Compare a matter wave v dz (r 7ek-:f;vk;25rnwelir$\Psi$ 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 compatic.desvu,fr p; ,n8g1mu k9y8jble with quantum mechanics, particularly the uncertai, g, 8je 1c8f.py9kmrs;uun vdnty principle.

Explain why it is that the more massive an i79)q/eua iweecto.7p(y q h8 object is, the easier it becomes to predict its future position.oe(yqit aucew /i.7 8e qp79)h

In view of the uncertainty principle, why does a baseball seem to have a well-xwrr l,)hg696zu8ax ydefined position and speed whezr 6w )xa9ygu8l,6rx hreas an electron does not?

  Would it ever be possible to balance a very sharp needle precisely on dcklg+g4z-sa3c e4 n)its point? 4+s3lzcaeg) n d 4k-cgExplain.

  A cold thermometer is placed in a hot bowl of soup. Will the temperature read 8:g(9u/j lkzpogdfa .ing of the thermometer be the same as the temperatu pjz8o(.uk/gg9a f:dl re of the hot soup before the measurement was made? Explain.

  Does the uncertainty principle set a limit to how ( cp+nmua-8opwell you can make any single mcm ua+p opn-8(easurement of position?

If you knew the position of an jq oq2s1.s jd-k j3e2eobject precisely, with no uncertainty, how well would you know its mo j2qe3-dks1qj o2e.sj mentum?

  When you check the pressure in a tire, doesn't some air keep/n -/ws l q7*kmu).btlam(ea)+ ainevitably escape? Is it possible to avoid this escape of air aplnwqmlua/7 -)(*seak. /eaet)k+ m bltogether? What is the relation to the uncertainty principle?

  It has been said that the gr9z/)wkxy j(ngound-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 in its en g)zknx(jy/ 9wergy form? Explain.

Which model of the hydrogen atom, the Bohr model or the quantkv6v y554e(wqsajpw1um-mechanical model, predicts that the electron spv41je yq6avw kw (sp55ends more time near the nucleus?

The size of atoms varies by only cwbbc4dn whd42q r/(.b23u jsa factor of three or so, from largest to smallest, yet the number of electrons varies from one to jnhbc d/ 2w34b(2 drwbcs u4.qover 100. Explain.

Excited hydrogen and excited helium atoms both radiate light as they jrhh6 +t xx+zci3o1. zhk b;:saump down to za6k: hio;h tc3.+x1srx +bzhthe $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table look if there iqmemqf /vfua/wl5/5 v e6b,z/fl99rwere no electron spin but otherwise quan/wmfq/5v9 /meif 5/ 6fqllavu be9 ,rztum mechanics were valid? Consider the first 20 elements or so.

  Which of the followiij -e.fc6hbglw y3.ps a:go0:ng electron configurations 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 uran g8f17+bsouoj2p eao:ium atom (careful scrutiny across the periodic table on the inside back cover will provide useful hints+7ea1p28 jg bo: sfoou).

In what column of the periodic table would you d* se,favp+h / 84, 7epomc-ht*tf(tlhrg5 ge expect to find the atom with each of the following configurafh4t a8ld et*pm*rghg7h5ee,t ,o+fc/sv p-(tions:
(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 exhi6+zkmp 3frc w9bit similar properties?

Explain why potassium and sodium ex9gvu 5d,g; esghibit similar properties.

The ionization energy for neon b7bzriib1)p afl728 p: w+hym $(Z=10)$ 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 photon nature foro jncr 1u4wnja7q7x: + light? 7xj7 o+jurq41:nnwac

Why do we not expect perfe2nthfz-:xwp ; 4ul:gmct agreement between measured values of X-ray line wavelengths and those calculate;fgx tn:2p -mwl:z4 hud using Bohr theory?

