Compare a matter wave h /si q,kd224s qao0hy$\Psi$ to (a) a wave on a string, (b) an EM wave. Discuss similarities and differences.

Explain why Bohr's theory of),u o e7icbb )h8s/gzp the atom is not compatible with quantum mechanics, particularly the uncertainty prig8h bbu ) ciepo/zs,)7nciple.

Explain why it is that the more massive an object is, the easier it becomes to bu ezus n5-+-yw /ihe:predict its -ibe/-szy:n u e +h5wufuture position.

In view of the uncertainty principle, why does a bas ibiqv:**dy9rmz74 rn eball seem to have a well-defined position and speed whereas anizr7i:4 q*bnvd9 y* mr electron does not?

  Would it ever be possible to balance a very sharp)b -eo,zcs:etkb q+t9 needle precisely on its point? otk-eze:bb +)9 s,tqc Explain.

  A cold thermometer is placed in a hot bowl of soup. Will the temperature readingrwfm3.t(tqf(a3u . xt of the thermometer be the same as the tea tfmt 3qr( ufx.w(.t3mperature of the hot soup before the measurement was made? Explain.

  Does the uncertainty principle set a limit to how well you can make m*. qqz) 8 v+o81rsefinaqd9c any single measurementdqe8vfn +q c.q 8zsa19)i*mro of position?

If you knew the position of an object precisely, with no unv0ar8n 0wp mq 6m;u**b 9yrwmfcertainty, how well would fuwm08y0aqv nmw *rm;6 9r* bpyou know its momentum?

  When you check the pressure in a tire, doesn't some air inevitably escape? Is gn,yuynt1 pp4, rvo((do8o+kqdznq .v 1a8x+ it possible to av (kndnqgx1+o8z8 vat.o+ , pou,4q1p(nryd vyoid this escape of air altogether? What is the relation to the uncertainty principle?

  It has been said that the ground-state energy f:sfdh :i9y1 tin the hydrogen atom can be precisely known but the excited statff:h :ydi1 9tses 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 ato5k pkqfs2ldt5+f )m 5im, the Bohr model or the quantum-mechanical model, fis)f 5k5tk +5mld q2ppredicts that the electron spends more time near the nucleus?

The size of atoms varies by only a factor of three or so, from larges;8mf9gbcty*xk eg 8;w ,e3e irt to smallest, yet the number of electrons varies from one to over 100. E8f8gkxb e*r9 mi;twe;g c 3,yexplain.

Excited hydrogen and excited helium cjp m-ztr.2(c* st1zuo)dns- bd4f,watoms both radiate light as they jump down to *.ds)cdrj-o2z zbf -,tmc tsu1 n4w(pthe $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic tali(w :sep( /rjble look if there were no electron spin but otherwise quantum mechanics were valid? Consider the frj ws/i epl(:(irst 20 elements or so.

  Which of the following electron configdm4z+k:njbr yn9.b -durations 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 atom (careful scrutiny zs;h dn7/5vb f 9u+vn au7c9nbacross the periodic table sbb9z+;c / ndu 7nvan79hu5vfon the inside back cover will provide useful hints).

In what column of the periodic table would you expect to find the x bk6l358heqmgd cn+2j8ckh7 atom with each of the followingk 3n2qbcdk6 j 5x8g8h+mel 7ch 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 exhibtb0lep zl5 )z0p/w ,ywit similar properties?

Explain why potassium and sodium exhibit si* :it*+6r5 uptrtblvc milar properties.

The ionization energy for neo 7 ub(1(saemtjj 6hhy2n $(Z=10)$ 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 f7 muvbdv)o tk57m9ji 1vt4/h oor light? otmhb5iv7udv1o/ 7mvkj4t 9 )

Why do we not expect perfeczkx yh-irzgla )f10l7 p8 m f9oyw6:l0t agreement between measured values of X-ray line wavelengths and those calc:-1lla fwxl oi 8fzzk6y gp9ry0m0h)7ulated using Bohr theory?

How would you figure out which lines i1pl uve;a-3k pn an X-ray spectrum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electronz:ukp*bjt+ fpsulm z1v6 s+78 transitions deep within an atom to produce shorter wavelengths *8++jvb:71pfpk ltuz6 s uszmthan transitions by outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ fjh 9: z:2 j5memqwga::ww2iylrom ordinary light? How is it the same?

Explain how a 0.0005-W laser beam, photographed at a distance, can sej u*y89uxz3elroq kc3w9h2u ;em much stronger than a 1000-W street lamp at the samzju *w29qo 3y9u 3euc;hx8lkre distance.

  Does the intensity of ligh9oqn1h:rmu lh tgjr4xe,40q*t from a laser fall off as the inverse square of the distance? Explain. 9qhg 1 0rnl4jm*tqo4e,xh:ru  ----待选择----yesno 

  The neutrons in a parallel beam, each having kinetic 0f 8 k:vqblv6v 2h9tzxenergy $0.40 \, \text{eV}$ 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 speedm3akh nqc*8z mv59/j/ dwue -r 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 t ib3uy+ w0,-qagey 4zdo 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 typicallyv9-wtb dk, 7ca $10^{-8} \, \text{s}$. 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 m(85a.j5zw hz mwc 3tmnuon $(m=105.7 \, \text{MeV/c}^2)$, 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 contaie+;k4 ngalxq a:ps3iys*q ,lx-uyt:1o( )xzkned in a typical nucleus of 4pe y uk)xlsz-q 3sx+aa:;(qki*y1 ,ogltnx: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 tbjl0r7q(u 6jxsp/x- bhat if an electron were present in the nucleusqb0/ubx7ljxpj 6 (-rs $(r \approx 10^{-15} \, \text{m})$, 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 posoa df,4r +oxffc;,j3 h b2-u1 /ljgjypition 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 bk iqw)c+iiuo;5 e: 3zbl5rk,w e in the $n=6, l=3$ subshell?    elections

How many different states are possible sd1/vowuc * /ngd:: i3n8tncl7 th.5ha weu.v for an electron whose principal quantum number is t n8ti 7g1 :3ehlw*ova.uc: //h.wsduvdnc5 n$n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each electron in).ftkhc7 . utv5pi k: qe7emy4 the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each electron in the ground state of nitrog2 7++ss w uzmbltgpa(,y(3uxjw . s,jken $(Z=7)$.

