Compare a matter wave ck)t,dpp;o /p$\Psi$ 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 quantum mechanics, lt 8qaxq:fyh(/b1- daparticularly the u(/baq8tfh 1dy x:-aql ncertainty principle.

Explain why it is that the more massive an o x6- sbj ; xtoy9wf8n63xi2amv/ y1cstbject is, the easier it becomes to predict its futur6b xs -y nj;t9maysc ivxx8f1 6/t2wo3e position.

In view of the uncertainty principle, why does a basl +58 tkuwavh:eball seem to have a well-defined position and speat8 hv:wkul 5+ed whereas an electron does not?

  Would it ever be possible to balance a very shar qf/(r5mp z:ap* :qst+trtjn1p needle precisely on its point? Er z( q1mt p5t/fq:psrj:ta+ n*xplain.

  A cold thermometer is placed in :6- ud.u1:ge;prhm g o/n*9lql/qz nauj l6gua hot bowl of soup. Will the temperature reading of the thermometer be the same as the temperature of the hot soup before the meauhqrq/;/djg .go:6*u llpn:9a eu g-nl m61uzsurement was made? Explain.

  Does the uncertainty principle set a limit to how well you can make any singr e kr6lj9*d v8/jebh/le measurement lkbjre 8h 6 ed//*vrj9of position?

If you knew the position of an objetln9vm h;2*ngkfbj+ : 34ptxtct precisely, with no uncertainty, how well would you ktf:*n+3xlm htj g;2 vt9k4nbpnow its momentum?

  When you check the pressure imk:ch n;qd3d1 u,yde.n a tire, doesn't some air inevitably escape? Is it possible to avoid thi,enkd1 y;dmquhc:.d3 s escape of air altogether? What is the relation to the uncertainty principle?

  It has been said that the ground-state energy in the hydrogqd 97+ yfx9r5qawbcx 3ep3 8pkl 5kb(fen atom can be precisely known but the excited states have some uncewdxkab8f+3(p9 bq9r clpy exkf73q 55rtainty 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-mh jrho)/f33fq w-w2l dechanical model, predicts t-l/3 wdq fh 2h)3wojfrhat the electron spends more time near the nucleus?

The size of atoms varies by only a factorkfvx4 d.3-4 ;oe i zf;fd2dqfk of three or so, from largest to smallest, yet the number of electronsxfd ;-;dk4 df 2okfev q4i.fz3 varies from one to over 100. Explain.

Excited hydrogen and exco n16s50phrrx+(m res ited helium atoms both radiate light as they jump down to th6mp50rosxreh1s (n +re $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table look if there were no electron spin but otherwise cmvr7;:b wjtal7 hf; fp 4q;1uquantum mechanics were vali: lb4jpt1; f ;wmh vcuaq7;r7fd? Consider the first 20 elements or so.

  Which of the following electron configurations are not allowed: 0oeixc+o*ys6 fet7maog j)7 2
(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 uranium atom (careful scruip a.x vg2:9ohtiny acrosvoxa :. 29pghis the periodic table on the inside back cover will provide useful hints).

In what column of the periodic ta7k,ye iii*opa sa x-6md6k7) ible would you expect to find the atom with each of the dyii7i) p6-,maxes 6kk a*oi7 following configurations:
(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 iodin0ruwktb 8w vy/:u/j ;ee exhibit similar properties?

Explain why potassium and sodium exhibit sim/wi4- ozv-;sylsaalka u177 milar properties.

The ionization energy for ,d1qy. xdx 6d3 jx)l qt;obpx9neon $(Z=10)$ 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 nat( 007fy qvbav td+p1mlure for light? tad( f701v yq+b0pm vl

Why do we not expect perfect agreement between-0hbn29 i wwu2gwooa akl /kfkng7 5;+ measured values of X-ray line wavel;2 kwui/bafkw wga +kl0 9h-2 noog57nengths and those calculated using Bohr theory?

