Compare a matter wave.cqu6aj7a3 iy1/bz e: y(rnvf $\Psi$ 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 withoj j ;tnf2.+ofbbgu,.gpn6 6y)rvry8 quantum mechanics, particularly the uncertainnp8+jogg2f.fj by;bo6,trv u)ry .6 nty principle.

Explain why it is that the more massive an object is, the easier it becomes tok sgay ty)gcjt)8q4u5n b/fpix6 r94/ predict its future pogr94q 6itu/sg xf5y)/8p 4kbcy )nt ajsition.

In view of the uncertainty principumq7s 6 1tfv6wle, why does a baseball seem to have a well-defined position and speed whereas an electron does 61tsfv mqw 67unot?

  Would it ever be possible to balance a very sharp needle precisely on its pvv j5d b6eii013ah db3oint? Explaiai15bdji vh3b ed06 v3n.

  A cold thermometer is placed in a hot bowl of soup. Will the temperature readin 1/uhdp(fg c*qg ofh/1 dfp *(qucg the thermometer be the same as the temperature of the hot soup before the measurement was made? Explain.

  Does the uncertainty principle set a lie(dk l;n o,7kymit to how well you can make any single me, o(yn;7edkk lasurement of position?

If you knew the position of an object precisely, with no uncertainty, how wn 0snq0 (p6kla7qtg d.ell would y.n0pgq0n 7l6dt saq (kou know its momentum?

  When you check the pressure in a tire, doesn'if +tzebwz;5l1)x2;p, g/ z bt5us pbat some air inevitably escape? Is it possible to avoid this escape of air altogether? What i1bw5fx elbbaz,2z ;ti5u;sptz +/ )gps the relation to the uncertainty principle?

  It has been said that thef3 2qb6 w6mf+.xlnaxj ground-state energy in the hydrogen atom can be precisely known but the excited states have some uncertainty in thei+bxa flf26j. q 6wmxn3r 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-me/k oq5ma7 ve(/,kti vnchanical model, predicts that the electron spends more time ne kq,7o /mk (/vei5tanvar the nucleus?

The size of atoms varies byr+xzb, 8e6yzz only a factor of three or so, from largest to smallest, yet the number of electrons varies f ,+yzrbz 6xez8rom one to over 100. Explain.

Excited hydrogen and excited helium atoms :q4u*poub rq.,v. ut kboth radiate light as they jump down to thep*tvu k.oqu:.q4, urb $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic tab 8ysym+-x/1cm2vlqg cle look if there were no electron spin but otherwise quantum mechanmyvc2 sq+lym-xc 8/1g ics were valid? Consider the first 20 elements or so.

  Which of the following electron configurations are not allowd4*ny,jns3 cik nz+r8 ed:
(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 confi+-dcgc9/:vai gyej t7 guration for a uranium atom (careful scrutiny across the periodic table on the inside back cover will provide useful hints dgaj:/cge7v 9-+city).

In what column of the perird5d 3inyo s5wzl,6 0eodic table would you expect to find the atom with each of the following ciw0 ezly35so6d r,5 dnonfigurations:
(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 exhib0e+giztr x ;9bit similar properties?

Explain why potassium 4g pb)1gag/)hnw/ 3wg;4-ld saa--h zlvh cwrand sodium exhibit similar properties.

The ionization energy for*.69 qv kbzofh*bpnr*1ahrs+ 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 photon nature peckgp;m3d,4 for light? 3 e4,cppgkd;m

Why do we not expect perfect agreement between measured values of Xtyhc8dk2 jwphl f)p r fb8y:v 23qc6+;-ray line wavelengths and those calculated using Bohr theory?v2l cktpjfq + 6crpf; )y3dy:2bhw 88h

How would you figure out which lines in an X-ray spectrum corrko /my)bv7 uo;inpm j-1s2 *l9xd/izxespond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect elelxwp:w fwp-;u bcqvedo:( a3a2 dc 066g5tzp(ctron transitions deep within an atom to produce shorter wavecqw05 pfdtvg;-( p (wb6::az3xec26uolw ad plengths than transitions by outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from ordinary light? o7jdm/n.6ln sn7-hup How is it the same?

