Compare a matter wave4* zr6fu8monvx/u cs2t-7vpq $\Psi$ 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 compa79sbqq 9t lca x0vtfyg9 g2+h-tible with quantum mechanics, particularly the ut+9qla07xgyb tgh9 q-c9 fs2vncertainty principle.

Explain why it is that the more massive an object is, the easier it beco0gna4*w ko+8bb ru3zhmes to predicab*ur+h n8g 43kwzo0bt its future position.

In view of the uncertainty principle, why does a bamo tx,a(aq,fviad/3g), md efg- m o9;seball seem to have a well-defined position and spee e3a,dg f m gv(foxqa-);ma/,t9omd,id whereas an electron does not?

  Would it ever be possible to balance a very sharp needle precisely on its pointg 2jdy:c8pl wj 02:dg:u/nqqb? Explain2/2gg:0ddc:pybqnu:j qw8 lj .

  A cold thermometer is placed in a hot bowl of soupyw-4o q zn1n d4)nmg7nhoxyyup,0o:7 . Will the temperature reading of the thermometer be the same as tq0ynpy7 -z7, xg no:44hm)dyonnow u1he temperature of the hot soup before the measurement was made? Explain.

  Does the uncertainty principle set a limit to how wellwo w+dpz3ch-qgio 1 /; you can make any single measurement of + c/3qzd i1goohw p;w-position?

If you knew the position of an object precisely, witzu m2 a1h5w584qnlsf gh no uncertainty, how well would you know itsu52hlmnqfa54g w1z8 s momentum?

  When you check the pressure in a tire, doesn'fzjeod np 3/ -app3k95t some air inevitably escape? Is it possible to avoid this esc5ep3n/z pad3k jp-of 9ape of air altogether? What is the relation to the uncertainty principle?

  It has been said that the gks*9 59 +1h:t*nyfnelo e n; esfn7cpsround-state energy in the hydrogen atom can be precisely known but the excited states have some uncertainty in their values (an "energy width"). Is this consistent with the uncertainty principle in its energy f ;s7ye n5n9sth9snf1e* l:e pon+*fckorm? Explain.

Which model of the hydrogen atom, the Bohr model or the quan77rwp4hdz,x(ilvl p;h 87 ;)-s5h ekf cdsak atum-mechanical model, predicts that8ph f7 a-dh kdw7cp (5aklxz 4;,v)l i7srhes; the electron spends more time near the nucleus?

The size of atoms varieszq, ib(bizof jy( f5,, by only a factor of three or so, from largest to smallest, yet the number of electronsb, iiq(,f5o ,(yj zfzb varies from one to over 100. Explain.

Excited hydrogen and excited helium atoms both radiate*/uwf 3rgf pt42uwf+g light as they jump down to fu pu+g/tgf*4 2wwf3 rthe $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table look if there were no electron spin but otherwis 3oestd2jq j8,6t:xj ie quantum mechanics were valid? Consider the first 20:ijes8 jtt, o62dq 3xj elements or so.

  Which of the following electroh b+i a4un9xy dibr773n 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 configuration for a uranium atom (careful scru .ley+j 8kee68m fzo;2si,rf mlj7w9 ptiny across the periodic table on the inside back cover will provide u kz7e2lp;fl8r of +ejey 9wimm,6sj8.seful hints).

In what column of the periodic table would you expect to fintqzebqs g1g8ykgugh6g9ey( 3/,fl4*d the atom with each oftqghzfl 61ugy*g(/sg,3ge8 b e9qyk4 the 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 wqp: lw+ + ww5b2of/seexhibit similar properties?

Explain why potassium and sodis i lk9j .0sode8-eabf,(;atoum exhibit similar properties.

The ionization energy for neon ap5pgy:zzj+ 7$(Z=10)$ 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 n )d m0tk-u(pg- j+n* iskuaew-ature for light? ws0mnkej-g-+ puu-t)d ia *k(

Why do we not expect perfect agreement between measured values of X-ray lino0) e8pa c6z zkb/ )a9uj)ihone wavelengths and those calculated usingh8za)n6 ce/9)z u0)bp kaoijo Bohr theory?