How would you figure out whiap4 l1nv w ubl( f1wf,ed144vvch lines in an X-ray spectrum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transitions deep within an atom to:3 lrcp5crqo; uo 65e,+hunlv produce shorter wavelengths than tranr+: 6cn55roh3 ou lp ucqlv;,esitions by outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from ordinary light? How is it t01xvr06 (l( uifg vnothe same?

Explain how a 0.0005-W laser b40bp/7lp s mmeeam, photographed at a distance, can seem much stronger than a 1000-W sm04mp /ple7b street lamp at the same distance.

  Does the intensity of light from a laser fall off as the ina mj 1effo l3 uhtl6o)yb7(1uzu: +3kkverse square of the distance? Expeu fo )k7yolftl31 (a:uhu kbjz6 m+31lain. ----待选择----yesno 

  The neutrons in a parallel beam, each having kinetic enenu 5 g e1+cayfc,ucx:.v:y2mxrgy $0.40 \, \text{eV}$ 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 speed uzw3p 9 ,pge.8 2adwsvof $(6.560 \pm 0.012) \times 10^3 \, \text{m/s}$. 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 ofgu;z ;vtm txhy. +t8w. $1.0 \times 10^{-8} \, \text{m}$, how accurately can its speed be known?
$\Delta v $=    $\times 10^{3}\ \text{m/s}$

  An electron remains in an exu ld .jd+-iuti0 ;kvco 3ob/0tcited 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 ofp 7 . yz v)ix:5n*keerx0j,gyy 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 possiblebbqx)e)-kc ar+io5w+ energy of a neutron contained in a typical nucleus of radraweik)co+b)5- qxb+ius $1.0 \times 10^{-15} \, \text{m}$. [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 princh st;-*a )ca8gebu -s1 t;rgshiple to show that if an electron were present in the nucleu*18 tb-actssu;s-)h; aerh ggs $(r \approx 10^{-15} \, \text{m})$, 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 positiog/p* lz/+yir jma ;bk51k7xecn 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 b(2qo vkw9+uaitmbt olx0 u +75e in the $n=6, l=3$ subshell?    elections

How many different states are possible for an electron whoseg :q)c*nj74 nojriv5;yu2xk) hqns,l principal quantum numb gij lc n4yknx qo5* n,qsr)2:7;)hvujer is $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each electron lpf foi-2 rm:/in the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each electron in the gry+ lnjcr/3fcqvo5u:8 ound state of nitrogen $(Z=7)$.

  Suppose a certain hydro+df 5es 3 uc2r,j5aoqzgen atom 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 theye79f + fo(4wcguh0xf angular momentum 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 haq3(,odxiys i+ 5p,korc 1lo2ms $m_l=-3$, 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 electrosjlx*4beh2 7 lns in a "g" subshell.    electrons

What is the full elefb2 *c0*u lk ;hfl9mgqctron 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 for (a) selenium (Se), (b) gold (Au +ce0 (ouulk c9rs8suddr0ix0m) o (x;), (c) radium (Ra)? [Hint: see the periodic table inside thu(r+x d0)ie(oc0sulku9d; x 8srm 0oce back cover.]

  A hydrogen atom is in thqmr s h+qx3ky *kcv1(q7:ug8te $6s$ 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 elec)smse(b d ,a2ktron in lithium using the Bohr theory. [Hint: This eledsm a(,)2sb ekctron 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 momentu,ku(0dkr a w-rm is zero for a filled subshell.

