Compare a matter wavep1;xr0fgw p,62 dds zq;k. bo5a9quy s $\Psi$ 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 quantum0/ xbbmy,cg/v mechanics, particular gc/bxb/v 0,myly the uncertainty principle.

Explain why it is that the more massive an ok a .tcez1i(1-u(e qttbject is, the easier it becomes to predict its future pa k(1 (cut.t1ietq- zeosition.

In view of the uncertaint /lwpke+m; 5shy principle, why does a baseball seem to have a well-defined position and speed w+5mhkpsew; /l hereas an electron does not?

  Would it ever be possible to balance a very sharp needle precisely on its enyqw( p9 eiuvt j6 a*7/9(dszpoint? Explain d s *qjie9(ptwz 7en9/u(avy6.

  A cold thermometer is placed in a hotmzu2 10v.n qau bowl of soup. Will the temperature reading of the tma. 0qu1u2vnzhermometer be the 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 sinanct)ybv x38vtnv14ba s3 08wgle measurement of positioc1sbt3a4n w)3n vx08vtab 8yvn?

If you knew the position of an object precisely, withx3*yrhrf v q,6gj +4eobs)a1v no uncertainty, how well would you know its m)4fq*vxsbr6 jra1 yve o +,h3gomentum?

  When you check the pressure in a tire, doesn't some air inevitably escapekrz;viduz d4x5 t81 9h? Is it possible to avoid this escape of air altogether? What isddzit 945 8; xzuh1vkr the relation to the uncertainty principle?

  It has been said that the ground-state energy in the hydrogen ath-hh(l/t ozv3 y/to,6h yug 8oom can be precisely knt //h,touy-(vhz8oo6yh g 3l hown but the excited 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 hydrogen atom, the Bohr model or theu2:jc of8gk,f quantum-mechanical modelf j:gkfu82 o,c, predicts that the electron spends more time near the nucleus?

The size of atoms varies by only a factor of three or so, from largest to smalffn7 vp-qd91 /a2z wgglest,vwzafgg-1pd9q/f 7 n2 yet the number of electrons varies from one to over 100. Explain.

Excited hydrogen and excited helium atoms both radiab .gxfebpf4zl,:1 riu eehg/l1c) k+t/r4a* lte light as they jump down to the /) f, agzebhcr *k4+tu/l 1ifl:1le4.ergpx b $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table look if there were no electrondi+9gefojz86 2 4sk8df yu- -ye mvd-m spin but otherwise quantum 9ys4dd6 k2m-jv fe8-om+fygu-d8 zei mechanics were valid? Consider the first 20 elements or so.

  Which of the following electron confwjtn+1ecwc d2:87ognzcdc, p3 py ,l-igurations 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 ura(l5zm tqqbi7 1nium atom (careful scrutiny across the p15 t7imqqlz(beriodic table on the inside back cover will provide useful hints).

In what column of the periodic table would you expect to find thznoghgc-ly87gn +:c ) e atom with each of the following configurations: gz -lgh :+)y8c7c ongn
(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 propert*dsi-m/7xl;nayo 6oir:q hx 6ies?

Explain why potassium and sodium exhibit 8dl;4z lz qw4uov-3oiu* ko yl+g v./zsimilar properties.

The ionization energy f8vc * (lge2/xe8jf nzpor 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) / q2(xo gyj2wjimply a photon nature for light? ojj 2x/y2(qwg

Why do we not expect perfect agreement (ma kkzcj23k*between measured values of X-ray line wavelengtk*3m j( akcz2khs and those calculated using Bohr theory?

How would you figure out which lines in an X-ray spectrum correspond towyog:,) o 2bnt $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transitions deep within an atom to pf9f, xvd,o y wy63lyq qmp-)r0roduce shorter wavelengths than transitions by 3vqdl6y9 xr-wm0 y, )fqo, fypouter electrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from ordinary light? Hotp./qy .tti d0w is it the same?

Explain how a 0.0005-W laser beam, photographed udak s 0g* 13v6x1o)-anw j0vf.5nl9l pril wkat a distance, can seem much stro- xn pl a*lsk)0n61 u3 vwo1lvrik0fdwj9g5.anger than a 1000-W street lamp at the same distance.

  Does the intensity of light from a laser fall off as 4p6+ r/kma( qityt up8the inverse square of the distance( p+q6tkyi4/p8m urat? Explain. ----待选择----yesno 

  The neutrons in a parallel beam, each having kinetic energy e-n0r z 2 xfqofly)qf 9xoh/9;$0.40 \, \text{eV}$ 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 a23p(q5 vzrrhn0lkj e3 r3l-fm 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 posi2 qwj2 z8mycu+g 5k8gstion can be 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 excijg(t3)dx2is-an n al3 cd2 (o7lps v6tted 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 the mass of a muon c tp )wz(p:jh; biy28k$(m=105.7 \, \text{MeV/c}^2)$, 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 energyo;s/n*s n)0tbesnm;5w cya4l of a neutron contained in a typical nucleus of ;s5 *4naetcm o by lsw;0)nn/sradius $1.0 \times 10^{-15} \, \text{m}$. [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 i)cg rx/.hynywm2 wg v9i35 0xin the nuc2hg 0y3gcii 95 nmxw/r).wvyxleus $(r \approx 10^{-15} \, \text{m})$, 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 a9 k71yxdwa7zgwg;jak)o :+ji $3.00$-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 h7 4f jsd 9b+4/c0ixod7bsjqzbe in the $n=6, l=3$ subshell?    elections

How many different states are possc-pyhlcx 8w -yivw-t .6c)98byh;p uf ible for an electron whose principal quantum number is p hw8;-yy6vc8x-u.tcbl)pch iy - f9w $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each ele/6q0f qty7s ,zknv,s5 6oj vuactron in the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers+ r;rtl 7gd;gt for each electron in the ground state of nitrogen $(Z=7)$.

