Compare a matter wave) oja+7k9czgvx, l f+e $\Psi$ to (a) a wave on a string, (b) an EM wave. Discuss similarities and differences.

Explain why Bohr's theory of the atom is3,f tah qfn0(y not compatible with quantum mechanics, particularly the uncertainty princ0qfy( af 3ht,niple.

Explain why it is that the more massiv.c *um:cbsj u26l s/aze an object is, the easier it becomes to predict its future posit /buc.j cul2*mz a6s:sion.

In view of the uncertainty principle, why doj+dfiz:0a:v7;.yzbxo p m(58 qgvv coes a baseball seem to have a well-defined position and speed whereas an electrx: 57pag;0fyb8q.vji zzmvv o:d(o c+on does not?

  Would it ever be possible to balawzz/q2ef7 kw ;by ),7f6ap+ ,y srykqsnce a very sharp needle precisely on its point? Expfqssy kzaz7q2y/yr w,f we6k+)b;,7plain.

  A cold thermometer is plac.ri rgd,;vtx rry r60g91 o9qned in a hot bowl of soup. Will the temperature reading of the thermometer be the same as9r vryd9r6ro g.q ixr,1n;gt0 the temperature of the hot soup before the measurement was made? Explain.

  Does the uncertainty principlhz,1 rrv6ww )pe set a limit to how well you can make any single measv61,)rz w hprwurement of position?

If you knew the position of an object precisely, with no uncertainty, 3ecb6xgyk,sx* 2yp d (+bo .uyhow well would you kno 3xbb2xycu.po(k * ye6+, gsydw its momentum?

  When you check the pressure in a tik7a80e5q.d8 db jlwnvre, doesn't some air inevitably escape? Is it possible to avoid this escape of air altogether? What is the rdebwvnaq l.8 j8d k570elation to the uncertainty principle?

  It has been said that the ground-state energy iz n7*blga6+v+ n1ypvmn the hydrogen atom can be precisely known but the excited states have some uncertainty in their values (an "energy width"). Is this consl1y vaz7 +b 6m+vgnn*pistent with the uncertainty principle in its energy form? Explain.

Which model of the hydrogen atom, the Bohr mod ryi2h)m7ey+s0ixte9 el or the quantum-mechanical model, predicts that the electron spends more time near the ne)r0s2ihixte ym+y7 9 ucleus?

The size of atoms varies by onl9cf;nn z86jnd y a factor of three or so, from largest to smallest, yet the number of electrons 8 z9 jnfndn6c;varies from one to over 100. Explain.

Excited hydrogen and excited helium atoms both radiatew7 ojl597c 0nut cl6:s j(ahtu light as they jump down to the jusuc0j9:7hc7o l6t ltwa (n5 $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table look if there p q q 1*kdh.5ekr9k( zi fbzzlc78-o/jwere no electron spin but otherwise quantum mechanics were valid? Consider the first 20 el8qi 5(pqkockd/-z rezl z9h. kb 7jf1*ements or so.

  Which of the following electron configurations are qxh7a329 ci2*ly cihcz e.q-pnot 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 uraniu/3l: crnj (qvy y /u.4a2:qaoiubr0rum atom (careful scrutiny acr2 uny 0 lqvr oa u(/3ujyib/aq:c.:rr4oss the periodic table on the inside back cover will provide useful hints).

In what column of th 9t(uww+cosi/ .ophk*e periodic table would you expect to find the atom with each of the following configur/hp .ow+(io ut9c k*swations:
(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 propertienrz 6 e14lz 82m5ioi9qyflc0es?

Explain why potassium 4eg4k 6o-nkucsr9q mrc+my-- and sodium exhibit similar properties.

The ionization energ+b6cwwmc o( j53 7gyozy 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. ne h:t/d br7p58dlt;g(siyk o :5i3uoBelow) imply a photon nature for light? 5 ;d7ptg8 :s:ny il 3 (ebtrd5khioou/

Why do we not expect perfect agreement between measured vas ste 0/rj5cv- n+qrh/lues of X-ray line wavelengths and those calculated/ /ssvqe 0hj5trrnc -+ using Bohr theory?