  For each of the following atomic transitions, sfibodl9s;gtz *p764u,2as 1wn s iv /etate whether the transition is allowed or forbidden, and if forbidden, what rud6o/wtzls12 9e , ivf7g*ps is;an4 bule 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 iskc,dp:7nks q-,2wqln-4 junc6arq ei*+ q:p x in 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 emi2 ,mc2-pki, ed01c ryhvikl7 otted by electrons striking the face ofrkcpm,0e, ivi c1 y2-l2o7k dh a $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremsstrahlung X-raysv.hr 5y a8)wdtxd96tn.rt( gw emitted from an X-ray tube havexa9t(r.tw vn)r.t5 d d w6h8yg $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoffiod3lh c1(4xwz; 4jf2):xm eg6o cghs 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 wavelength emittedd qbi6 evmh(c/fj2.7w when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength for anivrz 0nxtq0mdh7 :;o 4 $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the wavelengthxk6h *2xl*g 8xuqd bf4 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 unk4 r8c jt75hdkhnown material is bombarded with electrons. The wavelength of thj5dc8rkh4t 7 he $K_\alpha$ 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 re5k/ :n 3vbx 4kkqqov4xk1 rk0rtinas 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 lavcffgpya 7do/+ /*w ajf6 juu.,7n hh.b 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 aq 03o8otk f(ab gf;ki 3i+qur/ laser beam due to diffraction if the beam emerges 8/k3gab3tq ro;(+ffkoi 0 quithrough 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 tj+uz5v; gvbfxuh*c4a4 wt -8yhe He-Ne laser?$\lambda $ =    nm

Use the uncertainty principle to estimate the position uncertainty for the elecg4)gitr)xf j+tron 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 Boh )ftrjxg4+)gi r 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;jm(c ttyn1f : 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 onc99tu8q7 l ixu82z- qnq7svi f an electron orbit, estimate the average distance from the nucleus for an ev8 79i9 tul-unqzx 8is7qn 2qclectron 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 operatesjtnbi*y r* ebmhvayg)7;f+ 47 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 outzpkv 6(*a1gygermost 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(Fiyrdpn05u +nu1pjr1 p+g. 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 given vat;hj9fw89 acljfqorb go 88jg6 .5v(x lue 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 val;on*qa8gf o6 eue of e6n;8*gqfaoo $n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energk8yetb,v .+l0dw(c9 lze eq +ny 45 kV is shot through two narrow slits in a barrier. The slits are a distan8c eeew+z l90v+kyqbtln (.,dce $2.0\times10^{-6}\ \text{m}$ 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 i0tch lo 5z( kotwp/d 6gfh,8n5s given both by the Bohr model and by quantuhg8kt , 5/zh5npc t0ld6 fowo(m mechanics. 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 speed ofys3 8ekt(, )fvowh3aw $(22 \pm 0.22)\ \text{m/s}$. 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 nml:p/;fo ) v5giooicf/a)a ( yu. Careful measurements show the linoi)5goofl ;a )ua /p y(/fvc:ie 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. They are then directed at ta(,i4m o feqisfl45z2wo slits 0.70 mm apart. How far apart will the interference peaks be on a screen 28 m qize,f(4m f4a5 2ilsoaway? $\Delta y$ =    nm

An electron and a proton, each initially at rest, are acceleratedj;*1ryw.cvdq : jc+xk 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 c qj1wxkj*;:+ cd.r yvmomentum.

  If the principal quantum lw ( dc,mtzr0ey51 ;gtnumber n were limited to the range from 1 to 6, how many elements wouldwrm0l(dctg5, ;zt1e y we find in nature?   

  If your de Broglie wavelength were 0.50 m, how fast wouls0 wc2ihr; hixt,k8k /d you be moving if your mass is 75.0 kg? hxkit0h kr;i /8 wc2s,$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 withwx*kxk cph 068 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, determine Z and the element x*kxkc0hp68 w in question. ----待选择----lithium berylliumboron 

  Photons of wavelength 0.154 nm are emitted from the surface of a (5a3 pg7-+ 6.jx iieav wqwegrcertain metal when it is bombarded with high energy radiation. If this photon wavelength coraigwwr5.ea+je-(i7 q pvxg6 3responds to the $$K_\alpha$$ line, what is the element? ----待选择----nickelcopperzinc 

Show that the diffractive spread of jg-ecs6 1ul4 qk,vnrr59u ca7 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.]