Compare a matter wave8e rcx)wdoafu q8 ((i y1i/f)y $\Psi$ 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 + 6,y.q+vp*knpm wyxz with quantum mechanics, particup*pyqmnyk, vwz+x.+6 larly the uncertainty principle.

Explain why it is that the :ue xsnb()x7h w1uv* pmore massive an object is, the easier it becomes to predict its futu7xn u *e)bvwp u:(1xhsre position.

In view of the uncertainty princrom:g g4tr -he1 *c+heiple, why does a baseball seem to have a well-defined position and speed whereas an electron does not?h1 rgm tech*o-re+:4g

  Would it ever be possim+g; +ft3d2sky. 8uwkjf;e es ble to balance a very sharp needle precisely on its point? Explas e+kk e8.;;jtg2yfm wd+3fus in.

  A cold thermometer is placed in a hot bowl of soup. Will the temperature reas0v(ki *km li+ding of the th0msk(+ kiv*li ermometer be the same as the temperature of the hot soup before the measurement was made? Explain.

  Does the uncertainty principle set a wix5x by (g7s.limit to how well you can make any single measuxwigb7sx(. 5yrement of position?

If you knew the position of an object precisely, wiryw nsg.-f / qobwm4*7:,gll dth no uncertainty, how well would you know its momentum 7g dfb ls:yg-w.lr4w m/oqn*,?

  When you check the pressure in a tire, doesn't some air inevita(3wye +5v jltelmj -b.bly escape? Is it possible to avoid this escape of air altogether? What +e vtw-l .l5mjbj3e (yis the relation to the uncertainty principle?

  It has been said that the ground-state energy in the hydrogen ato.4x)fw kv7 cdjm can be precisely known but the excited states have some uncertainty in t7 dvfc. kj4w)xheir 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-mechanicoob)6v+ju9u ez v gr1(al model, predicts that the electrorog)1 ujoz6+v ev9bu(n spends more time near the nucleus?

The size of atoms varies by only a factor of three or so, from largest to s:sfkubr u9q-d9u vl9r .tu 3l:qg q7k/mallest, yet the number of elec b: r:gu.3 /vuulur9dq-7kq9lk qfts9trons varies from one to over 100. Explain.

Excited hydrogen and ex/ zup 7b9f zrtp,0p3qdxe3h6acited helium atoms both radiate light as they jump down top 6t70 q e9uzapzxhbr/3,f3 dp 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 electron spin but otherwisk5a aav8(+l 5 dy q3bmvri+zu(e quantum mechanics were valid? Consider the first 20 eleme dkaqu5r8(a v+z+35ailvymb(nts or so.

  Which of the following electron configurations are a+nigroc)h:- -( rvz 5uxajik- v,( klnot 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 n tzo7)lclv 2x5b*r b-(careful scrutiny across the periodic table on the inside back cover will provide use*bllrbtocv7 n z5) -x2ful hints).

In what column of the periodic table would you expect to find the atom wit(lr/fsfa3x:j.q7/erxkt 9m vr 1y6 lxh each of the followy6/ x3(:tr xlx m.71vjfksaqrl/e9r fing 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 similar pr2 szwnw5 s5b:uoperties?

Explain why potassium an8a ut/j0 xgehn/ozm rsk -:/+xd sodium exhibit similar properties.

The ionization energy for neonur v)tw*j b6 -)ssbc:f $(Z=10)$ 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 light? 0ze.++3ej s0b .xusgvzmu q)eez0)j g+ .es3bumu seq+.v 0xz

Why do we not expect perfect agreement between measured*vr e :yxidsy:*a5 acn4-vn:g values of X-ray line wavelengths and tha-iayg:x4 n* *ced5vys r vn::ose calculated using Bohr theory?

How would you figure out which line pe;4zdez:d 4pdhp -s :x5jwm8s in an X-ray spectrum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transitions deep 0ab(1fpta)gx z 1(1gjmgvo 4a within an atom to produce shorter wavelengths than ta gm((pa14gvj)1 0txaz1 bogfransitions by outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from ordinary light? How is it the same? ovi)lut6 y 0rh932evn

Explain how a 0.0005-W laser beam, photograp5nim rst4; 3ui.tm0 odhed at a distance, can seem much stronger than a 1000-W street lamp at the same distance4m0 5dso.3t; nriutmi.

  Does the intensity of light from a laser fall off as the inverse squajs kf7rvk);6h d370nnr mq)at re of the distan7sn h) r;nf630kvajr)d qt7kmce? Explain. ----待选择----yesno 

  The neutrons in a parallel beam, each having xxjra3 du32,ckinetic 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 obhmul/,az sb 2q0 ea,1*sqfo)f $(6.560 \pm 0.012) \times 10^3 \, \text{m/s}$. 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 to8va oj.g/1rd b 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 for typically 1ts *zz0c2epnxg5r4h $10^{-8} \, \text{s}$. 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 uncertaintpfp1oipgy )1 gs*s(l;g ) rhu,y 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 possible energy of a neutron contained inta8w *:gm.wxa a typical nucleus of rxamaw 8g *w:.tadius $1.0 \times 10^{-15} \, \text{m}$. [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 ey 1mnq7h s7q9okz 13omlectron were present in the nucleusmm73o kn7 ys 1qoq9h1z $(r \approx 10^{-15} \, \text{m})$, 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 wrt:-t tt+6aag2pt v-l0ne p :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 c ag lwsp.rkj-1l.*o e9an be in the $n=6, l=3$ subshell?    elections

How many different sjg(8t*jxn6;p 6ua i kxtates are possible for an electron whose principal quantum numberpkn *86x; juji 6tg(xa is $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each:5tfddu2gx s7 electron in the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each electron in the ground stz7/auxww2/2ymq 7 m jsate of nitrogen $(Z=7)$.

