Compare a matter wave twv/96 j myyoms 3(s*(sbbo4g $\Psi$ to (a) a wave on a string, (b) an EM wave. Discuss similarities and differences.

Explain why Bohr's theory of the atos33*ij)itw ge(hj*-l*j ng vxm is not compatible with quantum mechanics, particulevw j3h i3i**(-ljs*t )gn gjxarly the uncertainty principle.

Explain why it is that the more massive an o4y:9b9mir 8 (joom tmxbject is, the easier it becomes to predict itsommm( 9j4bxro9y:8it future position.

In view of the uncertainty principle, why does a baseball seem to have a wdv0xx1f 0i z,y9i*dmkell-defined position and speed whereas an electron does nz*9d xidvxiy0 ,m k01fot?

  Would it ever be possible to balance d1c4tdj cer(jmxk6jy:+ , 1mp a very sharp needle precisely on its point? Explaijcpxt jd+ :dm(1 ryk4j ecm6,1n.

  A cold thermometer is placed in a hot wl a/q5qw 56a:gxw +hrbowl of soup. Will the temperature reading of the thermometer be the same as the temperature of the aa5/gwh5 l6:rwqwq +xhot soup before the measurement was made? Explain.

  Does the uncertainty principle set a limit to howci;zlz fvc87+e *) jhrsc5c;6gjryy w*qx30z well you can make any single measurement of p7 ;jsyx 0h*;c cl)zfierzg w*5+ c683cryqj zvosition?

If you knew the position of an objec3kxpb;p4ip)3 v wi h*(5 lgelzt precisely, with no uncertainty, how well would you know its momentvp4giz (pewb5)xpk*ilh33l;um?

  When you check the pressure in a tire, doesn't somek,ni;-pb q*)ere *x /mc9zqni air inevitably escape? Is it possible to avoid this escape of air a9qn r*;*q/)i ebxpmkz,nei- c ltogether? What is the relation to the uncertainty principle?

  It has been said that the ground-state energy in the hydrogen atom can be p x 6clov r6n1/b7oi0qyg1 no.lrecisely known but the excited states have some uncertainty in their values (an "energy width"). Is this consistent with the uncertainty principle in its energy form? Expla nqb7cl0n 6v1o 1/ox.6lg oiyrin.

Which model of the hydrogen atom, the Bohr modeh,hn4a1ufz 8 ks(ii f-l or the quantum-mechanical model, predicts that the electron spends more time near the nucleus ahfinhi4zuk18,-(s f?

The size of atoms varies by only a frl (amnrps rl 8oyx64d177s:factor of three or so, from largest to smallest, yet the number of electrons varies from one t7 sllm1nxp: s48rfayr6r d( o7o over 100. Explain.

Excited hydrogen and excited helium atoms both radiate light as they jump4(ex ir;r6na k down to thek na46rx ;eir( $n=1, l=0, m_l=0$ 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 js ,dn xlcto2c32ry8q .nu:g*e quantum mechanics wernl:crou x 2cs,8 .y*nqgt2jd3e valid? Consider the first 20 elements or so.

  Which of the following electron configurations are not allowed:ratb(o qta *f 2(mf.4y
(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 sr 6sw exo/39iaya(q;a uranium atom (careful scrutiny across the periodic table on the sas/e 9a63oq(ryi; wx inside back cover will provide useful hints).

In what column of the periodic table would you e+oq k:v-o mf6lxpect to find the atom with each of the following confo+- l:qfmv6ok igurations:
(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 i 4qv0:z jf+fnproperties?

Explain why potassium and sodium exhibit similar pj9 7a1r 3vkrxqroperties.

The ionization energyryu + oc48 m nmeqlv0m.t:ze:5p-gg)v 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 wavelengtht9p : pyegehl4rxrl f6gnn (;/p0i52f in (Fig. Below) imply a photon nature for light? pr 9f /(er6n; yhxi:l lpef54gnt0p2 g

Why do we not expect perfect agreement between measuredor4i waz+pb1 l( b(*om values of X-ray line wavelengths and those calculated using Bohr theory?ar4*wlz(obp 1 b+( oim

How would you figure out which lines liyy; (xnf* y6in an X-ray spectrum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transitions deep within an atom to produce shorter walb fso9 j8.1qo9n you;velengths than transitions by outer electro onuf9j1.ybo ;9lq os8ns?

