Compare a matter wave k4 k5l zs rdgw:uxv92-$\Psi$ to (a) a wave on a string, (b) an EM wave. Discuss similarities and differences.

Explain why Bohr's theory of the atom is nfbh;edp 0hl 3ef-ymo6c 3 g7tb;f3eh5ot compatible with quantum mechanics, particularly 5;dhhhbmf f; ylbt63 e33e-fo7g c0epthe uncertainty principle.

Explain why it is that the more massikrvc k*9ge z80ve an object is, the easier it becomes to predickr8v0egcz9 k* t its future position.

In view of the uncertainty principle, why does a b7jfm 4h,3 dsgyirqmci,63ie 7aseball seem to have a well-defined 3gjsimf 6iie377d cq rm4, y,hposition and speed whereas an electron does not?

  Would it ever be possible to balance a very sharp needle precisely on its ( 82bnm p* 5jw:1jt aqnx-rlbu/omi/ gpoint? Examtblq/-2 81gu b:ri5(xop*/ wjn nmj plain.

  A cold thermometer is placed in a hu p.0i7 w-lk ;iej,+1dh5dtlqod si9 uot bowl of soup. Will the temperature reading of the thermometer be t. +s 0 el7p kuiwi;hiqd-lu,djtod519he same as the temperature of the hot soup before the measurement was made? Explain.

  Does the uncertainty principle set a limit to how well you can make any single m-e qa*0b 5md-zdjp,jq easurement of posit qzj0d-ab pm q*d,je5-ion?

If you knew the position of an object preciselk 2 7ts1htgsd6efvuhp(6.a6zy, with no uncertainty, how well would you know ade( fktv1 phs2 66.6g zst7huits momentum?

  When you check the pressure inbb3)mrb n .x.f a tire, doesn't some air inevitably escape? Is it possible to avoid this escn)bx r3b. bf.mape of air altogether? What is the relation to the uncertainty principle?

  It has been said that the ground-state energy in tlr2*y9edruz9 wbk* *khe hydrogen atom can be precisely known but the excited states have some uncertainty in their values (an "energy width"). Is this consistent wi*e yrbl9*kzdw*k 92ru th the uncertainty principle in its energy form? Explain.

Which model of the hydrogen atom, the Bohr mo 1ivjr 6ubitlwh /yvn(x24f0-t5e c+kdel or the quantum-mechanical model, predicts that the electf5 jw2l-et6 nkbcx4t1ivvu ry(h/ 0+iron spends more time near the nucleus?

The size of atoms varies by only a factor of three or so, ) /y ymo tjfa(7:t6zeffrom largest to smallest, yet the number oft (7oyjafe/ f)ty :z6m electrons varies from one to over 100. Explain.

Excited hydrogen and excited helium atoms both radiate light as tentcuqvq- 7w6;q 4 k:sp* ::frn,xcgjhey jump down to 4:vqck 7;sn: cu p ,:efnrwq* t-j6gqxthe $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table look i iqncf i2 j,syvi51dv-f68ur*f there were no electron spin but otherwise quantum mechanics were valid? Consider the first 20 eleu qin syd-ji,f562 i*cf1vr v8ments or so.

  Which of the following electron configurations are not allowedo7rw7kpk x0sxk wkij175q f2 cp):dr4:
(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 aked85wzko*/w5 n7ork (t nmg0tom (careful scrutiny across the periodic table on the inside back cover will provide ukmkn wkn(0rwodgeo5z *5/7 t 8seful hints).

In what column of the periodic table would you expect to find the atom with e- dg6)u,h bmzcach of the fz ud h-6gm)cb,ollowing 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 exhibit similar properti.tnfj1r,7z txes?

Explain why potassium and so/ qvgb8 sr,e5pdium exhibit similar properties.

The ionization energdva1ljh *x- p8lfd8w ;y 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) ilcp a*5w 0s roo0g5pt58 rihc:mply a photon nature for light? 8li0pcgt hp:*ar0c55 rs5woo

Why do we not expect perfect agreement between measure7y c,;9w:ow xoj)y: zmmo kl(kd values of X-ray line wavelengths o(yj)9;o7m, :o:ky xz mlkcwwand those calculated using Bohr theory?