  Suppose a certain hydrogen ae+ ikr 03c7waqoh8x 5ptom 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 thj/ hpq th(.0cj 9tos5ue $n=4, l=3$ state of hydrogen.
L =    $\times 10^{-34}\ \text{kg} \cdot \text{m}^2/\text{s}$

  If a hydrogen atom hp( r297egtpfmas $m_l=-3$, 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 elecsfd1 n hnhih4;0(f 4lntrons in a "g" subshell.    electrons

What is the full electron configuration wj, 7q4ge m:xeg(zw1 wg.op7ma 3(iv,dbzfk/ 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) seleniumh.cm b2(g)hgziofll)0 8rgb. (Se), (b) gold (Au), (c) radium (Ra)? [Hint: see the periodic table i0zbf.gg )82ho.(i rbhglc) ml nside the back cover.]

  A hydrogen atom is in the x7 ck n/-p*qm(p*l fcb$6s$ 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 tht2id l)ej4vbp2 ) x9ive third electron in lithium using the Bohr theory. [Hint: This electronjvti22) b ved )p49xli 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 subshel l)skyyrwj1tzwsvjlsv**8 *i2 ;fu. +l.

  For each of the following atomic transitions, state whethe2c q9h6uq(1wnsv c2bh-m3tp r r the transition is allowed or forbidden, and if forbidden, what rule is behup 2(- rchnmb 16tc9vq3s2wqing 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 ue2yt wfr 1x6g3h+e4 va $6d$ 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 striking9mmjzpk5 p8d8tt+mz.x0)n q k4*dl co the face of dkcx+jp)tpl*on8q m8t5mm z0d 9z. k 4a $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremsstrahlung X-rays emp f/9pvq3 a)evtj.)bkp7 ecv2itted from an X-ray tube have f3c7vp va)t jeekp)pbq9/ 2v.$\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wa6anewk(6nhfsm+4 6s pi4 k0fvvelength $\lambda_0$ 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:./7ipewiogxh1; je 8ecbpv m,*y; uh when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelengt+.(/sp cu 6llt/2h,ljpet ox dh for an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimaten-pepv(isrh:7h53oz 9 j7 f rnep5 t/e 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 xgdd6e x uw964an unknown material is bombarded with electrons. The wavelengx6ddxe9gu6 w4th 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* a ;,tjx+g1xq u.szx /;t8gqhrtp jk1 detached 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 physics lab mif /vo:u7q( ly(iln ,xqght 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 s o*7f28;; pfgek rkw,fm7okbxpread of a laser beam due to diffraction if the beam emerges through ae87por kwg7f2kmfox ;,fb* ;k $3.0$-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 x,,w 5,jqvlw-m gvyvg+8d uho(u* .zqthe He-Ne laser?$\lambda $ =    nm

Use the uncertainty principle to estimate the positp3 jzkr*tg hl2ax)klr4qui 9ry807s7ion uncertainty for the electron in the ground state of the hydrog 0943 iht gr*y7psqzx8rj)rakkllu72en 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 momed oel*jrts79z5 o;,cxntum $L$ 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 electron orbit, esti r05wya pz5y9d:jhj-smate the average dis5jy ps:50 d-ayrzjh9wtance 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 xm s5co6hf :zbb5/(zjt $95 \, \text{kV}$ 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 )hb2fhhvo;s/jq gi3,) rhmw7 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 th neq kzagdg(vw.k+r ;26ia)p 5e 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*ll(h 4d; xwyvqf..xikt1v318piu y)p1 fvf t 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 possible foro tid(:nh,6 hmhupe)udzp7*5tfe, 6w a given value of ,5fh en*:p )dt,d emt u6h7h6w (izpuo$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV 28 gb+5dkuinb is shot through two narrow slits in a barrier2 idb85u kgb+n. 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 givox - qtc1ot ll;hjc:4(en both by the Bohr model and by quantum mecht1;4 cj to:xlc-lo hq(anics. 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 of vdcc atr7 p,635 (rejn$(22 \pm 0.22)\ \text{m/s}$. With what maximum accuracy can its position be determined? $\Delta x =$    $\times 10^{-37}\ \text{m}$

  An atomic spectrum contains a li3hc ni97li dgc k(xu*/ne with a wavelength centered at 488 nm. Careful measurements show the lin(7g h9i/*kui3cdn lxce 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 f6y*zd bw he+z9 kriz-z4ei;t31w aeqs8i+ 1dc rom rest across 550 V. They are then directed at two slits 0.70 mm apart. How far apart will the interference peaks be on a screen ie br86-eaw9qz k;4 sy 3 d+h1*d zicitze1wz+28 m away? $\Delta y$ =    nm

An electron and a proton, each initially at rest, are accelerated acrc /zf hi6cw3q 1srqj- ,qt.8wross the same voltage. Assuming that the uncertainty in their position is given by their de Broglie wavelength, find the ratio of the uncertainty in thesz6i1t hj rc3-q wcqq .,frw/8ir momentum.

  If the principal quantum number n were limited to the range from 1 to 6, hoyy(ootnmi7; ;m6mc 0 nw many elements would ot6ci nmy;m;on m(07 ywe find in nature?   

  If your de Broglie wavelength were 0.50 m, how fast would you be moving iff i7ta x3+g g;savnhz-g07.vr your mass isa+z;vvxs37 g f7gag i0.t hrn- 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 spectrump35nh. 2;tbvu 2odgoi of an unknown element shows a series of lines with one gv3phnob d;tu.2io52 out of every four matching a line from the 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 from the surface of w34 q: mji-uezd)ja9ha certain metal when it ei:9hw mq3j-da juz4)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 lasef gcr u1j h.y*y+96oqdr 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.]