How would you figure out which l.p)g okd5 2rg kss1pl(ines in an X-ray spectrum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron t9b ;;t rfu2xst75ro vi kqf -7:tvbp5kransitions deep within an atom to produce shorter wavelengths than transitions by ot55sr vk:iu t9f;2 ot rfbv; x-pkq7b7uter electrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from ordinary lig2/o/5cq,fuzz*5g 8r *-doz rxswd atiht? How is it the same?

Explain how a 0.0005-W laser beam, photographed at a distance, ca:24 p 37ne57fqlieulbbx (nhtn seem much stronger than alt(nbi7 h5: pf2qe eun 43xl7b 1000-W street lamp at the same distance.

  Does the intensity of pf+k:. b7shkylight from a laser fall off as the inverse square of the distance?ysbk.p+:7hk f Explain. ----待选择----yesno 

  The neutrons in a parallel beam, clei9hnim9 ln 2i. q2/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 8y4c535m sj k ku9qqv6izuyq :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 bkyqif2 -3/ib;3qz pwz e measured 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 stax(a w/ li15j6oj 8oslscfk af cat+h2e6yz:0)te of an 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 uncertaintm1eum qf7.2 q0uewm.b 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 in a typical nucleusxugr 20i:8 5kn2sht 8 dbdskn8 of radius 8drnsnk5bxdk:t 28gsu ih820 $1.0 \times 10^{-15} \, \text{m}$. [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 rb3ad9 .zr)i *rbv4xt an electron were present in the nucleus 9id*r4vb z.b)rt3xr a $(r \approx 10^{-15} \, \text{m})$, 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 of a3a ro *p13 0vni6dszha $3.00$-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 bpw8wfz2r-a0 vf/d sv7e in the $n=6, l=3$ subshell?    elections

How many different states are possible for an electron whose principal quantuhuq /y5cpu+/ hm nh/u+yu5 qph c/umber is $n=4$? Write down the quantum numbers for each state.

List the quantum numbers fombe:6 do-uix)kx p,:or each electron in the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for ea2s4j)4hclk wle gk,wr vk790h ch electron in the ground state of nitrogen $(Z=7)$.

  Suppose a certain hydrogen atom ha i+dx-8hgsql6kr9 ,kv s $l=4$. 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 momxdw*5zzfub (now hl9 u*m;yv87 yr)-h entum 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 hl+q jivruwc3)r3tu+7ig.br 1 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 gvik7 83c jr0qelectrons in a "g" subshell.    electrons

What is the full electron configuration in the ground state forh,1t .a tv9piqx//s ke 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) gold 4jaoze376g es(Au), (c) radium (Ra)? [Hint: se soez3e74j ga6e the periodic table inside the back cover.]

  A hydrogen atom is ir7s o7ztryf 3+5 2hefan the $6s$ 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 eu(p;ktdtk/ 8 kvzx0(nnergy of the third electron in lithium using the Bohr theory. [Hint: Tv;tkntk( pd8 /(zx0 kuhis 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 mo :fjkh6s;u/w :v7yk: h: kqrlhmentum is zero for a filled subshell.

  For each of the following atomi7siijacoeb4i3 2 7s1hc transitions, state whether the transition is allowed or fosh iso77b aice41j32irbidden, 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 a s.sjto+ kdj: 5$6d$ 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 b9uxfgy/uj /6mq 4tr(xy electrons striking the face of a4u grm/ xyt/ q(ujx6f9 $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength brems4 v 85it7ccqsre)g/vhstrahlung X-rays emitted from an X-ray tube have vsh8i45 7ve) /cqtgcr$\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wava39/ r7a+3 lpdljhilwelength $\lambda_0$ 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 when a Co m 06+mh*3bi-:y;uecl hjkb :iy wq vw8$(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength rg 6o3oq obx3/for an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory t ,)v8:0tqyl 7e hsg 5as1jrmezo 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 b(hm)i ,s6h u pten9ydd.byc/41nl ;rgombarded with electrons. The wavelength oh/limg dtuh1(6.,s4 db;nn)yr 9 pyecf 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 retinas puts out (3nvu c 5,.9tkqlbcs mo*5mxec-gvkopm9 u:-$28$-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.f;p-on(hik g lab 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 lam+. zw4ous26v; wfbf9vlvf t 4ser beam due to diffraction if the beam emerges throug fwlfz.6vu4 vs2twb 4f+ov9; mh 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 lase ) md+uefl3-jer?$\lambda $ =    nm

