Compare a matter wav l( mh 8q8fhb8-p gl6ozf39izwe $\Psi$ 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 compa xa1bnqa;l 4u+7vl1c 7mtb+nytible with quantum mechanics, particularly the uncert1x4vbtq7y+nm u bc;1aa nll 7+ainty principle.

Explain why it is that the more massive araarv,8: l-.nffc az/ n object is, the easier it becomes to predict its v fr:na,f -lc /z.8raafuture position.

In view of the uncertainty principle, why does e ayc yuna;(-,.nkw7ca baseball seem to have a well-defined position and speed wherey .,na-n(cecw7 ;ukyaas an electron does not?

  Would it ever be possible to balance a very q npf3)b.k6uq sharp needle precisely on its point? Exp6 qk.p)buqf3nlain.

  A cold thermometer is placed in a hot bowl of soup. Will eyv zl5j m9p.q4kfh)(the temperature reading of the thermometer be the same as the temperature of the5 (z4.fh 9jelkv)mqy p hot soup before the measurement was made? Explain.

  Does the uncertainty prxin+73rr3 w agysgv6c r ju:pz77++ gpinciple set a limit to how well you can make any single measurement of pos:pp3c 7g+r n6sz ruigx 7v3yja++g7rwition?

If you knew the position of an objezccu* lt24w0(:cp/y yn gk7sfct precisely, with no uncertainty, how well would you k:ykc*/ 40uczwgf l27psy n(ctnow its momentum?

  When you check the pressure in a tire, doesmsu/sj 29.6dazyl sv4 k v-f,m)desn1n't some air inevitably escape? Is it possible to avoid this escape of air altogether? What ie,j9am/s2 v.ud1 df-4yzsm n ss vl6k)s the relation to the uncertainty principle?

  It has been said that the ground-state energy in the hydrogen atom can b qfa86+yr0k kt itod7g.t o+)ne precisely known but the excited states have some uncertainty in their values (t7ito0q. ao+ kkfr8d )t6 g+nyan "energy width"). Is this consistent with the uncertainty principle in its energy form? Explain.

Which model of the hydrogen atom, the Bohr modelb3nt bot6y 8g/ or the quantum-mechanical model, predit38y/bo bt 6gncts that the electron spends more time near the nucleus?

The size of atoms varies by only a factor h)l(rn+f ykd6zi tk4 nc/0qvc..f.r2ymtj 9bof three or so, from largest to smallest, yet the number of electrons varies from0yjkdvr fnb(c).hq .cmyzi4f6l/ r. tk+ 29tn one to over 100. Explain.

Excited hydrogen and excited helium atoms both radiate light as they jump doy xmgvht scww2+;7si ( ,-+4b-h lazimwn toc s,i w2bzmi+lga-; 7m( ts+yh4w-v hx 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 no electron spin but otheras rafxw mj .u0,b001wwise quantum mechanics were val 0f,w0wjr.ua0 sx 1mbaid? Consider the first 20 elements or so.

  Which of the following electbag w8l0r 4nusr;g /4iron 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;w: z 754x xiqip(ulqrium atom (careful scrutiny across the periodic table on the inside back cover ; 7puri xxq:zw4 i(q5lwill provide useful hints).

In what column of the periodic table would you 2ak.j vb 6i), rv6vqxxexpect to find the atom with each of the following configurati,vv2 .vaq6ir6bjk )xx 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 exhibit s.u dr1ilx0m no.l37v aimilar properties?

Explain why potassium and sodiw x)/dgvz5c +jum exhibit similar properties.

The ionization energy fh 4n.(ot s0o tlrd92lwor 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. . c9,cosowg7(, jgkpof.j-bdBelow) imply a photon nature for light? ,p(bjjw - gcs..oko d 7f9go,c

Why do we not expect perfect agreement between measured vr . 8jqxcq117xjem0rzalues of X-ray line wavelengths and those calculated using Boj qm x87qjx01r1 c.zerhr theory?