How would you figure out which lines in an Xr0p1u xjo3ymn/7 gs 5l-ray spectrum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transitions:kx). yg qk6 hro3j;/iis.jfc deep within an atom to produce shorter wavelengths than transition hc/ofiq x.jj).63i:y grskk;s by outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from ordinary light? How is ity hmgy; txfui+i)t*; 8 the same?

Explain how a 0.0005-W laser beam, poc6a :pic.z 4ho 8 vpqng+)n1lhotographed at a distance, can seem much stronger than a 1000-W street lamp at ta g:h z)lq+n8. 4vc6pp o1cionhe same distance.

  Does the intensity of light from a laxl:9kxt -: e oop(g1hrtz*d4lyc 4/kwser fall off as the inverse square of the distance? Expl:dw4llo-t/ o:x9pt(* kxg ker4 c1zyhain. ----待选择----yesno 

  The neutrons in a parallel beam, each hav8 ef) k7nyv9 xvmpm1+ting 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 withsbnc r*tfc-, xxg j;8: 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 ca:4np+7hn9jatej /zkn n 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 inh,-iv9og *iv5 chs/vk 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 the mx l qq.a.-zd 3xq2(3)zipgektass 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 typical nucle.xxmxx 2cvm yd0rx63/ us of radius 02vxx.cx ym36/rxdm x$1.0 \times 10^{-15} \, \text{m}$. [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 cf05qf5rwbl/jz j:bx nr/ /t s-,:nnuelectron were present in the nucbz/ fw::j/ r-bxn,l0 ft55rusc /qn njleus $(r \approx 10^{-15} \, \text{m})$, 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 posi vkss nyf86qn/ :kb f(*v :tyra/let-.,w ;zeztion 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 cb8adl)ukry* :an be in the $n=6, l=3$ subshell?    elections

How many different states are possible fvx*+0fkvc;hl3e fn j jz-)dir7t l+*cor an electron whose principal quantum number isj3vrc dfi - jv 0x*z+e;th7ckl)* n+fl $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each electron in the ground state ofjz0,*k*u axxcg g*br 7
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers oigg. 1pw8txdc)y 7a8 for each electron in the ground state of nitrogen $(Z=7)$.

  Suppose a certain hydrogen atomlxg/ x: nsh )twbw m15c*h;(en 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 elecjs;xt4hr5u;tr1it 85w . r jthtron 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 has g*v1u+m)1nr w3 pt x yq0k.w:i aqpi8n$m_l=-3$, 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 yisc n,; -hhb t7z0dp4"g" subshell.    electrons

What is the full elecch+j:g8 ( i:pw4jcbmv;wjs k8tron 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 config+f buo(dw kood783b+kuration for (a) selenium (Se), (b) gold (Au), (c) radium (Ra)? [Hi3o+ukk o 7o fb(8d+wdbnt: see the periodic table inside the back cover.]

  A hydrogen atom is in thecl9b;s(5 .enxysoi,3 nwh kt5 $6s$ 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 th dw 0) 4(b fgre7ml2r)-sbbxzzird electron in lithium using the Bohr theory. [Hint: Thisms2e- f7w0bbd)rl bz)4zr( xg 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 z-aj r:4r vyo-zol 3 o.oxm.be5ero for a filled subshell.

  For each of the following atomic transitions, state whether the transition is6qt0yssem -0 3etnj e/ allowed or forbidden, and if forbidden, what rule is being violated: t3m j0n0yq6-e s/ etse
(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 xp3dssid n).z 1naf; 2$6d$ 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 b)oe 8gi ixp/de1v+ w.xy electrons striking the face of pxed)owv .8e/ 1g+iix a $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremsstrahlung X-rays emitted from an X-ray tqkpwwj*our 1gn,:o 6 /-y8pkxube have /1g 8wp :u ,wpokr-xn*qjo6ky$\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoffth *r-wnf q7v2 wavelength $\lambda_0$ 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 wavtmic -/ nkayl*wgid-h.ax77y4.b v)nelength emitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelengopi1v/vj2t q):gsph-i i.m,wth for an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the wavelength fohdazn /ad-* ip afz/99r 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 elects :nngr6.:hnuk 7d r/orons. The wavelength of n gnr6uo drk.:hs/7n: 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 retinas puts outdc +gt fy3d1gzgf4; f9 $28$-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 )*x( 0/nbbpucjb)/v1iin g z cmight 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 due tojrld5;iyb q,t:,s1tdv,:xd) v7vx tc diffraction if the beam emerg yvxxd,)vl ,:tbi:;rd7tq,j 1 vsc 5dtes 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 Hd3rllu5: p *)gik 6pofe-Ne laser?$\lambda $ =    nm

