Compare a matter wave h6( riqni1 +xk$\Psi$ 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 rr h5yeq/o9*i/gt ( pqquantum mechanics, partyhig/*qo/rerptq(9 5icularly the uncertainty principle.

Explain why it is that the more massive an object is, the easier iyc0dps( + 3*aysxwb +ot becomes to predict its future posi owbsa+s* p(yc30 yx+dtion.

In view of the uncertainty principle, why doesta)9j62bn* yapt4 vc zwvhz:( a baseball seem to have a well-defined position and speed whereas an electron dop2navytcb zj)9w a(6*v :4thzes not?

  Would it ever be possible to balance a very gg 1fkt8m4zel,0.q;nn z dt/zsharp needle precisely on its point? E8gfzt4z qm l/.tg,nz1 ;0edkn xplain.

  A cold thermometer is placed in a hot bowl of soup. Will the temperature reading: -pz4sty, wqite2 g5u of the thermometer be the same as the temperature of the hot soup before e: i24pq-5sugzywtt, the measurement was made? Explain.

  Does the uncertainty principle set a limit to howdwub2vv ;0) yw well you can make any single measurement of 2;w wvdybv)0uposition?

If you knew the position of an object precisely, with no uncer)vdg,e2dl;prm:y nik7tonc af/86a :tainty, how well would you knod :nem lr6a; n,yv8gcdp2i tko:/)fa7w its momentum?

  When you check the pressure in a tire, doiw2u xtko1-4 tn )(nouesn't some air inevitably escape? Is it possible to avo1o)n tx4ut(k2o-nuiw id this escape of air altogether? What is the relation to the uncertainty principle?

  It has been said that the ground-state energhsr qx. . : m-gs/om5uz,qoex-y in the hydrogen atom can be precisely known but the exs:o srg mx-.ex 5q-/o.uq h,mzcited 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 hym8uw-jntp655fq j c7mdrogen atom, the Bohr model or the quantum-mechanical model, predicts that the electron spends more t8w-7 mtqu 5fjjc 5nm6pime near the nucleus?

The size of atoms varies by only a factor of three or so, from large50,-iurrjzrno ad f4.*mm p- est to smallest, yet er-.4np i r *0mamuor-,zd5fj the number of electrons varies from one to over 100. Explain.

Excited hydrogen and excited helium atoms both radiate light akmm iv2q79obdg 2;j /;hxg 5pas they jump down tog9 k;2aq g7/vjh; 2opm mxd5bi 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 otherwigy7im/elg+ (q8 f;yb zse quantum mechanics were valid? Consider themib;7z( flq+y8 gey/g first 20 elements or so.

  Which of the following electron configurations are not allorq8gej1d5 w ( puc46umwed:
(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 configurati9x- ttvrts)69mw,zy non for a uranium atom (careful scrutiny across the periodim)t-6s9 vrytz t,wn9xc table on the inside back cover will provide useful hints).

In what column of the grrhs u+u uf .si 9v593*1rqxs6mpr s(periodic table would you expect to find the atom with each of the following configuratio1q m5r sf9uugsxr 9r ss+ivh. 6pu(*r3ns:
(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 prop, cls9)tn(okp erties?

Explain why potassium and sodium exhibit similar prope y/cgs yac-agxgon *.6e1sr)3rties.

The ionization energy for ne-of h 9.jmil1mon $(Z=10)$ 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 ajzb( ;y398 7boiywvz photon nature for light? i 9owav8y b3;byz( zj7

Why do we not expect perfect agreement between measured values of X-ray line wlydbfz 65*ve fh+ig.)nzx ;9 navelengths and those calculated using B;nge)lz 56d*xhn9+ vzbf f iy.ohr theory?

