Compare a matter wave0 kb pmonw9y4die0b12sega 8- $\Psi$ to (a) a wave on a string, (b) an EM wave. Discuss similarities and differences.

Explain why Bohr's theory jj f 4j*kd/e94bsxkuxuei:uom*m,/ *of the atom is not compatible with quantum mechanics, particularly the uncertaintdk9x ujseum x4f*mu,4e*//: *b k jiojy principle.

Explain why it is that the more massive an object is, the easier0w8kiesqwmca28 ) b5 i it becomes to predict its sq8ica2 )i e0wm kb85wfuture position.

In view of the uncertainty principlq/fj5/zetj6+ : wrcsbe, why does a baseball seem to have a well-defi/z/5e sqjr 6+bt:c jfwned position and speed whereas an electron does not?

  Would it ever be possible to balancec (cm:hun1 k99gcwj 7r a very sharp needle precisely on its point? Explain(cg:ucrj9chwk9m71 n.

  A cold thermometer is placed in a hot bowl of soup. Will the temper r-e:wl llng +)-kqoy7-rfd)wature reading of the thermometer be the same as the temperature of the hot solroly:gd-ewn- )7r wl)q f- k+up before the measurement was made? Explain.

  Does the uncertainty principle set a limit to how rc8fbx31 xd7 oqm),iowell you can make any single measurement of positioi)d,crxfo1qmx7 8 b3on?

If you knew the position of an object precise t,csvdke8) m*ly, with no uncertainty, how well would you know its momentum?*s ,dve)8 mtkc

  When you check the pressuvn/f/(ia .-wsi wsw*si. 9 kwbre in a tire, doesn't some air inevitably escape? Is it possible to avoid this escape of air altogether? What is the relationv9.kwwb( saf/s w*ni -s/ iwi. to the uncertainty principle?

  It has been said that the ground-st bvf l/q0j 6(5z*ajhzmate energy in the hydrogen atom can be precis 5mjf alh60bv*zzj (/qely 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? Explain.

Which model of the hydrogen atom, the Bohr model or the quantum-mechani e uhlhd38miw/d*y7 n1cal model, predicts that the electron syd l 1 3eu/d7i*8wnhmhpends more time near the nucleus?

The size of atoms varies by only a factor of three or so, fz 1n0 zs zld,cuw9fvso*(j,o+ rom largest to smallest, yet the number of electro+co nwz*zz 1o0s( s,lf, dv9juns varies from one to over 100. Explain.

Excited hydrogen and excited helium atoms botje jrv /no5yxm, x-9*jh radiate light as they jump down tjoyvej 9, *rx5jn-/mxo the $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table look if there were -ef8qpi g +e4(3tem3 n7b0rfiqjf q 1nno electron spin but otherwise quantu3e- n 30qf7 1fqfj+pmrin8q4 gt ee(ibm mechanics were valid? Consider the first 20 elements or so.

  Which of the following electron configurathu9u00 8hxc6 rxawid6ions 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 configuratijiffpr:m7 dp5wm +e3 .ux 7iv)on for a uranium atom (careful scrutiny across the pe)f 3xd iu.7mp5fmvwi r pj7+e:riodic table on the inside back cover will provide useful hints).

In what column of the periodic table would you exks r2 28 oo tt1me2bg4qdd2.shpect to find the atom with each of the following configurati2k2q trdt2hoe4ss2gb18o .md ons:
(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 -5h n. 0stu dadu8i4nhexhibit similar properties?

Explain why potassium and sodium exhibit similar properti75juq surxeo4cfn- 7:y 0,tq,vfych-es.

The ionization energy c9xw fw2x6 hd3goa:;po /a0bmfor 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) imply a axm*q8ne 2l ;0ov w1pdphoton nature for light? 8m ae21wd0pxol;qv * n

Why do we not expect perfect a w0 0etcmot.j)xd)qywy-+p3 2t 9y suwgreement between measured values of X-ray line wavelengths and those calculated uet 2p3y0y.x)0 s9+www)yc udj -mttoqsing Bohr theory?