Explain how a 0.0005-W laser beam, photographed at a distance, can seem ma,zqh l/c): ajuch strq: j )/aa,hczlonger than a 1000-W street lamp at the same distance.

  Does the intensity of lig lxiz.w w2qe5ot7/(f.zdpo; tht from a laser fall off as the inverse square of the distance?w l5qtz p/w;2idoe.7 xt z(.fo Explain. ----待选择----yesno 

  The neutrons in a parallel beam, each havinavq)c; *f*0sfmj:ew qg 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 witx)-1lk3r6l pe ;uf aoxh 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 4nu92*r-qidpq e sm q0measured 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 for typicalq8qrqy8e d.ntag,7*al2 hk: qyki - 5zly $10^{-8} \, \text{s}$. 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 mikd+ oe fs)s9,uon $(m=105.7 \, \text{MeV/c}^2)$, 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 in 2ys/cl q/9uzm a typical nucleus of radiuszmly/9us / 2qc $1.0 \times 10^{-15} \, \text{m}$. [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 prinb*jr8ztni cf 6i7x 0.aciple to show that if an electron were present in the nucleus jn 6ri7cfax zt.i0b8* $(r \approx 10^{-15} \, \text{m})$, 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 o(,rsyt+ q /1trbin-x mf 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 itilb -/ 2zfjnfmpy6 sq:8.x j8n the $n=6, l=3$ subshell?    elections

How many different states are possible for an electron whos0j-,csgfda 8h70 4wt, lz dg-akec mxhimg ,(-e principal quantum nu,7-mfsd c0ae0tx- a j,-ig 4 hz,8gdlm(gc kwhmber is $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for + oah91/uly w( hqtnh7mz0d:teach electron in the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for eiu, .mn -gaa9nq 0( n(bww tjjas:r+v,ach electron in the ground state of nitrogen $(Z=7)$.

  Suppose a certain hydrogen atom hd ad(8, ea7yz k(tbib)as $l=4$. 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 electr/lz m((,t- k uilry/ison 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 * 2l vsfl otncafz,*13$m_l=-3$, 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 elec7 q ,g hvwcccl6qga,u*o f8-c c(n2:pxtrons in a "g" subshell.    electrons

What is the full electron configuration in the ground state for elements 8 yr :u2q2hvvywith $Z$ equal to (a) 27, (b) 36, (c) 38? [Hint: see the periodic table inside the back cover.]

What is the full electron configurat*.eggo w *1avaion for (a) selenium (Se), (b) gold (Au), (c) radium (Ra)? [Hint: see the periodic tabwovge .*a*a1 gle inside the back cover.]

  A hydrogen atom is in th*hioqd y;d,: ze $6s$ 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 thjyh8y+kmhesv . 9 35m+r t8gmwe third electron in lithium using the Bohr theory. [Hint: This electrts8khm+ j.mh9g er8yymw+5 v3on has $n=2$ and "sees" a net charge of approximately $+1e$.] The measured value is $5.36 \, \text{eV}$.    eV

Show that the total angu-aan6 +jz: xsklar momentum is zero for a filled subshell.