How would you figure out which lines in anj-3e urn( yot,1hz 1mi X-ray spectrum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transitions deep within an atom to produce shoufy54,4vm/ ;ado: f kcmwys2crter wavelw4da,fsm yuv :ofk;cm/y4 2c5 engths than transitions by outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light diffedrp18 nqg3mi c ka1.7gr from ordinary light? How is it the same?

Explain how a 0.0005-W laser beam, photo62nex8mqs a) jw890eb /ncstggraphed at a distance, can seem much stronger than a 1000-W wgnms09et/)c s b8n x6q82ja estreet lamp at the same distance.

  Does the intensity of light r,b(/d1d xa+e 4h(-vpp u yznmfrom a laser fall off as the inverse square of the di(e14h+ mzu-d y,( vrdxp/pbanstance? Explain. ----待选择----yesno 

  The neutrons in a paj t.+/it5e eklrallel beam, each 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 traveling with a x1 hg4*19l(gd ne* u wfcz(hayspeed 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 accuracy+ea qp(ej z,i7:wbpd6 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 ans03j2* lq/r)iwdxw da 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 .ax w+yrg*s.y 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 energy of a neutron contained ifb-5 uzy cx/ ,aaw-+9mvrsi:tn a typical nucleus of radiusr- t:/5y w ,a- v9ucsazbimfx+ $1.0 \times 10^{-15} \, \text{m}$. [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 pry ;i i k.cdqebib/qeo9:ko( 47inciple to show that if an electron were present in the nucleu 9bye(47 /dko.;: o icbieqkiqs $(r \approx 10^{-15} \, \text{m})$, 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 the8cz lewz+(lp5au9n- q 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 the4fz/4c 9k m dx9zat+ (6sapw5iy giad9zd5y+k $n=6, l=3$ subshell?    elections

How many different states are possible for an electron whose princiii6/ 7rdetyf28ed.xy/ s ezxl+(ce3spal quantum number is .ys7e3dxy2+cesz li/f(6t xe/re8di $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each elec 4uje56ti,qj:,k yxswtron 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 nitrogenyj uq4/hq mc9e3e36ywiq/ em 3 $(Z=7)$.

  Suppose a certain hydrogen atom hybl p ;1 dwgi0z(qyb) b:z9*tpas $l=4$. 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 ele98+ws e+cq,0hkv wzco ctron 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 ha-2ugy :w8v)u hgl)r jxyhilx: ):i5zys $m_l=-3$, 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" subsh h7 k8itp(h bd4( ag;zzmy60nmx mc,,dell.    electrons

What is the full electron configuration in the g)dutjy,7;ux 3xp x7efround 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 for (a) selenium (Se), (b) gold (Au), t)avftv*3 otcaobds/7 7x(*v(c) rad7xda7tt*o * s(v acof t/v3)bvium (Ra)? [Hint: see the periodic table inside the back cover.]

  A hydrogen atom is ini+5 l)1mhxtz .4dc fxl 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 (zf47n .,qub rlykoqn y*gk4 :energy of the third electron in lithium using the Bohr theory. [Hint:g:oqz q4( run,4klby 7n*. fyk 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 subshel fcv 0zed()p1vl.

  For each of the following atomic transitions, state whether.odvxby (:5s 3kyb:vs the transition is allowed or forbidden, and if forbidden, what rule is being 5 s:b vkx.bv3ysd(y:oviolated:
(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 ifc3d ps9asv(vc((fqp bx * bzo+d13g+n 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 374m*y: 8bcfv dex w ato.;hzpelectrons striking the face of cve7y fp ;*dwh3o:t4xzm b.8 aa $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremsstrahlung X-rays emitted from a zde q;x(*8arln X-ray tube haverzq 8(de;a *lx $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff waveled o5k:;qi(hih .l z*.wg8p2ek+ddbsungth $\lambda_0$ 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 emzd qxr;8 sh 8gz:75cmm48mpg mitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelengtvdko- 29cyi/m8o+g2t yr5z ae)1t dkgh for an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory t p4udd jdic(mov:d0ls v05q6j) n7l3oo estimate 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 majh4 l 3sfva67 6c atm+cwe40ttterial is bombarded with electrons. The wavelength of j+ 7sawcle34tfc0vhm6t t6a4the $K_\alpha$ 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 t; o)uc45s4ro) w v)nyccjaj0 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 : ,npbkd6 -lis0ud ox7physics 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 due to diffraction if the beam0.wn(xq h :ed pn 0*bgh0iz1qc emerges through a .1 :nqq0eng *zhhw0 cdp(xb0i $3.0$-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 lasya4e0 rsf1w 9uer?$\lambda $ =    nm

