Compare a matter wave j 8i6u;jdx*4uikiy0 xwa.t a-$\Psi$ to (a) a wave on a string, (b) an EM wave. Discuss similarities and differences.

Explain why Bohr's tqcmul(. -;ir jheory of the atom is not compatible with quantum mechanics, particularl-j .;lciu mq(ry the uncertainty principle.

Explain why it is that the more macn- 8*oz9ap-r w8aqoom iv )c4ssive an object is, the easier it becomes to prediccr -*) a4pc8no8am q9oowv-izt its future position.

In view of the uncertainty principle, why does a baseball seex9 6cl yz*cpi8m to have a well-defined position ayp lx6c9i8cz* nd speed whereas an electron does not?

  Would it ever be possible to balance a very sharp needle precisely on io01w+d-: p ne0dryatrts point? Explain1tpd :oa+r-r0 d y0wne.

  A cold thermometer is placed in a hot bowl of a:v2 ck rxvm ,alta)y 578zxx.soup. Will the temperature reading of the thermometer be the same as the temperature of the hot soup before the meas,8 ax).t5v c:mxx72kl y zavarurement was made? Explain.

  Does the uncertainty principle set a limit to how well you can make any , 3pxcj1tdl;gsingle mex 1p,l tgdcj;3asurement of position?

If you knew the position of an object precisely, with no uncert7 jsg h/+d, f5sen4uivainty, how well would you know its momen niu5e+ 7/d4fssvh, jgtum?

  When you check the pressure in a tire, doesn't so6ihe4 xaag/8 okej 8z4me air inevitably escape? Is it possible to avoid this escape of air altogaeakz 6o4h8 j4eix/ 8gether? What is the relation to the uncertainty principle?

  It has been said that the ground-statese2o1- vl+ zcv energy in the hydrogen atom can be precisely known but the excited states have some uncertainty in their values (an "energy width"). Is thz eo +v2c1vls-is consistent with the uncertainty principle in its energy form? Explain.

Which model of the hydrogen atom, the Bohr model or the quantum-ma3o 5:vz;ftvo)/clbhfh-/v dg ;x mh0 echanical model, predictoxao /gztbf;vvf3 0h;:m/h- c lh )dv5s that the electron spends more time near the nucleus?

The size of atoms varies by only a factor oq7 ix uzb11a,gpuw,6mf three or so, from largest to smallest, yet the number of electrons ug q71um6,w pxba,iz1varies from one to over 100. Explain.

Excited hydrogen and excited helium atoms both radiate light as they lzz6/arq dr)hqs4g -:jump down to thlg-z:qqr6d sra/h )z 4e $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table look if there were nokxt.o9g6 x,1ql mx db) electron spin but otherwise quantum mechanics were valid? Consido.x gt9xlm 6 k1x,qdb)er the first 20 elements or so.

  Which of the following ele( anmcmw9nn5h.j7 3vp ctron 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 a4po8/wg mc vd3 uranium atom (careful scrutiny across the periodic table on the inside back cover will provide useful hints)dogpc8m /3w4v .

In what column of the periodic table would you expphj-k4myl :s1 s./qzpect to find the atom with each of the following configurationspsyk hm/ zl4j:p-.q 1s:
(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 properti*jlfig2h. 9g r4l za;qes?

Explain why potassiu)rw- l1uz0 t-zrpw .qim and sodium exhibit similar properties.

The ionization energy f.as u 3lk+n: owtxf)o5or 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 wa s-ms:p7 /i z0)tpy,3jsclxgavelength in (Fig. Below) imply a photon nature for light? mtp0p3 ys iag -/zx:,)lcs 7js

Why do we not expect perfect agre mgar8 ;hlut;4ement between measured values of X-ray line wavelengths and those calg at8 r;;ulmh4culated using Bohr theory?

How would you figure out which lines in an X-ray spectrum corw;syu s6u fx57 c wush0.em),lrespond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron t-e2 -r y r;ajgwu2glz46zans0ransitions deep within an atom to produce shorter wavelengths than transitions by orwaz a22-lzg ;4-6njgry0eu suter electrons?

Compare spontaneous emission to stimulated emission.

How does laser light diffemn-nhw5udf; :r from ordinary light? How is it the same?

