Compare a matter wavey-k c1 j.uy,4tfc,a)wzm.o n q*hflh/ $\Psi$ 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 compatdmx6w,i 8hdq.vnp n5 0ible with quantum mechanics, particularly the uncertainty prmdw8dv 6 i,h0x5n p.qninciple.

Explain why it is that the more massive an object is, the easier it e n77wl ocu;r4h3dmk9becomes to predi ;9h77cen3mdowur4 klct its future position.

In view of the uncertainty principle, why does a bh16tm/j.lku fz+ u *uhaseball seem to have a well-defined position and speed whereasz flm 1jh*6t/kh uuu+. an electron does not?

  Would it ever be possible to balance a very sharp needle precisely on its point?fa +if-zx.c8o Eo 8fax- .+fcizxplain.

  A cold thermometer is placed in a hohc/-s) hkk+rj.q o(ilx 9w7co t bowl of soup. Will the temperature reading of the thermometer k9/- h x.k) srochl7j+c(wiqobe the same as the temperature of the hot soup before the measurement was made? Explain.

  Does the uncertainty principle set a limithsrv)cz . -.gn3p .g +dnpz)bm 0ar1af to how well you can make any single measurement of posn gv3-s) mpagrh+ f.a1) zdb0cp..rzn ition?

If you knew the position of an object precisely, with no uncertainty, how r0t 9j1h4tewy well would you know its momtw 9r 4j1yh0etentum?

  When you check the pressure in a tire, doesn't some air inevitably escape?ybux9/qubw-xu0 jwsy:4 4) mo Is it possible to avoid this escape of air altogether? Whx0qymu- )sob xjuub 44w9/:wyat is the relation to the uncertainty principle?

  It has been said that the grsn cy3f6n5n0oound-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 innn6 sy5fcn o03 its energy form? Explain.

Which model of the hydrogen atom,aw 6.n ob7uesv:n**ha the Bohr model or the quantum-mechanical model, predicts that the electron spa7o.bn:hvw *su ae *n6ends more time near the nucleus?

The size of atoms varies by only a factor of three or so, from largest aqoq zcb68+ ,64oa5t* rdhjsi to smallest, yet the number of electronssc 56oatj8 +dbq*ho4arq 6 iz, varies from one to over 100. Explain.

Excited hydrogen and excited helium atoms both radj(u u watj4qydw5am( **hi:,yiate light as they jump down to t(,4 t * iqawamyudyjh5*:w ju(he $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic j+l z59 3zsnb*p4u pbpk +tt(3sjs8gftable look if there were no electron spin but otherwise quantu+jbtksj*f+z3 b(ps3l9gnu5 pt8 z s4 pm mechanics were valid? Consider the first 20 elements or so.

  Which of the following electron configura-:e 2gxrcjh11cgl kl;tions 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 elex ,8yginylaa+h x+4 t 4w-d/hectron configuration for a uranium atom (careful scrutiny across the periodic table on the inside bacl4ged-y,4/xwh y a+ +hax8tink cover will provide useful hints).

In what column of the periodic2mnjo,s4e,ok ll1o g9 table would you expect to find the atom with each of the following configurations: ,gslo4onjkl, e291o m
(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 ra7 pu/01q vu,c4ue j6tygu6qexhibit similar properties?

Explain why potassium and pq pu tpod3k4-h9d(/usodium exhibit similar properties.

The ionization energr( qs 90/e :tjmipw/;dpyam7 sy for 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 photow:upnj)zsg7gn 1k88o*ngr jc u uod m4 y/0(a5n nature for light? g/ur j5y4jz 8)gupodg km :7n08 n(na 1wu*soc

Why do we not expect perfect agreement between measured vairw u s3 l1a/hi7;03beaux3hw lues of X-ray line wavehwe wah7 /u l 0b1r3iai;3sux3lengths and those calculated using Bohr theory?

