurementby(xan-i 2z-ks 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 restv4 ms :( :bafe.wbw vch./v.cj across 550 V. They are then directed at two slits 0.70 mm apart. How far apart will t: b.va4m.ewccv/:j sh(w.vb f he interference peaks be on a screen 28 m away? $\Delta y$ =    nm

An electron and a proton, each initially at rest, are acceleratu0pby(q(vjp .ldwk)ah ) ,5j ued across the same voltage. Assuming that the uncertainty in their position is given by their de Broglie wavelength, find the ratio of the uncertainty in thej ujw v )p0,(dhub)p.ka5(qlyir momentum.

  If the principal quantum number n were limited 56d7ck xg60-lpgna d 5q h6vabto the range from 1 to 6, how many elegv dabda0n 6 cqk5xpgl66h-5 7ments would we find in nature?   

  If your de Broglie wa6j)wgc 5be xr4velength were 0.50 m, how fast would you be moving if your mas x 5)c6r4ebwgjs 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 shows a seri p1hs: qdh (0evlg-3otes of lines with one out of every four matching a line from the : pq3ht0gvodes h l(1-Lyman series of hydrogen. 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 emitted fro3r e xf0;5bzgdm the surface of a certain metal when it is bombarded withgd rfbxe35; 0z 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 om .mdjx*l/i)kb fr*l+ f 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.]