How would you figure out which lines in an X-ray sp s, l+j,zjmc+pectrum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transitions deep within an atom to produce shorter wl+bf:om fi8w 0avelengths tha+b 0mfl8 :ifown transitions by outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from ordh(s)gx/o3 n bv*wzct1( e+5koa9j g toinary light? How is it the same?

Explain how a 0.0005-W laser beam, photographed at a distance, can seh st-a516kn *ejvx 5wiem much stronger than a 1000-W street lamp at 565 sat *ixv1e-w jhnkthe same distance.

  Does the intensity of light from a laser fall off as the inverse square of the t 1 vs0ug zby)1 2uth09dxrfj5distanu11y z2usrtvhd )0xfg905j btce? Explain. ----待选择----yesno 

  The neutrons in a parallel beam, each2u3mky u2wxi 0 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 travelinmnhg(0vor( p6eg 26c0 k gfx9jg with 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 canqux9-p/h:k vbqskr :u6j o+ t/ 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 typicn37on6+l;gabui- b9uj q hs31x- zalxally $10^{-8} \, \text{s}$. 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 mftk2ky kt ,ez vo01*1o+vc5ckass 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 ay,;5jmtr:of ice3,m5 mo m- sv typical nucleus oo-mv3m yr,5,tfj:im5 ;emscof 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 sho3k j ic ;xjr7qwa0-1jcw that if an electron were present in the nu ;jc r-10k3xcaqjijw7cleus $(r \approx 10^{-15} \, \text{m})$, 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 positi- qitka +4ulz;on 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 wha/ f ep2r5q0hc v24fin the $n=6, l=3$ subshell?    elections

How many different states are poss,-xapbx4 3jctfx, 2npible for an electron whose principal quantum number i,xxn b3x,4p 2pa-cf tjs $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each electron in the g7iio5imx23vru2;0 o/9 bs2n qcflx z rround state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each 6 jzhnqs):s3e electron in the ground state of nitrogen $(Z=7)$.

  Suppose a certain hydrogen atom haspdwu:al2l1 2q $l=4$. 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(n k8930zq snccqrdp ( 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 hahyg 5n qt2vf9/s $m_l=-3$, 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 electutpu7,2 pn1;1 fejrnj flp9g*rons in a "g" subshell.    electrons

What is the full electron configuration in the ground state for elemenywr 0x; 82ok abe2lsse/ut+n tndx-,0ts 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) seua5iwtt; xg5/+ ic6etlenium (Se), (b) gold (Au), (c) radium (Ra)? [Hint: see the pe ax gwie t/+655u;tcitriodic table inside the back cover.]

  A hydrogen atom is inu)tev 0tyu;f jzub(yue0 l+19 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 using the Bohr t cr.umj9qix 5qe/ 63sqg+4l ieheory. [Hint: This esr.+x qlj43qe6ie9u/c miq 5glectron 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 monmrefyu8bi 78l7zbq /+j;t zi w9 an9/mentum is zero for a filled subshell.

  For each of the following atomic transitions, state whethe2pf.2klzsv , lqde90cr the transition is allowed or fop2qc, zsel20 k9.d lfvrbidden, 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 ingws e*9x0 p)v+c o7)gtqla qc. 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 faceztf1n 35cc75jy yn1h u of a fzyc7nt nhuy1j 53c15$33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremsstrahlung X-rays emitted from an X-ray tube h cd4d7 rz1hb*wave 7 zd4*1hrw dbc$\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wa7fcp z .uipr7mnf(,i, 6hou2e velength $\lambda_0$ 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 wavel, n/z-d/17r ,dwarj9r w xpihvength emitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength ;,s9qvfn+nfj2qvw na b7z.h 2for an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to est 3s*m zm7kegm( ekp31jk7vrc2imate the 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 unknown ma3z5uzfnf *nsbzx* lb i*14ul5d o ,t1dterial is bombarded with electrons. The wavelength of th*b iub 5x1t1*fo,4d3z lnsnl*du f5z ze $K_\alpha$ 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 deta2c)-s+cae kog2mv-h/ ewk 2jj ched 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 physics lab might h:phziw;ku(e/z;scj/ 5kfd3+ )wug k(f ii vg0ave 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 bec 5-sa 83px4dpin0s hram due to diffraction if the beam emerges through5x3c8n apd- 0pss ihr4 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 laservn4/vj2n-ud/ji* ua lhz)4 mm?$\lambda $ =    nm