How would you figure out which lines in an X-ray spectrum corg 6mn3cs7.gsx respond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transit(fve v4 y9-:jx0-sa )v dbclwjions deep within an atom to produce shorter wavelengths than transitions by outer el fe4jacyv) sb v d(9-x:0jvl-wectrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from orw2c -jg)uhw. rdinary light? How is it the same?

Explain how a 0.0005-W laseo)kr 2xczv5:l r beam, photographed at a distance, can seem much stronger than a2ov )5kl:rc xz 1000-W street lamp at the same distance.

  Does the intensity of light from a laser fall off as the invezjs;*y ln .o)erse square of the distance? Explain.ojy. ;szle*n)  ----待选择----yesno 

  The neutrons in a parallel beam, earr5m .c-fqc2 mov9mq ;ch 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 woty( .v +in+i j3tii4jith 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 measured to an accuracy o ly6m0 k))rahtf $1.0 \times 10^{-8} \, \text{m}$, how accurately can its speed be known?
$\Delta v $=    $\times 10^{3}\ \text{m/s}$

  An electron remains in p :7lbg ) l 4g;ncnl/tdf6axo h6jn)+kan excited state 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 uncertainkjpxc 9y5 : i0 :kdtsg8 bsxw.;ks/;bcty 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 possiblbuu+-* fro8db u5 dj3ze energy of a neutron contained in a typical nucleus of ra-j fuu +u*3b or8bdzd5dius $1.0 \times 10^{-15} \, \text{m}$. [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 thatb*, z9xu7wwb;bxei d2 if an electron were present in the nuw,xbw*d9 7 xze;i2bu bcleus $(r \approx 10^{-15} \, \text{m})$, 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 w7;emlsfj g fkx*6+8c +5rkcba $3.00$-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 in t8jsd)5h5ju mij-osk6he $n=6, l=3$ subshell?    elections

How many different states are possible for an elecp9tz./m+ifvn tron whose principal quantum nui v9.pn+ mzf/tmber is $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each electros:tr*onk, t- 5dtw)n on in the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each electron in the groun ust.5p)568tvvy qhn od state of nitrogen $(Z=7)$.

  Suppose a certain hydrogen atom hashz.sz(: na eg )0eaz4s $l=4$. 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 t.mpqij3t8rw65 9y wng he $n=4, l=3$ state of hydrogen.
L =    $\times 10^{-34}\ \text{kg} \cdot \text{m}^2/\text{s}$

  If a hydrogen atom hasxyn;rixq bt4 ( ,rab9. $m_l=-3$, 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 7emsh miexw;o+yt** 4.y6j wmelectrons in a "g" subshell.    electrons

What is the full electron configuration i.e dh/t og)y diy/5g/fn 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) selenium (Se), (b) j9fwpbz+7 ;li k7i7.dt u4drlgold (Au), (c) radium (Ra)? [Hint: see the pe9i4 ;fl k7dl7utwz+bdj i r7p.riodic table inside the back cover.]

  A hydrogen atom is in thj;0b6p9g lh8 nls8nqxe $6s$ 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 th dys-ae 01uf*zu )j; aoojr)z5toa0l+ e third electron in lithium using the Bohr theory. [Hin )aazjfs or 5zutej0)a-*o1yo0 dl; +ut: 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 mo wvyig* :vg ua 3cm* chj6o84crmw3z:+mentum is zero for a filled subshell.