Use the uncertainty principle to estimate the position uncertainmttb(kp ,:ds4 ty for the electron in the ground state of the hydrogen atom. [Him,:kt p4b(dts nt: 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 values for the angular mome1btuy9u)de*-e xn g4z, dvol7r/ ,rbw ntum $L$ 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 elic.4q4 .ylbj l k(uva2ectron orbit, estimate the average distance from the nucleus for an electron in the innermost .jyclq(iv 2ak .l44ub$(n=1)$ 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 mfzg j, :9acf 5:hi;)/wbugrk 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 the outermost electron in borons zk 1 6e4dr/za,weai9dvo3j8 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 sh mh:olusj 80h6ow that 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 different stak7fxnn.rc:z( x:sfkq-0 g -q ptes 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 fj t4a-4rhv(.8selq5 lymi mv zg8/ zi7or a given value of 4 lv it7.ylz z- gqh5ea48r(m/8 msjiv$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy n f(n ,yz;w,tms hqd;-45 kV is shot through two narrow slits in a barrier,; dn;, hnzs(y-mtq fw. 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 .tq(2q8fvxj c model and by quantum mechanics. Compare the resulf c(v.qjt q2x8ts for $L = mvr$.
$L_{\text{Bohr}}$ =    $\hbar$
$L_{\text{QM}}$ =    $\hbar$ or $L_{\text{QM}}$ =    $\hbar$

  An 1100-kg car is traveling with a speuns4nud01a6 o ed 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 waveleng5w(7m k1 xt cqzgnwi(v3y 1f6bth centered at 488 nm. Careful measurements show the line is really spread out between 487 and 489 nm. Estimate the lifc51im(w z 7ygnw(1ktvqb x36fetime of the excited state that produced this line.
$\Delta t = $    $\times 10^{-14}\ \text{s}$

  Protons are accelerated from rest across 550 V. Theco *;ckwa2:)wmfd4 yg y are then directed at two slits 0.70 mm apart. How far apart will the interferenced;c km: gc )2*o4wywfa peaks be on a screen 28 m away? $\Delta y$ =    nm

An electron and a proton, each initially at rest, are ach+ugpegzz5y8 b.. hg(celerated across the same voltage. Assuming that the uncertainty in their position is given by their de Broglie wavelength, find the ratio of the uncertainty in their momentum5hhb(zg..8 pezg+ ugy .

  If the principal quantum number n were limited to the range from 1 to 6, how d,x: sbs1o(/ k3p. v-zb,xg4zu raflumany( uab:/vzk1rp 3sxodf zl. ,4gux-bs, elements would we find in nature?   

  If your de Broglie wavelength were 0.5ai4nd vt;n w 20ws+*2klwl5tm 0 m, how fast would you be moving if your m*i5k +t0 4dltwwm2l2nwv;a s nass 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 linawf:87rthfh c8 cy9*1vrd: 1qv y(zx ces with 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, determi8 9z cvvdy(f8 :hrrc1y1t hf7c: *qxawne Z and the element in question. ----待选择----lithium berylliumboron 

  Photons of wavelength 0.154 nm are emitted from the surfacsc:pz804nde(px qv bne71x v;e of a certain metal w(n 7x ze8 spv; cnx041devqpb:hen it is bombarded 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 diffractb: ei0t na*qr:ive spread 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.]