Use the uncertainty principle to estimate the position u,hoon(okl a: ot(xo f28 w,t/cncertainty for the electhtowkooo(/ cxlo(nfa82 t ,:,ron 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 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 arhfq ziwf/i846/pq, g nd smallest possible 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 of an electron orbit, estimate v6)b y1 cljqlugj d2:ex85;cz the average distance from thg c el:)y6x28c1jdubv ;zljq5e 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 operatesdx6 3+lqe :hzs 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 o+y6v eq5dpb* . fcf,.pf g.tlzutermost 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(Fik r n7uy15dvbf +gx46bozy)h0z lo(*pg. 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 states for a g*9l-- r do,;yqs7 i)olpm obkviven 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 electr )p sz 06s(rct65fq5kdv o7n*edtv:h,vb jp3uon states possible for a given value of 0)3voh 7qs5d*6utfvc6,pn5rvtk (es b:zjp d$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with enexi3:d;e*71 .ba krnw+idzr qsrgy 45 kV is shot through two narrow slits in a barriei +qriz nb3s1w k ra;*x.7:dder. 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 a* :l e7mq4eilu 25jqpstom is given both by the Bohr model and by quantum mechanics.eseu 5qlji lp:2m*74 q Compare the results for $L = mvr$.
$L_{\text{Bohr}}$ =    $\hbar$
$L_{\text{QM}}$ =    $\hbar$ or $L_{\text{QM}}$ =    $\hbar$

  An 1100-kg car is travelinq .zmxsm: j 1dr:sq3-ig 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 laynv0rev4w *3ce(x8 o ine with a wavelength centered at 488 nm. Careful measurements show the line is really spread out between 487 and 489 nm. Estimate the lifetime of the excited state that produced th34enor80 *wvyav x(ceis line.
$\Delta t = $    $\times 10^{-14}\ \text{s}$

  Protons are accelerated from rest across 550 V. They are then diay)oj2vo 2w,n9p/ d2 y ) cc5fx3fuqwsrected at two slits 0.70 mm/ofaxwn yw2qdj f2 ,o us2 9)y5vc)p3c 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,u,r;;*o4ee7* j cvnvl; vojn/5taa jz are accelerated across the same voltage. Assuming that the uncertainty in their position is given by their de Broglie wavelength, fina;n;eco **e/; ,4vjot5v jrn 7uzvjlad the ratio of the uncertainty in their momentum.

  If the principal quantum num82inom*fujponc77 f 9 ar.b0aber n were limited to the range from 1 to 6, how many elements would we cf28 ajo ar mp79*bnfi.oun07 find in nature?   

  If your de Broglie wavelength were 0.50 m, how fast would you be 6 js)t+ wioza+moving if your mass isao)j +si6tw z+ 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 eleme ujkk0p1.fg-y uh4wze f4+m d0nt shows a series of lines with one out of every four matching a line from the Lyman series of hydrogen. Assuming that the unknown el .zmk0 4k0efd+p1ugyj-hfwu 4ement 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 emitted from thqjl .ie5 0izdjgq6u*9e surface of a certain metal when it is bombarded with high energy radiation. If this photon wavelength c 9jqu5d*ql0i geiz6j .orresponds to the $$K_\alpha$$ line, what is the element? ----待选择----nickelcopperzinc 

Show that the diffrac:0wjxk;rm3 wf)k ant*ugo x( .tive 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.]