  For each of the following atomic transitions, spft22wb) w19yvwa43 kqt v a:ktate whether the transition is allowed or fovaktv w4y 1) 9wwk3t2b:pq2afrbidden, 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 in7(fqh5 jb7 etx 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- m -enp6+avcjr15q fz2wavelength X-rays emitted by electrons striking the face o jre12 qc vmn5azp-f+6f a $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-waveleqp9ypb, 9bhtdw9 2blfn2;s2ingth bremsstrahlung X-rays emitted from an X-ray tube have9l2b h; 9 p,b2 fdnwsq29tbipy $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wave/g 4iv f7g)0xo.v huxolength $\lambda_0$ 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 wavelengjw x pr3l+k0a -*thbbhd+n1v+)w 8ajvth emitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength pufwt 7.6jt)y for an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory toy5w,-f1i(4 supu mtcf 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 t,dw: lj )(x+von0amahl4 n*ean unknown material is bombarded with electrons. The wavele +d(j):mvohnw*xl 4en 0 aal,tngth 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 det +a0 e ci e,k 1w-z*9)b8m;ghqfqopxbl)mdx r9ached 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 might have a power of vow wf5b f)scs79w-x( s*mgka *xm:,w$0.50 \, \text{mW}$ 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 diffr.hvj.hwv9m z 1action if the beam emerges throughhwmz9 .h1 v.vj 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+qc u-a 7gi5zq-Ne laser?$\lambda $ =    nm

Use the uncertainty principle to estimate m) x 3gt4vi,h.f5ee-mb wcuz z.betc1k, b((i the position uncertainty for the electron in the ground state of the hydrogen atom. [Hint: Determine the momentum using the Bohr fe,xme(3euzkbtvhi.z ),c-tw (.m5 bbi14cg 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 t0wpg3 hncrmofcc :zc(x 1r:z. 8pz*7p )mqvw3he 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 electron orbit, estiml 61 zmv/i 7jjyef55h,f7xm mwate the average distance from the nucleus for an electroh,f7jv5izmm 1l7w/y 5 mex6f jn 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 ahy*y); ap7f skt $95 \, \text{kV}$ 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 o irt 8 r8wk2:6;ygzwdwutermost 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. Bia *ldf/cz 9avqsc. l+7) uqgn.x) yn/elow) 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 s- r5;w5lh5iy nduf qk7swt-e5tates 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 electrqxm,36/zphc2k4.h ziwtr u/; h;pwx lon states possible for a given value of ;p w xiqk24lc6pw.; m t z/xhhhz3/ur,$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 4x l0kpyw 1 rc rb+tb:he4(l.+f5 kV is shot through two narrow slits in a barrier. h+bb0l y(xfcr+4er .w lpt:k1The 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 bothxr 96us;sa6 tiy3ubo; by the Bohr model and by quantum mechanics. Compare the;bya6u3ts9x rui; o6 s 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:m/* w:c.+g ex v6mdgqygzq4 yn(a 0h1kwydp+ 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 wav4*bzjhyu3h x ;elength centered at 488 nm. Careful measurements show 4hh3b; y*jxuz 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 fr-f0wy 1vh jmale +8pp;om rest across 550 V. They are then directed at two slits 0.70 + eafp1pj hv8lw-;m 0ymm apart. 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 accelerated a fkm0f1cfw.nyul(ti.0vnb4 -b /s9 tscross the same voltage. Assuming that the uncertainty in their position is given by their de Broglie wavelength, find the ratio of the k1n w 9-4lc/ft(0fbv 0ytu. fsb nism.uncertainty in their momentum.

  If the principal quantum number n were limited to the range from 1yx o4yv+t7o *f9gcb9a to 6, how many elements would we find ig c v9otyo*yx97+b4afn nature?   

  If your de Broglie wavelength were 0.50 m, how fast would you be m 1lp1zi w- ;*63y: zgwscnspoxoving if your mass is 7 :p1 z3w6syzwlc ;ngxpio*1s-5.0 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 wit(e(oles(t v+p h one selt +ev( (o(pout 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 3t/ljl j+)r1wj 5;hzx b lm(dm0.154 nm are emitted from the surface of a certain metal when it is bombarded with high energy radiation. If this m/3 txjjz )+jl l1rlmhb 5;d(wphoton wavelength corresponds to the $$K_\alpha$$ line, what is the element? ----待选择----nickelcopperzinc 

Show that the diffractive spread of a laserp3u6;8h r/qz)io leatlp *wwhzs83 i1 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.]