  Suppose a certain hydrogen atom has r, sa+r iczh tj*rtf+mw3 yvy 4pty8x.6e/ 6v:$l=4$. 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 mj 93*qebxln3 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 ody4x6d y r:3mv/lzdm oh jf.:1y6i:r$m_l=-3$, 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 electrodrw ck:l.jb ,)g08qbsns in a "g" subshell.    electrons

What is the full electron configuration in the ground state 2h) uefp+za8yfor 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) g(*jp44( s7s x :y9o v (gqeckpylkt4etold (Au), (c) radium (Ra)? [Hint: see the periodiysckt*se 4etg (4lv7 p: jqxo(yp k94(c table inside the back cover.]

  A hydrogen atom is in the)/ 8w df19ym3omedrre $6s$ 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 thirddrcrb i :n*3kr7 zin ,,q4n)lk electron in lithium using the Bohr theory. [Hint: This electron nd nnk)b qr:i l*cr,kr4,3z7i 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 filledwqpb+0*xzh y dq2 ,1o1a6 gdrw subshell.

  For each of the following atomic transivwln on 6(. mtims9:y*) +ljfstions, state whether the transition is allowed or forbidden, (svnjftiw *sm.6 nly:) ml o9+and if 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 isei tz/p5 me)3w in 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 4i6q1t mczo)9kk+vpf X-rays emitted by electrons striking the face of a zm+cv6q4 opt1 kkf )i9$33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremssh +:gs 2if2p. qjf8mrztrahlung X-rays emitted from an X-ray tube have:g8q h2j.+2s mzfpi fr $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff d;do2n/o0hhm 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 wavelen)9hg)0d: iwlsl o,cuy gth emitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength for abt otw74pyn:bbo6 r -xo3 yy)hk)n3x .n $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory toac naw b492pw wxeko:e6:07eaj 5kk:s estimate 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 electrons. Theocr4/ p)05owfvr uf(l wavelen w)v(fofup450r l/ orcgth 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 retin7,,pv9u zwq mtas 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 /je kodv78q7d 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 4y0 0vjxh*wwb diffraction if the beam emergeswyw 4b0hv0j* x 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-Ne laser*tw ;vlbo0 e.vb0 q* qnsc7.tx?$\lambda $ =    nm

Use the uncertainty principle to estimate the position uncertainty for thkwwx q /f +yf.hif:f3gv :ql99e 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 anywhere between this wy:wf x3ffh99fl/v gk:i.q+ q 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 angulalyr. 7tewexn 00un4 tn a;yq4009enl pr 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 electron orc5yx k+e.a9qwqa: h xma ,uz:;bit, estimate the average distae m5qhy: aq ,x:k c9xau;aw+.znce 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 operat9 + /ijxcni,p(,d o0ikqhzokvh u4;ok ++wi p+es 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 out,69bsall;2+ zyv nd8azb3 rnmermost 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 shov1 p-fb3d ,4uvyzx/ lcw 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*l4k nb.f91.tt jnjva 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 for a given value o3 n8b8w s/ -aje6jgxqmf gew -bsjq6x883am j /n$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV is shot through two3:cj 6h54.d xl mqn-yno9vmbb narrow slits in a barrier. The slil5c nxb.mjb-d:ym9q vnoh643ts 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 hydrogenn*1 e9i xxig/i:utu/y atom is given both by the Bohr model and by quantum mechanics. Compare the results for itu/igx*/i 9:xnu 1ye$L = mvr$.
$L_{\text{Bohr}}$ =    $\hbar$
$L_{\text{QM}}$ =    $\hbar$ or $L_{\text{QM}}$ =    $\hbar$

  An 1100-kg car is traveling with a speed oviz0et: i4 gcnod +3h2f $(22 \pm 0.22)\ \text{m/s}$. 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 ad( pe(zh v;/fmt 488 nm. Careful measurez (h mp/;fv(dements 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 rest across 550 V. They are then directed at 9; cs 5ghco*cbtwo slits 0.70 mm apart. How far apart will the interference peaks be on a screen 28 m away? *9c5h;ccsgo b$\Delta y$ =    nm

An electron and a proton, each initially at rest, are accelerated across thevx*1fg5o / ua6w9i kfy same voltage. Assuming t vo agf y95kfiw*x/u16hat 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 6oyyupmrq/7 -9, r v+r*wqeno to the range from 1 to 6, how many elements nwpry+qqve7*m, rou/6o-9y rwould we find in nature?   

  If your de Broglie wavb6j m8szu7k d/elength were 0.50 m, how fast would you be moving if your mass is 75.0 ku6 zj78sb d/kmg? $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+q3trsu 3or7m sm1v+e of lines with one out of every four matching a line from the Lyman series of hydrogen. Assumqsr3t17re+ u mvo+3msing 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 fu zc7k 2;vd,zvrom the surface of a certain metal when it is bombarded witk7v;zc ,vu2zd h 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 8;t7ryqtm- zuu-k8npql- v9olaser 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.]