Compare a matter wave or;44dy*jzxc$\Psi$ 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 quxd; lq,8pjdk4 antum mechanics, particularly the uncertd;djx qkl84,painty principle.

Explain why it is that the more massive an object i5d(-fps tvac1 /c pz.es, the easier it becomes to predict its future positi1t( fcz/ec.5v dpspa -on.

In view of the uncertaintz1zt iq 5ec)8ty principle, why does a baseball seem to have a well-defined position and speed whereas ait8tzqe )51czn electron does not?

  Would it ever be possible to balance a very sharp needle pe/k +0aui7plv/6a4y nby/l(k/f ytj erecisely on its point? Expla k+klan0 4//b/yvuy7ti e6 f/pjylae(in.

  A cold thermometer is placed in a hot bowl of soup. Will t z7d wgrwo1+/dhe temperature reading of the thermometer be the same as the temperature of the hot soup before the measurement was madewod r7z/d+wg1? Explain.

  Does the uncertainty principle set a limit to how well you can make ani7ci-j l.1 jksy single measurement of posit.s7ic ljj1-ki ion?

If you knew the position of an object precisely, with no uncertainty, how well hi;h+gjnz 3s ,bnba+: would you know its momentub ,3i+ha: znbjsg;n +hm?

  When you check the pressure in a tire, doesn't some air inevitably er)lolt5ae1a.v1b z3a drk;aq9m4d ljoq 4 71escape? Is it possible to avoid this escape of air altotz lo5ael1aakad4bq3 qve9 m1j.7lrr);14 dogether? What is the relation to the uncertainty principle?

  It has been said that the ground-state energy in the hydrogen atom can be (1vb6rel6xrf precisely known but th6l(1r ve xbfr6e excited 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 hydrogen atom, the Bohr model or the quantum-mechanical modelxx ur, u*hve5b53+niz, predicts that the electron spends more time venx+,ir z3u h* 5xu5bnear the nucleus?

The size of atoms varies by only a factor of three or so, from largest tox 47 4fqz)mpng*pi uk.p -i9pv smallest, yet the number of electrons varies from one to over 1ik 47)9umpnp fgiz 4pvp.*qx -00. Explain.

Excited hydrogen and excited helium elyx8 e0, )gdyatoms both radiate light as they jump down to y, lex0)dy8gethe $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table look if there werbln/k)ze i3;i3 xzw: 2s1uegx e no electron spin but otherwise quantue z1)lkzx: 2ibesiwg33/ n;x um mechanics were valid? Consider the first 20 elements or so.

  Which of the following electro(n ulte5igf07 43l+79ppmt k r cw4wixn configurations 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 configurat2 wl 5bk oczyb c;ndu-*ry/r)9ion for a uranium atom (careful scrutiny across the periodic table on the inside back cover will pro-czk/rby 5*; lc)y9dbuwnr o 2vide useful hints).

In what column of the periodic tabotch.-q ww7 x6le would you expect to find the atom with each of t7 o -wqcwxth6.he 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 exhibit siw: 1.gvk l3(v ng:g1nhz+ssjimilar properties?

Explain why potassium and sodium exhibit similar +ptusat )hn l 2qx(- 6*mp+ich4gx3i6rf, koi properties.

The ionization energg4rfaa3 ;ns 9whm0v1f y for 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 photonh9(y7+z lm9s vwf.emrxz1/rw nature for light? (m9w1hlryz wmv+/s xfez 79. r

Why do we not expect perfect agr4gcum xfkwn,e5vbe,q 4 yg67 6eement between measured values of X-ray line wavelengths and those calculated using Bohr ,m46,7 ygev f56cwx 4eunkgbq theory?

How would you figure out which lines in an Xoq 3kq*j 1h- 1a-f(j m6/ /frocgcfltu-ray spectrum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transi mjzmzbo;y-5 8*gcyi2tions deep within an atom to produce shorter wavelengths than transitions by outer electrjcybzmiyo;5 8 2m-g z*ons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from*+vdvz:k oq6 j ordinary light? How is it the same?

Explain how a 0.0005-W laser beam, photographedp-2 5laisb3)r8u1z5q gv qrxs at a distance, can seem much stronger than ssva)82zpg ru1lb 5ri3q 5 -xqa 1000-W street lamp at the same distance.

  Does the intensity of light from a laserzjdf6;)boy -wkv d7ono,0ju lk9 )y8c fall off as the inverse square of the distancec-zu) lb8y )wj7d ,vk yd0o;foo69 knj? Explain. ----待选择----yesno 

  The neutrons in a parallel beam, each having kinetic energynqmo0-tay3h 51-r inh $0.40 \, \text{eV}$ 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 travelinga)0 k4 y((r/ff 3en4 b2ydsghmyu mp;d with 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 mez1ek mb.8ta8p asured 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 at d/muprydb aw)i4s7/ 6om 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 uncertaint8+4xq 5t4oe; usqtajqy in 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 possiblep cyw;4uhlm *( energy of a neutron contained in a typical nucleus p4 hul(yw ;cm*of 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 the nph,bljrin19j tsj ,tmw 2 ,9l8ucleus jh lm2 ,ljs8 n w9t,9r,ji1pbt$(r \approx 10^{-15} \, \text{m})$, 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 position10k+ vx7l o .ruznh(pv 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 bqraoqo3(r wj8ab(;r.w k-pc 2e in the $n=6, l=3$ subshell?    elections

How many different states are possible jau98z. n i4iva7hj5v for an electron whose principal quantum numb j aha49ujvi5v.i87nzer is $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each ec0;q)e*ca r(:m, 1t9anrj.i.xwfsird v 8nah lectron in the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each0s bee k(9-dba ltopf62zl92 s electron in the ground state of nitrogen $(Z=7)$.