Use the uncertainty principle to estimate tl7udatdc:q 3k/x p*,nhe position uncertainty for the electron in the ground state of the hydrogen atom. [Hint: Determine the momentum using the Bohr model of 3ac:,q/p*d 7udlxt knSection 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 anwgdqg8vi.nqhf )s7l.pa / ;) mgular 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 thzc:xhq :4) ekye radius of an electron orbit, estimate the average distance from the nucleus for an electron in the innermky:ce4z xq: h)ost $(n=1)$ 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 acblb ro5 p3 e9oq+s)8) aeq eaa2ts;7$95 \, \text{kV}$ 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 jb tle *4i-zd2*bt2p aelectron 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 Mosele4:6oypz8+rm8u l gmeny 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 num(b*sjkuzbtoj+f4t5qk3 k5o2 ber of different 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 difa)8 q6c tdr3g2ixhkw*gm.4w jferent electron states possible for a given value of w*t. 6irjcgq4 dgh w2m3k8x)a$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV is shot through two narrow slits in a bar1 ,qfdjin02a4t8excf9cpjt ) rier. The slits are a distanc9pd1j,cti2n)f 4q 0ecx8a tjfe $2.0\times10^{-6}\ \text{m}$ 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 hydrog/u ) *bi-ko-+ nfwolghen atom is given both by the Bohr model and by quantum mechanics. Compar*iuk+ -)obl -nhwo/gf e 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 sndayy 3a a)3fd1 rpvr; 46od.tpeed 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 wavelengtx h5pzgxx1g ig5 f4msezk/ )+gq*a6 r5h centered at 488 nm. Careful measurements show the line is really spread out between 487 and 4k 5zg zeh) r+a5q /*mxgx61g4pisf5xg89 nm. Estimate the lifetime of the excited state that produced this line.
$\Delta t = $    $\times 10^{-14}\ \text{s}$

  Protons are acceleraai33 rnh:9fed 9b hfh/ted from rest across 550 V. They are then directed at two slits 0.70 mm apart. How far apart will the int/r9bifhf n:a39h3 deh erference peaks be on a screen 28 m away? $\Delta y$ =    nm

An electron and a protj8;z y1c6vzj gs3srt 3on, each initially at rest, are accelerated across the same volzs18v3ygj tj z;c 3sr6tage. Assuming that the uncertainty 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 to the range from 1 to 6, how mag;rh n 2-wlhqz4s+h 3gny elements would we find in h;n-s2hz+grw4 h3lgqnature?   

  If your de Broglie wavelength were 0.50 m, how fast would you be moving yizg*o2j qa*qpl (y46 if your mass iyg(46 q a2zoy i**qpjls 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 seriebg7 lt n*jm0y*s of lines with one out7*n tg*0yml jb 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, determine Z and the element in question. ----待选择----lithium berylliumboron 

  Photons of wavelength 0.154 nm are emitted from the sur5 v-gzt. )wdmw91glhfface of a certain metal when it is bombarded with high energy radiation. If this photon wa9tw .5v) l-mz dgwgh1fvelength corresponds to the $$K_\alpha$$ line, what is the element? ----待选择----nickelcopperzinc 

Show that the diffractive spread of1sq / m)mmdjq)2x -xid 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.]