  Suppose a certain hydrogen atom c3xkan lsi19*tzhe3q(nkg z g nuh3u9087wg4 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 momentum of an electron (w6o- fdbfr3g8 fy y6rin the $n=4, l=3$ state of hydrogen.
L =    $\times 10^{-34}\ \text{kg} \cdot \text{m}^2/\text{s}$

  If a hydrogen atom h7pmjmcltt+ ;+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 in a ifw0tb1ve -z31wl nhn8a ss7xj 90- pb5 dz.sv"g" subshell.    electrons

What is the full electrou b;h7rp ykhem:n(a1p h 9)hv)n configuration 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 configuration formb5l;.p+m. pm hn4iyj 7j*h li (a) selenium (Se), (b) gold (Au), (c) hp.n5pj7l l; 4mhjyimm b i*.+radium (Ra)? [Hint: see the periodic table inside the back cover.]

  A hydrogen atom is in the fs)5gec;jx ,o+r r7/w6nuxdj $6s$ 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 elect256 5 psi zwgnlr/ t:ea;85gl 9dsmnwtron in lithium using the Bohr theory illmw56 drae:/n sng58wtg 2s;z59tp. [Hint: 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 momentum is zero for a filled subshell+akf y0,- 3k /c/8ze otv fy*dzgf2hel.

  For each of the following atomic transitions, state whether the transitb- uh:qr(w +m mlz0vjbo;m 0g)ion is allowed or forbidden, and if forbidden, what r0mq o+ bl)hv(gzb um0;jw- :mrule 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 inn.(phctluwl -/br9k0/s,z t o 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 emitted by electrons striking the jrvhor/9. n :wface of a :r/hv nw jo.r9$33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremsstrav (q2ea v,(s bjg-a ;-axr5nur0t 9svhhlung X-rays emitted from an X-ray tube have ,;-a2h-tnqxsvv95r sgeu (j baavr (0$\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wavew bgam z;v0o:e 40yp6,dn*wsmzq m1b /length $\lambda_0$ 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 wavelen ,i n:jnn7vm73z0 ayrngth emitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelengc dj3 c;+w. bq ,pujp3xp27jt3aikd a-th for an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the:wu1 0zeghz/2a .pno q 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 unknowx1f7.budo xd ,hy5o5bn material is bombarded with electrons. The waveled1hx5oxdb7 y.5ofub ,ngth 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 detau ou+ya/ 15+;5t0mr unr lnh4 q:vproyched 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 physncxz*lc0; -dn 4qsg)aics 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 angularf+-j7; qw qt 4pzdrx(i spread of a laser beam due to diffraction if the beam emerges throuiw(zxf jq pq;7r4 td-+gh 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 la kpo/:t cojn85cm+p(v ser?$\lambda $ =    nm

Use the uncertainty principle to estimates,x xdetj (v14)9urbi the position uncertainty for the electron in the ground state of the hydrogen ai)xjxdb,vrus4 t( 1 e9tom. [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 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 va3f8nre acet8m, htk; /lues 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,mb )f- tap)9:9qiud4j tyu p9o estimate the average distance from the nucleus for an electron in thp9op y)9qf:9-m4dtbjauti u )e 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 operatoubj1 67kpkleju. e(6 es 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 (binding) energy of t2yiuvr a/nt hdz 042f+he 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 w b+t zp xymegk0+*v;ysvk,-3 gritten $\sqrt{\frac{1}{\lambda}} = a(Z - b)$, where $b \approx 1$, and evaluate $a$. a =    $\ \text{m}^{-\frac{1}{2}}$

(a) Show that the number of different states for qmxrg(,1t h479x(lftik1 et qa 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 ahz.q(ed-:np t given value of ht. -pnd:qez($n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with enk* e l5 a1;wqr gv.pga 6ouwwf1sa::i7ergy 45 kV is shot through two narrow slits in a barrierks1:65 voe1qagauga f 7l:wwr i ;p.*w. 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 by the Bohr model amq ygq e,8e8q(nd by quantum mechanicm8 (eey ,gq8qqs. 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 with a sfwg+vypir.uun+0s, *o1.vinei 2 x 2tpeed 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. Carect;, qzcdq:6) +kndnq ez;v2 xful measurements show the line z zndtxdc;qq,v2+)ec :6n kq ;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.,na z8 7x c2*io8mpsmd They are then directed at two slits 0.70 mm apart. How far,nc*iszmp72xm8 8 d oa 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 acc5ev ab.l90oom elerated across the same voltage. Assuming that the unb9lm ao5o0v. ecertainty 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 6tdh/a 5v)a2ws tquuqfb5).gto the range from 1 to 6, how many elements would we find in nature?duqa v6fq2 /btt)u)g5shw 5a.   

  If your de Broglie wavelength were 0.50 m, how fasdvz;n i sa),k+48gl szna8x *ht would you be moving if your mass is 75.0 kg? dash lkz,;s*nn )4v 8azxg8i +$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 wm geqei;qz9boz(;, xr. j: w;dith one out of every four matching a line from the Lyman series of hydrogen. Assuming that the unknown eliwm (;qg9 bz; ;qz,je.ore:dxement is an ion with Z protons and one electron, determine Z and the element in question. ----待选择----lithium berylliumboron 

  Photons of wavelength 0hv2o/0 ov)ibp(/jsje y*fgzqw8 k)+v .154 nm are emitted from the surface of a certain metal when it is bombarded wpo w /h80vjyk*f)v/+ beio)gq(2 svzjith 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 )xkl/+c .cskz5bk+ z3jxeu r. 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.]