  Suppose a certain hydrogenayoi) ,b0y,umn2,wcq 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 on)662td ( 6txzfd z yqt)nxz3mf the angular momentum 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,e 78dt), :rhu ho.(l -n )xjjlmo6xkcy frs5c $m_l=-3$, 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 ele fw;e:m gd5mbum76q1bctrons in a "g" subshell.    electrons

What is the full electron configuration in the grounx 57a )q fcfylz5 hltpb/k4e6rf*y4g* ys7 w0sd 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:cm(, g*(2i3xzh3n kk 1 wesybbjb6eh), (b) gold (Au), (c) radium (Ra)? [Hint: see the periodic table inside the backnk6kmye3cxsw(3b ,2 z:e(h g1ij*h b b cover.]

  A hydrogen atom is in the )ejrlqbp d)2 d7d)x ,y$6s$ 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 in lithium using the Bohr theru /nxw;b0oj kpau.q ,,, yxih 6ggj 5lva:y46ory.q4pj ilu; nughxa5,6jr.,gv6,yw/ 0kbao yx: [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 momentu35zpeawubq.q:q+ f x,m is zero for a filled subshell.

  For each of the following atomic transitions, state whe,woxe)078d bn ;hrsgkg 5c pzy+g-)evther the transition is allowed or forbidden, and if forbidden, what rule is bevy wk ))e0bso5cg8,dr;+xng7z gphe- ing 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 i1-qpn)zvx2kz :l lfu+ n 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 strikjc, rkkw/d 51 fygq-c+ing the face of arjcfgck1/w+ 5,yqdk- $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremsstrahlung X-rays emitteqb oo7mg-7:qtwj1i1h d from an X-ray tube havetg7o: qmo1ib1j -hwq7 $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff9in2vhv197 tgxcx hq ( 0knk8h wavelength $\lambda_0$ 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 emitvim5)hfazf;24m qg3j ted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength fr,z w3smo) 8zror an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the waa/6zw yg8m6u-o tzp s56hdz ,vo,xvc +velength 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 i a0ytl8 c:n,ucs bombarded with electrons. The wavelength of t:c0t8 ,l uanyche $K_\alpha$ 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 puts- 7)a dwz8 69uuffrfyp 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 mightpmxa4 ja58.b jl9m9qc 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 i rqhozi. 3els(h0/ (vvf the beam emerges(3v0is(hl/v.hrze qo 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 tvze(ed dx dcr qv,ur84h1oc3n:/4tw , he He-Ne laser?$\lambda $ =    nm

Use the uncertainty principle to estimate the po * u 32e(wta2xsxp) p1tf.z+d3uieuqnsition 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 anywhs)iuw2un2 (+xue f3eqpt*dz 1tpxa.3ere 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 an i rr0r+di7-wtc7c0rwd 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 electronkwe;d6 sud5 m4 orbit, estimate the average distance from the nucleus ddm 56u4 ks;ewfor 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 operate0bd;s:isf;vr 5,7df m.taknq s 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) energyq 97 /z5w45hvbqpr aab of the outermost 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 u gkdza)us:z v8,jqiqj 2(37gshow that 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 number of different states for a gqu(k.umk5 w +r5sg d;xmp2tq c g98ks7iven 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 elecz 9wtnekg3jn .pn8*1wtron states possible for a given value of k8npz9g n jw* .31nwet$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV is shot through two narrow slits in a bar/3) elv8ctptzj 8bx m5rxv 3lt /cte)5 j8bm8zpier. 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 7e;9ugg8w*mkxu mq 0d r)t:wk is given both by the Bohr model and by e: xt09*urkg)mwugm78;kdqw quantum mechanics. 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 speed of3ct4lv,wu : x01i qw q65lbdpi $(22 \pm 0.22)\ \text{m/s}$. 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 oup0gqf t(c4h8 1,tbtat 488 nm. Careful measurements show the line is really spreac8 1otf,b 4(g ut0pqhtd 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 across 550 V. They are then,tl*ls,ej+ pjgd5: .u 6uwvpg directed at two slits 0.70 mm apart. How far ad,wp*5 lju+v gjt, psg.u6:lepart 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 across thent pw9lc tgee8:tw(, 7 same voltage. Assuming that the uncertwe( g8 9w,t:lntcep t7ainty 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 th+rfe;td3/c ka+ (k5biv -c l.kctsfn;e range from 1 to 6, how many elemkif+ne.rk -c sbk+;3tf/ ctav (c5d l;ents would we find in nature?   

  If your de Broglie w i0 41yjb) hzf1,j4hobtp h3zoavelength were 0.50 m, how fast would you be moving if your mhtbz1 p,bh 0)hzi4j3 yo14fojass 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 unknoz+s *.v c3 h,v s3lctofa/54deieu0jg wn 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 elemea3igo. se+cl0t,s evzu/3v* j4cd 5fhnt 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 from ti81b: 99sja bfut5ajj he surface of a certain metal whe jsb985a:9u itfb1 ajjn it 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 spread of a laser beac/h9,ldgfq k+g4i +mjm, $$\theta \approx \lambda/D$$ , 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.]