How would you figure out which lines in an X-ray spectrum corresk8 a ;ci8sg- i8pitaw:s/pt6 kpond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transitions deep within an atom to produce st.j 4bdu* svj5ha:vc- (sj: xehorter wavelengths than transitions by ota jv xhv(:jsbed:js5*. u4c -uter electrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from ordinah u 29ydmiledjt:5. 0zry light? How is it the same?

Explain how a 0.0005-W laser beam, photographed at a distance, can seem much k;;w. : h(vp fwf o0vk*tcl uikz*x-9ystronger than a 1000-W street lamp at the sam.( ;yv9i-ufh 0pw: z*fk; l*kcvxtokwe distance.

  Does the intensity of light from a laser fall off as the inverse square of the( *z0wpvs pgh; distance? Expl z(sw*0;hg ppvain. ----待选择----yesno 

  The neutrons in a parallel beam, each having kinw 24f2l od5ezhetic 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 speed4b7)rzgr :s7yd gm- hu 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 coxfywqs hfp,05s8 x2-an 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 excited state of an atom for tyr4hur6ox 6gs8 pically $10^{-8} \, \text{s}$. 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 fw fr ojbp-4,t+tp l4v-ro+1f +2eqximass 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 typie6p)qgy 0cs(wmgaw.1 cal nucleus of radiwcw1sag6p) (qg.my0 eus $1.0 \times 10^{-15} \, \text{m}$. [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 tr. z ow *pqgpy bjnrf:,t/p+1jf 9*z8that if an electron were present in the nucttbpjg8q*z /+,zpp:1 rfn.j o9 y* wfrleus $(r \approx 10^{-15} \, \text{m})$, 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 l5g2(txb;vbiy4k ec/ 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 ca/pri2 yx69b nvn be in the $n=6, l=3$ subshell?    elections

How many different states are possible for an electron whose principal qua z,qyzya8 (64pveet (vntum number i4pvzyq( t6eaezv( y ,8s $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each electrn0 ljzr0 *oob+on in the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each electron in the ground state of nitrogej-y ;sk n 30oajd+ uo01q*3wplk:e8oml/ zqis n $(Z=7)$.

  Suppose a certain hydrogen atominl:hf f xweam-l43o;oo)53w 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 ,m4w+pjzh (,qr ) xzcfin the $n=4, l=3$ state of hydrogen.
L =    $\times 10^{-34}\ \text{kg} \cdot \text{m}^2/\text{s}$

  If a hydrogen atom haj5 tv8y)525h0a fs8iigi9 wf4f2vouq ,ug ecvs $m_l=-3$, 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 "g" sun86is*z*c o.bdie 8*x hq7 jlvbshell.    electrons

What is the full electron configuration in the ground state for elemenf0,;c)pj+p9g pwxdupg7 (v r cts 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 (Au)/ -f * do(mknruyrdl/ur(lw6:, (c) radium (Ra)? [Hint: see the periodic table inside the bawdrrrm(/ -( o uy/lu:klf6d*nck cover.]

  A hydrogen atom is insmvs7 5jlcmy0a9b57o 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 lithlf3opwewr( tkdy22 l zg/ 6ju,j-qv11ium using the Bohr theory. [Hin z3ldw wf1-p qyjvrkt2g62o ,luj1/e( 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 angular mome:x fki(ic1qo4 ntum is zero for a filled subshell.