Compare spontaneous emission to stimulated emission.

How does laser light diayagj h b6 l*xo0s)y1e6sno4(ffer from ordinary light? How is it the same?

Explain how a 0.0005-W laser beam, photographed at a distance, can seem mu p3dw-fx*0;q9mvs: t0bb5vqg9jrr zk ch stro9 mzv*jpv9q-d 03;bq rrtx5fw:kbgs 0nger than a 1000-W street lamp at the same distance.

  Does the intensity of light from a laser fall ofucdq 5 6)9ac8wk1r*v2meq2qwv ,xzs(/zbo c bf as the inverse square of the distand qr6w1ub5aq2x/c vkq w )em( occzb*,vsz928ce? Explain. ----待选择----yesno 

  The neutrons in a par.9iop;/ rqpr1.ne8)q y/h 4yg2+exrwftf u eaallel beam, 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 we nai0t eizsr.wun.., * 7qa-dby) b,hith 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 measuh053v27n vs 75olkegasql7vg red 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 s8hkvja6y(s(:a p 6ht7z- kdwatate 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 uncertainty in the mass of ceiw:ow) 6 tan,*au c/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 coi6 xpirla/ of;pe66p/ntained in a typical nucleus / lea;r6x o/f6pi6ppiof 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 that if an electron were present in ec5,mq1 k5urfs3(jwd*s k4 rz the nucleuj rsf 1k ,mqec 543*k5szrwud(s $(r \approx 10^{-15} \, \text{m})$, 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 p*m;d.yl*w e9x7e kb1 pe (wrkaosition 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 cantirz v+:o rv- b 01h 7l7mokjn*vmg-6w be in the $n=6, l=3$ subshell?    elections

How many different states0r fq;;coph-04d fusq are possible for an electron whose principal quantum numpq;qs0ffcoh d-r0u; 4ber is $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each electron in th2 p8ygg f8ei*a.e;zvfe ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbertcnj/u-,d*6;rfmm a+;id z*a pv o g.ys for each electron in the ground state of nitrogen $(Z=7)$.

  Suppose a certain hydrogen atom hasj0k.frnlt 1chtl : )w8h; hk:p $l=4$. 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 elve0z v 2wq)3mvectron 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 has9 sex2m-i)r cnv1qjl4 $m_l=-3$, 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 in a4a d(uw*4s,xie6cvv xbg o -*jw e84it "g" subshell.    electrons

What is the full electron configuration in the ground stat re(b3 3vuvkds50a, gfe 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)k.2a +drb1l dc6 jg+ky gold (Au), (c) radium (Ra)? [Ha cd 2. 1k+rkjbylg6+dint: see the periodic table inside the back cover.]

  A hydrogen atom is i/b-m9ue.eyy2c 7 ezhhn 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 energy of the third electron in lithium zxbllb.gwb 9r;4rd 0/bw eqh7dp b.4)using the Bohr theory. [Hint: Thiqde ; 9l.x bdbw4wpr/ .bb0ghr7b)4lzs 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 momesdf1z 1gxe. trd9w:stv-fb45 ntum is zero for a filled subshell.