Use the uncertainty principle to estimate th3rv ,f p2bfo5ne position 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 anywhere between this value and zero.] How does this result compare to the Bohr radiu 3ro,fbfp5 v2ns?

  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 an 1))/:fwe5kk tn/ajv xeqw:u jgular 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 orbit, estimate the averag f.2e+c.ad udi pd9w9pe distance from the nucleus for an electron in the innermost cw.i du+p ea92d9d f.p$(n=1)$ 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 0s9* hw a o7 awy)u3i),k 4db2(zrhrrgcb4yroat $95 \, \text{kV}$ 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 elexb*kk0 7ryprpbyb7dv l76q5ggy / 6 8nctron 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 z h-znkh)4mf4Moseley 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 giveblcb 7:/wyk /hn 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 statesy:i1dl d0l 6fx possible for a given value of l1x0 y :fd6ldi$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energj67 ssho0fi f.; sqsy8y 45 kV is shot through two narrow slits in a barrier. s sq6f.07;ssyihj 8of 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 t)wki) e,zws n+he hydrogen atom is given both by the Bohr model and by quantum mechanics. Compare the results fi) snzww),+e kor $L = mvr$.
$L_{\text{Bohr}}$ =    $\hbar$
$L_{\text{QM}}$ =    $\hbar$ or $L_{\text{QM}}$ =    $\hbar$

  An 1100-kg car is traveg : kd4fr:dcw+ling 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 nm. Carefh( 9wvtkg3 ,ueul measurements show the line is really spread out between 487 and 489 nm. Es9e( v3ktwhug,timate the lifetime of the excited state that produced this line.
$\Delta t = $    $\times 10^{-14}\ \text{s}$

  Protons are accelerat9 py.n3uo twalt2oh3 t 3tv06yed from rest across 550 V. They are then directed at two slits 0.70 mm apart. How far apart will the interference peaks be ony t0y33.wvout olp6 h3at t92n a screen 28 m away? $\Delta y$ =    nm

An electron and a protgaqqdi8j o;x33 *(pw4( q ezdlon, each initially at rest, are accelerated across the same voltage. Assuming that the uncertainty in their position d;xpi( qdz8q4wlq j(goe 33a*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 the range from 1 to:y6 wc(m yz9(z28fb gp xm:5pgt xkn* 39sxesk 6, how many elements k5sz6t(w:mxc( :92x xyn m*g k9pgpz8esbfy3would we find in nature?   

  If your de Broglie wavelength were 0.50 m,1yk 3wi)m 1cx;e1f qaq how fast would you be moving if your mass is 75.0 kg?yc11fqem;1xkaq 3) wi $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 series1)hl31 l /d ihb3sxgw*f5gwutf y u*s( of lines with one 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, deted1sl w ti3)5/h(gus blgfxw 3uh*1f*yrmine Z and the element in question. ----待选择----lithium berylliumboron 

  Photons of wavelength 0.154 nm are emitted from the surface of a certarasg*j4i 8f8 .tjp-, a:;ztxao *zuly in metal when it is bombarded with high energy radiation. If this photon wavelength corresponds to t;o8zstyfp:84il*g raa.j tx-z,jau* he $$K_\alpha$$ line, what is the element? ----待选择----nickelcopperzinc 

Show that the diffractivn;qona ee/ ;qokf3.u 2e spread of a laser 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.]