  For each of the following atomic transitions, state whether the transitiono 4m+; u7u vebquklx.- is allowed or forbidden, and if forbidden, what rule is bein ox47q -klu;m.u+bu evg 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 a,.sb zmf4-i md $6d$ 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-wave+2zmiz-ll hw8 length X-rays emitted by electrons striking the face of a 2l lm-8zizwh +$33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength brems;ndj c6w 0kyjq 71)ecyp//: mgb zjh6mstrahlung X-rays emitted from an X-ray tube have 7)jbjg /ekjmn cydw:6p6h0qm ;zc1y/ $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wavelengi.u6zwv,zmzd gxy0( bh*ib 7 0th $\lambda_0$ 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 emitt* ug4- 2zxew6zhj jg2fed when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength for an 8q4. ixnod1(c: waq1dmeh z1q $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the wavbr2 iw m+t* 0xpzt/ m:3euv-fbelength 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 maz7 0lpau8 s6jvax+wq 2terial is bombarded with electrons. The wavelength ofs ap+z8x2qawj 0 u76vl 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 reti qb;rs/c5( hxycnb)*e; ft8d cnas 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 q3(li ,)01p e: .qw lqmts5oh2ddxkb cmight 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 sp/e ;8ed 4vwgakoqby27read of a laser beam due to diffraction if the beam emerge8b7vk2/eag deqw4 y;os 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 v w-,plnm 4* c,irfoz8He-Ne laser?$\lambda $ =    nm

Use the uncertainty principle to estimate the position uncertainty for theyyxzs..q)+ddxd5c4 w 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 vac .yd qd.ys+x)dwxz54lue 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 a5x6v 0i.3 nud5 5nz ua- qfcg:sn+uygxnd 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 elecw(jhisshm ./ 4tron orbit, estimate the average distan m ./jhh(wis4sce 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 auvg* 5 tngkx4wv0l7 x or;;0tht $95 \, \text{kV}$ 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 ;nx(-g h;tvu4k e fkl0d0s/rfboron 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 t 8j4zc7q82ner fz cb8po show 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 given vai 1 d, sxlddxk(09e a-;eoa4low7akk)lue 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 elecmg9 (a7f zks1ytron states possible for a given value of 7ksazm9f 1(gy$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV is shotabr e59;gmx zu)6xxdyck+ i5zxl ,go+2lt i 0- through two narrow slits in a barrier. The slitsxi52x 6laxgo e +cgdky0 x-iu;95)zrmbl,z+ t 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 given both by the Bohr mo ;s r3hqo,l1mkdel and by quantum mechanics. Compare the resultso, 1lmh q;rsk3 for $L = mvr$.
$L_{\text{Bohr}}$ =    $\hbar$
$L_{\text{QM}}$ =    $\hbar$ or $L_{\text{QM}}$ =    $\hbar$

  An 1100-kg car is trave4sv.uafcnxo-4 /cm w 4ling 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 n4zq(c go+v)dctlx n (*m. Careful meavn(g)4l(od z*cq+ cxtsurements 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 direuet1rehb+ fj r:xpo -p cg),2.cted at two slits 0.70 mm fjr u +cero2t.xep p:, )1h-bgapart. How far apart will the interference peaks be on a screen 28 m away? $\Delta y$ =    nm

An electron and a proton, each initially at rest, are accelzjdiifp 1om2c,qety4y qk(; 0, ; qd7uerated 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 the1yfc; z4ip q di2 y(j,q0;qeu, mktd7oir momentum.

  If the principal quantum number n were limited to the ranwu:dh,vkd uz; 4 kljg5m9 0)fppv3kt3ge from 1 to 6, how many elements would we find in natuhkp3lm guvd,t;z4 3)vf9:u 0kp w5djk re?   

  If your de Broglie wavelength were 0.50 m, how fast would you3flazfj(jwu+t;f8,4 bi par ,k6k+he be moving if your ma,fk3r4 lbjfu( jpaefz,+ +ti k w6ha8;ss 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 series ofwopyu+;x z.h0 lines with one out of every four matching a line fromx.0wpu; +yzho 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 nmt1to b6/o1)ovo(zidb- zn qy ; are emitted from the surface of a certain metal when it is bombarded with high energy radiation. If thqvo1 t-odtb oy6nozi ;z/b(1)is photon wavelength corresponds to the $$K_\alpha$$ line, what is the element? ----待选择----nickelcopperzinc 

Show that the diffractive spread of a laser )77x5bt iwsq3/cx h7fx9j(fi* qxz v0w(sprzbeam, $$\theta \approx \lambda/D$$ , 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.]