  For each of the following atomic transitions, state whether the transition is81h1 o wnj,b4neuuc,s allowed or forbidden, and if forbidde b wneu1 no1hcj,84us,n, 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 id.g1kuql 4qk2g rp ( * lba1sm01ginc t;w7sc2n 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-wave :gfkq n/t3btm5v-*c wuvbz+6length X-rays emitted by electrons striking the face of a u5gzn/qw3f+ :b*vt6t bm-v ck$33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremsstrahlung X-rays emm/x p /na0mk3citted from an X-ray tube haveaxmn3km 0cp // $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff +mu-n/ 2kmj glf3uz( rwavelength $\lambda_0$ 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 e3s/(o dr(2 va:auqx aji-5z fpmitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelengthr 4/xd srrh+g8gqv1a, 3nl6dh for an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the wavelength j6o:h5 tzdda3 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 isjc5w3t8tl ,d 0eddt1e bombarded with electrons. The wavel5dtcjwdl 8dt3t,1e0eength 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 detached rej5j x5m0e2q gn5pczrfs +l5 i8tinas 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 physihf7zi-4kq5 bowxp ly 934s:/jsx gzu 0cs 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 laser beam hb)6gng.5t x:t nqe4 d9 *ffbldue to diffraction if the beam emerges g5*. fqttf6bb: hxle9)d nn 4gthrough 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-lqr r(x*x.. fxscf 2u2Ne laser?$\lambda $ =    nm

Use the uncertainty principle to estimate the pos y xj(no.2 te,hcc6s7kition 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 ths7kcyexj t(.on ,26 che 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 paww ;bk-z8 kn dvs*j80i:sf3mm7a4 xtossible values 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 5zpid6thixtxi0,- l5 of an electron orbit, estimate the average distance from the nucleus for an electron in th 0ti,-zx 5hldx i5tp6ie 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 at y aspx)6yu;d.$95 \, \text{kV}$ 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 outm t lyl1e8-e9m7jyje4ermost 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 6 gxw bfm,yl*r99g;br the Moseley 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 diffe mho4mmg0 ml/+rent states 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 vs ; +ft(iaq :4o7qrwyvalue of 7 taqr(4vs iw;o :fyq+$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV is shot througx fxf3q.(q5 1 j2egcgph two narrow slits in a barrier. Th(cjgxq f3q egx1 52p.fe 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 : hji0hjmn ter3l5pr e./ d2y1hydrogen atom is given both by the Bohr model and by quantum mechanics. Compare .l rhh 3iy20d npr: 5jeetm1/jthe results for $L = mvr$.
$L_{\text{Bohr}}$ =    $\hbar$
$L_{\text{QM}}$ =    $\hbar$ or $L_{\text{QM}}$ =    $\hbar$

  An 1100-kg car is traveling. 0a 8kba)0ei*zjsu pp 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 wit1a8dhg.ver,qs8cs.7)g5 n fxk w:l o wh a wavelength centered at 488 nm. Careful measurements show the line is really spread out betw)xh,naf8c . gs1w:ervwg.s 7qdkl 85oeen 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 sw:u7a2tvno1 across 550 V. They are then directed at two slits 0t :7s1wuon2 av.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, are accelerated 06 b ,/lkvbvx.3vogf macross the same voltage. Assuming that the uncertainty in their position is given by their de B06g/.vmbvblvkx 3,fo roglie wavelength, find the ratio of the uncertainty in their momentum.

  If the principal quantum number n were limited to the ra k8w:cy//-cyjymr.eu nge from 1 to 6, how many eky:c/m ywc-e uyj/8.r lements would we find in nature?   

  If your de Broglie wavelength were 0.50 m, how f):q01*q jgf ,wsoz gzaast would you be moving if your mass is 75.ofzzsgg,:w ) qa1 jq0*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 a;*4 jxx g.p7fi08l/9, eljgqg lbpocwith 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, determ,7bl4*j0pjo;fg .pi /9g8 q lxc xglaeine Z and the element in question. ----待选择----lithium berylliumboron 

  Photons of wavelength 0.154 nm are emitted from the sur4xtwmzh /w1cm2 vzxka861ex2 face of a certain metal when it is bombarded with high en12w 4tkxxmm81/v6zz a wc2xheergy radiation. If this photon wavelength corresponds to the $$K_\alpha$$ line, what is the element? ----待选择----nickelcopperzinc 

Show that the diffractive spreax hvi: ht2oz m*3cx(y;d 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.]