How would you figure out which lines in an X-ray spectrb+x;a76uh) r jkvbt82 p pds.,zypw3 yum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transitiony, d6f6msxx0c/ jnb80 dbvew5s deep within an atom to produce shorter wavelengths than transitions by outer electronsdbx dbx06/fm8 nvy5c0 ,jsew6?

Compare spontaneous emission to stimulated emission.

How does laser light differ c;ed5 2moahj9from ordinary light? How is it the same?

Explain how a 0.0005-W laser beam, photb(flazn2 6k/aw4 n r th;/ps*q9-sgccographed at a distance, can seem much strong-4;a kf*wsa(c62 s/ngz/ phrnlcb 9qter than a 1000-W street lamp at the same distance.

  Does the intensity of light from a laser fall off as tty+cbar ik+z qlhuxw4 ; 2*) 9v46zy/f 5jizdphe inverse square of the distance? Explai pvlij f+x9z6wd 5*q c/+y;aibu4)r4 kzht2yzn. ----待选择----yesno 

  The neutrons in a parallel beam, each having ,r)q9d+pk y dvverd) w jei9.6kinetic 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 w:3by6tcft+4 )8m.gbhf ka dwoith 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 accura.(t bz1vme vd+cy 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 i38v )oqxp6gq uo8bw/s n an 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 uncertainty in , nt5qu565u vyxl5lfsrt9 3ohthe 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 energyn ,ckz) (o,mal of a neutron contained in a typical nucleus of radiuk)(, coma,zl ns $1.0 \times 10^{-15} \, \text{m}$. [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 preseu5yc 7- g 5b2kcd)x2 batkht.e285xeyypyc a3nt in the nucl tc.2g3 b5x5beaypa y-2chy k82 yukdc57ext) eus $(r \approx 10^{-15} \, \text{m})$, 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 tydb895t76 (vbos mu( vphx:pr he position 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 in ,vgp(8l1nt1 sx u*a1e 9yop.idhc7plthe $n=6, l=3$ subshell?    elections

How many different states are possible focgq(jl. ni1 iu;jwfi32(y,af r an electron whose principal quantum numb jq;1 2 jnlaycfg3(, .iufwii(er is $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each elect7lkq2 si3oc 7 *ely5joron in the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for eack;x2zaxst6noy ;p*f0h electron in the ground state of nitrogen $(Z=7)$.

  Suppose a certain hyuq:pf /-qb-ofh2ivmh 9r+sx2drogen atom 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 angular momentu7s0v7c8d* itnen ) 9emx+jcbqm 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 hask6ztls06- aph $m_l=-3$, 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 electr +-3w xne:eisgons in a "g" subshell.    electrons

What is the full electron configurb/if ri2ghq)y s *ev/8ation in 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 confi0wj va h7jom+.guration for (a) selenium (Se), (b) gold (Au), (c) radium (Ra)? [Hint: see the per v+0jaw.jm o7hiodic table inside the back cover.]

  A hydrogen atom is in thc9h3 3r0,thtwjt re p9e $6s$ 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 2od28n,zpsv8j my2 otelectron in lithium using the Bohr theory. [Hinz j82so2yot mnvp8d2,t: 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 angulbq +e(xyt h448/ lltolar momentum is zero for a filled subshell.

  For each of the followinjqrg31gw2 xw+x pxm08g atomic transitions, state whether the transition is allowed or forbidden, and if forbidden, whx2mj1w 0 q +3p8xgxwgrat 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 iwb6l( f- uf)sy/r t zbpz.t:.$6d$ 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 by electrons striking thes ti f q oet2ubg6:qn46 kf5-:yppfl;/ face of a gqfi5ep;k6sqt:t y of 2 p: 6/n4bul-f$33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremsstrahlung X-rays emitted fromuop 64k1zy fu ap qn)v;:c)h*e an X-ray tube havez oc))4;q1a* n up:peu h6kyvf $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wav c0oo.um 1s7wkelength $\lambda_0$ 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 wavelengc ny/ qi/4f8vbth emitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelengs462 y-d 1ryg l jrsqgdo42jaqs3/ 9mwth for an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the wavelen(t7jwvym 4 k/*cbry*mq2mr 1sgth 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 bombarded with elex8 ,/8u q o0yz -3so(5wvdxmmgaps:ux ctrons. The wavelength ofwu:z/go0a88dx-xsu , m q(m3yv5s xop 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 re5vvi (t-6 vf6w *cbmartinas 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 qhx-.,)n4/x5)8ulw fjh bvcm+t m vp xof $0.50 \, \text{mW}$ 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 spread6 ceqkn vq75e3 of a laser beam due to diffraction if the beam emerges thr3n7ece56kvqq ough 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 lasevtyj2kg8k9d 4bvkd1 1r?$\lambda $ =    nm