How would you figure out which lines in an X-ray spectrum c pa8nbjsqx 4c7xgdc/v7w86hs5 7xrw5orrespond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transitions deep within an atom to produce shortergi+h8 mzoxafdzdz007b(v3 uh98g f d * wavelengthi a0g h89mdzdg0+7 (bf8oh* fuzzxv3ds than transitions by outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from ordinary light? How is * j*yln rb0:kzfm 7f :gbo1srx08i9 gait the same?

Explain how a 0.0005-W laser beam, p qhz6lf*1 0yngwxnu+ ;hotographed at a distance, can seem much stronger than a 1000-W street lamp at the same diqyuz f;+* 6gw0h1n nlxstance.

  Does the intensity of light from a laser fall off as t 8)zol jamw,k0he inverse square of the distance? 8)l wj,aomz k0Explain. ----待选择----yesno 

  The neutrons in a parallel beamilrag9qpcx*,+0;ls 2 tof m) dpbtq9 ,, each having 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 with a spe1x.s1rkk:e 8d cryj m :x7tu7s8xp7kz9 tra+ced 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 to an m y7 (2cwz;jpuaccuracy 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 aplpk-hb39u f7n atom 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 uncertaintytbqs/ -p88cvu 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 possibly rczl8:r7 cq0e energy of a neutron contained in a typical nucleus of radiucl7r yrzq 80:cs $1.0 \times 10^{-15} \, \text{m}$. [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 sh5f 6ivwv,j d3 a1k2 s*gzhntk 2)ifd2iow that if an electron were present in the nuclddt2hja3 5,k 1w2 kf)zv2vin*f6 sii geus $(r \approx 10^{-15} \, \text{m})$, 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 positiqq0c /fbklk;l1f26 ts on 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 e(m *jp xl /5.(arjwygin the $n=6, l=3$ subshell?    elections

How many different sd.b;( m3hoi+ju:mgldtates are possible for an electron whose principal quantum ndlom3jg udmb.:+ h; i(umber is $n=4$? Write down the quantum numbers for each state.

List the quantum numberk gbo qu7fn2t j;8hljw (ng/37s for each electron in the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each electron in the ground stateki:r 9*pjcct9.1oyc x of nitrogen $(Z=7)$.

  Suppose a certain hydrogen ic -qv97c n:soatom 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(s3u8mav+ksk(zq2 y l 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 hv3 be rnzo(s-;iro6.l +3zafl as $m_l=-3$, 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 ina db(3y tna2pb,+v,2qy ia( zl a "g" subshell.    electrons

What is the full electron configuration in the ground state fok4ha -rnbl45z r elements with $Z$ equal to (a) 27, (b) 36, (c) 38? [Hint: see the periodic table inside the back cover.]

What is the full elec w+vki,d+:ua0ckb:g pup x-7 ztron configuration for (a) selenium (Se), (b) gold (Au), (c) radium (Ra)? [Hint: see the periodic table i+bgp 7 ik :uwak-cpz+,0x d:uvnside the back cover.]

  A hydrogen atom is in the lr--5p yfgdz ;$6s$ 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 usingpo(54+johij e; ebaw0 v ks5:g the Bohr theory. [Hint:0k iehojo5sg+w bv4a5pj(e ;: 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 momentx k5y+h7svl3s r:3 hoxum is zero for a filled subshell.