  For each of the following atomic transitions, state whether the transition is a; zal. szy/m)m(s1ka ullowed or forbkzzayu1.;(ml) s /samidden, 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 sdu3d:kr* kzk)2ur4z 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-rays emits m85y jz4em;fh bct,y56arr1 ted by electrons striking the face of a ajt1 m8rf6 e5 hm ;zcy,4yrb5s$33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremffu*h*m0k (v/ g+,b say tn1i zrj7ui:)jaj9 usstrahlung X-rays emitted from an X-ray tube haves r1u i,an +)f(7mz*ug yfk*9/uivbhj:jt0a j $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wavelengmn- .4 3r*zwkv4 vfidzz u,d-jth $\lambda_0$ 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 emitm -s yon3smgs 3i95c5.tsb pg2ted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength for i6yhyb.5htqdl4y c wrv0g 9).an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the wavelength for anyd tz0+sx6iiva12 pw, $n=3$ 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 unknow.vi(3 evo-giq n material is bombarded with electrons. The wavelength of tqi. -vg vio(3ehe $K_\alpha$ 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 deaf9/nxhu 23 dmtached 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 physi3bn ( hwu 3;tzf ,ykx1c48x-z6vexb ykcs 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 l blym0bj7a;o)l x6m-vn, zn*a aser beam due to diffraction if the beam emergba0j6blvmy)l;7mz *-xn o, n aes 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 laslsn* goj6h 2)pada.2ber?$\lambda $ =    nm

Use the uncertainty principle to estimate the position unca29qcxk6 / hrg rky(6sm, g;cnertainty 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 radc ( m26kq9xrgkcg ; an,s/6rhyius?

  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 angular momentum.nna qs 2v:fzd ,cgs5.k.6/yn g*oxsp $L$ 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 te* - cy7z/vdvgegqd77he radius of an electron orbit, estimate the average disd*gg ve q/dey7vz 7-7ctance 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 7gx6mt lj(d5jx84ma bv *5vu/o egoe- 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 (bindin,2hwcb5rm*8s 5d she ig) energy of the outermost 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) can be k(dd r hn*r) fguhp)y:dq)o70 ly0)zkwritten $\sqrt{\frac{1}{\lambda}} = a(Z - b)$, where $b \approx 1$, and evaluate $a$. a =    $\ \text{m}^{-\frac{1}{2}}$

(a) Show that the number of different states:o e 5a)htqwp xqu*h37 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 n,3m.ety eey;2pu,ayu6sy7 e .,;epeu yyny6myet3 ue,7a2s $n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV is shyu 6jgsq .vw92ot through two narrow slits in a barrier. The slits are a dista9v2syugwj .6q nce $2.0\times10^{-6}\ \text{m}$ 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 thejll:k5 m d) 9fvs0c+zj hydrogen atom is given both by the Bohr model and by quantum mechanics. Compare the results m l+kcjsj9d :0 zf5v)lfor $L = mvr$.
$L_{\text{Bohr}}$ =    $\hbar$
$L_{\text{QM}}$ =    $\hbar$ or $L_{\text{QM}}$ =    $\hbar$

  An 1100-kg car is traveling wd70k2sff x/qp+rc/bv 1 vn n9aith 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. Careargkm 720c6m,mmu,l bful measurements show the line is really spread out between 487 and 489 nm. Estimate ,,uck27mm rm6g0mabl the lifetime of the excited state that produced this line.
$\Delta t = $    $\times 10^{-14}\ \text{s}$

  Protons are accelerated v 8ypvv48l,esfrom rest across 550 V. They are then directed at two slits sp,vl 8yev48v0.70 mm apart. How far apart will the interference peaks be on a screen 28 m away? $\Delta y$ =    nm

An electron and a protox 2 hh8ep(*p9io/vhxl n, each initially at rest, are accelerated across the same voltage. Assumh2( 8iepx 9ohp*hlvx /ing 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 limited to the range from 1 to 6, h wk1dtrgs *k0h;s5.wyow many elements would we find 5trw *s01g y;.khd wksin nature?   

  If your de Broglie wavelength wrx/w 7 1 tyvvm4cgbu*7ere 0.50 m, how fast would you be moving if your mass is g 41*v7c v7uy/txmwr b75.0 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 elemepe3g fxdv jed d(bzl4p7r 3614nt shows a series of lines with one out of every four matching a line from the Lyman series of hydrogen. Assuming that the unknown element is an 3d1xz b 6g pe(f7p43 edvld4rjion 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 from the surface of a z2jibhr5sy.3 certain metal when it is bombari23ysjz .h5br ded with 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 laser bea2h (5kbbk0c) ;ihsk qzbd 3a1om, $$\theta \approx \lambda/D$$ , 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.]