Use the uncertainty principle to estimate the position uncertainty for the elecra,z 2hj *go7htron in the ground state of the hydrogen atom. [Hint: Determine the momentum using the Bohr model of Section 27-12 and assume the momentum hzhg 7arj*,o 2can 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 valuhbc(j:dv;o.yup8i 8a7pt k 2qes 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 electron orbit, estimate z*qs zfo q57+vthe average distance from the nucleus for aq5z zqo*sv7f+ n 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 at5,d)2*w ub:mjj fgtso $95 \, \text{kV}$ 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 ou-e 1/pr-dpbkq cg.+i gtermost 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 wriay gf91 smdfk*;c 54v1n)kk fwtten $\sqrt{\frac{1}{\lambda}} = a(Z - b)$, where $b \approx 1$, and evaluate $a$. a =    $\ \text{m}^{-\frac{1}{2}}$

(a) Show that the number of different states foraig qwux9d-,, 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 differenc48i+k0hozeqlzt) m(z / je etx8(2 jlt electron states possible for a given value of +8 z/ c0)ztlteoke zh(ji(m2 j84leqx$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons wob s prxs3.:o7ith energy 45 kV is shot through two narrow slits in a barrier. The slits are a dir: sp7 sxbo3o.stance $2.0\times10^{-6}\ \text{m}$ 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 Bohn /zz6(d+t8eu8zqik0yhdk- tr model and by quantum mechi du0y zn6z8zk -tdh+q8te/(kanics. Compare the results for $L = mvr$.
$L_{\text{Bohr}}$ =    $\hbar$
$L_{\text{QM}}$ =    $\hbar$ or $L_{\text{QM}}$ =    $\hbar$

  An 1100-kg car is traveling wish g: m/k j( atoe:c1.qgw62yeth 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 wavelh: k1-zg6qjh 9ew4ocdength centered at 488 nm. Careful measurements show the line is really spread out between 487 and 489 nm.zhj9-dhwc 41 qok:e g6 Estimate the lifetime of the excited state that produced this line.
$\Delta t = $    $\times 10^{-14}\ \text{s}$

  Protons are accelerated from resttqwz q;bp78x 3 across 550 V. They are then directed at two slitx zbp8twq7;q 3s 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, each initially at rest,s8q9- 3jqka0kq/bp(lq*mv et8/rlx lzxb0 j1 are accelerated across the same voltage. Assuming that the uncertainty in their position is given by their de Broglie wavelength, find the ratio of the uncertaintkq/slq zx /8 l0avx q01bltb 8q-*mjp j3(e9kry in their momentum.

  If the principal quantum number n were d5w623 pu0oxco xk)vjlimited to the range from 1 to 6, how many elements would we find in nature?p6x32k dco5j)ouv 0 wx   

  If your de Broglie wavelength were 0.50 m, how fast would you be moving iffx:7 (aj ofk6a your 7:faj6afxk (omass 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 otqb,4;hcuh fak6 06elasb0)one out of every four matching a line from the Lyman series of hydr a f 6a,0o;)0h6k h4utcesbbqlogen. 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 nm are emitttireao v 6s0*(oj7/c led from the surface of a certain metal when it is bombarded with high energy radiation. If this photon wavelength co/o7lri *stjo06 (ev carresponds to the $$K_\alpha$$ line, what is the element? ----待选择----nickelcopperzinc 

Show that the diffractive spread of a 3mdve*do pe7 p-kpj9 )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.]