  For each of the following atl0gty9 rgp.er;fh,1p k.ptvi, v .n0 *)b dzmtomic transitions, state whether the transition is allowedl f.); d.ren.zprg,9g0 vb0 ,tp*ykmp1vttih or forbidden, 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 is in ow. -9n3bx5j3ldgye,l5hx* 5uwe wrlz*lgx2 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 g qm:ubsh*q+9-rays emitted by electrons striking the face osh*:q bgqmu+ 9f a $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremsstrahlung X-rays emitted from an X-r vfx5v2z3d h8bay tube have 3x8 bhvzvd5f2 $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff8c yxci+oy8gc )jpi:. 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 wavelengtqru 6mio-uq 3sf695t )0qvgxw;if nc3h emitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelengt1 3ij9r1jo+pr7mut bth for an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the wavelength for an 4ca9cy- y,.cnb /fvgv$n=3$ 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 mate;u(jd)y*no 6 k: iuyjvrial is bombarded with electrons. The wavelength of thu:nku6*vyj );( odiyj e $K_\alpha$ 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 puta0of4l0cub8 (b j) csts 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 i kmlzq(q0p2/-ig3a m zj 93hilab might 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 o/rzn+7v7e*-zlvtc x3j,ze vuik 3 0xlaser beam due to diffraction if the beam emerges thrj+, ckri3uz3/o-7zvv exv7t 0zn x*leough 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 ojplyq5.tzxk /f8 h ,/ozw+u6n * arc/qf the He-Ne laser?$\lambda $ =    nm

Use the uncertainty principle to estimate the position unce lga. etz;nt4ax)ne0-ib5w2a /tm(pmrtainty 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 anywhere between this value and zero.] a ;lb-gmwtx5 2.)a (titn ee/p0zam4nHow 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 angulat l-xkr3w1/;l ppx ge5r 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 electrow.p7wq1bvv) xm fcon)0cvx.60sg)pt n orbit, estimate the average distance from the nucleus for an electron in the in)tgq xvsfnvbm.w cw1. 6 o c7vp)xp00)nermost $(n=1)$ 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 atscz e48mkr l5- $95 \, \text{kV}$ 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 outzm9zqz oszpy++4s 6h6ermost 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 show that the Moseley plot(Fig. Below)lcn) 9o 8/pamcg+ h r*kap hmhouwovz9 05-*i9 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 diydqx5k;h1p2 h fferent 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 for a given value of+ nh8 x)qheg;a xqg a)+e8h;hn$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with eit )/w fprhv87nergy 45 kV is shot through two narrow slits in a barriewh 78v/)i pfrtr. 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 given both vl(:; crt,byl x,j 7lvi1po4kby the Bohr model and by quantum mechanics. Compare lx7;lt ky(r,4,1ivvlp: jb octhe results for $L = mvr$.
$L_{\text{Bohr}}$ =    $\hbar$
$L_{\text{QM}}$ =    $\hbar$ or $L_{\text{QM}}$ =    $\hbar$

  An 1100-kg car is travelink0m:;z -pd hvgg 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 witw7 / ytfhsp3js hw*2swyxx2i xc:811 qh a wavelength centered at 488 nm. Careful measurements shh/8s2cy*7 xw2q w1tf3 :xsixjysh 1pwow 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 two srh zv;)/v7jk9 z;0soowja *xo lits/h;zkzjov9; ox7 wa *r0s)o jv 0.70 mm apart. How far apart will the interference peaks be on a screen 28 m away? $\Delta y$ =    nm

An electron and a proton, ead1y8y nz p2ff3twf. naqt+k w6p+68ad wg+.k vch initially at rest, are accelerated across the same y tgffk66a318.q 2w+t f+w w8k+d.pvdynanzpvoltage. Assuming that 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 limi8 i5tud iww hs0py*;z-ted to the range from 1 to 6, how many elemwt-ihp w05*8;u iszydents would we find in nature?   

  If your de Broglie wavelength were 0jgyn6lp.g5b-iu ev d- 4wy r+y9 9z4hm.50 m, how fast would you be moving if your vy4695e.umlzgywgd9bph +yin j-r4-mass 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 of lines with on3 rdrto(alg 2kudmk(7o2 -f/ne 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, determ3( dfkokod/22ta ug-ml(nrr 7 ine Z and the element in question. ----待选择----lithium berylliumboron 

  Photons of wavelength 0.154 nm are emit:p/6m9bw i zocted from the surface of a certain metal when it is bombarded with hi 9:pbowz6/c migh 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 i v-gvvm )/ed b)kf-+e/ i*poblaser 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.]