Explain how a 0.0005-W laser beam, photographed at a distance, can se 7l* lfte(,kpt,u bji4em much stronger than a 1000-W street lamp at the same dl7,betku(, ilf *pt4jistance.

  Does the intensity of light from a laser fall off asb2(qg1jl k7f2wa :tw7ziu5k n the inverse square of the distanc2gak7lwt:wq (7k2i fb5j n1uze? Explain. ----待选择----yesno 

  The neutrons in a parallel beam, each having krp(,f7pc 0e wc,x1 wyfinetic 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 a1bo 5ufje t cm*xwrv3*am. ,:p 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 can 6zyhpqc q :j7 m r6t- * nmrxbfh5hm:)/cy5sm.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 -u2:tcf .naat 1*ducqin 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 mass of s3dpd oygtvx9p*2,9 c 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 afoqr*ivz1+r*k ihtsm e00*v 7zkih /7v f5 2ipossible energy of a neutron contained in a typical nucleus of hv*5 mrfsv0qiaei0h/*vft2 1z +ik*kzi o7r 7radius $1.0 \times 10^{-15} \, \text{m}$. [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 sh8xw(5 8(d2zvigg afl iow that if an electron were present in the nu i2g8i5 (z 8vfdlawxg(cleus $(r \approx 10^{-15} \, \text{m})$, 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 ofok.ze/gj-tj6b 5 n1a)(hox ,vzhbmd7 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 the fyfl s 1 y82bw.i0gkpo08ysd5$n=6, l=3$ subshell?    elections

How many different states are possible for an electr6lp,9m4a . xbph).nkf lvx5qb on whose principal quantum number isp 64.,bb5 xvpq9alfkx l.)mnh $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each electron in ; s 8.ddx:idil )zcd+ethe ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each elehwxcoyk7*qg/jxx2 0s 7f*dp 9q *e1ghctron in the ground state of nitrogen $(Z=7)$.

  Suppose a certain hydroglomc(d isgues,)/( 9nvu 6 s,xen 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 the angular momzbk 2k(7 kgu7bentum 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 hqici htig kl5okj h86ge(1 ;;7as $m_l=-3$, 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 gb nq1i6)5yuwin a "g" subshell.    electrons

What is the full electron configuratio b5eeadv a4/ ;q/,l4ni(a qz 2(1axesdi juq7yn 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 confh cv k8sgxf))/6ciky3iguration for (a) selenium (Se), (b) gold (Au), (c) radium (Ra)? [Hint: see the periodic table inside the backik8c 3c/v )s g6)xfkyh cover.]

  A hydrogen atom is inflx02 leygr*3k6 ( xit 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 lithcl1d. 00yk wl(s z :,zonnzq(kium using the Bohr theory. [Hint: This eld (o0k.y w0klq(s:c,nznzlz 1ectron has $n=2$ and "sees" a net charge of approximately $+1e$.] The measured value is $5.36 \, \text{eV}$.    eV

Show that the total angula:z 4xuc cew)lu92 0xfrr momentum is zero for a filled subshell.

  For each of the following atomic transitions, state whether the vj + x681zalsifsh ,jh9ix4z-transition is allowed or forbidden, and if forbi4 h+vjzizlx6h, -a isf s91j8xdden, 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 isa/k 3ep7jecgg4 0 ki0met ;i:h 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 em tqk05ygwcqq2q,) n0r itted by electrons striking the face of a 02,0q )nq5 cytgrqqwk$33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremsstrahlung X-rays emitted r2 qcwpzh8s 97g/h:.aru iwr0 from an X-ray tube have7w90/g:rw82ipcar hhzrus q. $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wavelength 9lm mr9d8 z0np4azygxxm;h .2i7p3 1wnpt aw8 $\lambda_0$ 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 wave .5yd365eubsam,5qoz s2k bi zlength emitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength fcptax, -to.d*bywf-1 z c 41nwor an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the xu6xoy*3( hh kxcoi( hk6 g,z pg:z:a7wavelength 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 matecybo* 8x .d:berial is bombarded with electrons. The wavelength of t:ce b*.8xbydohe $K_\alpha$ 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 7uz,bq5 jj:iepj96o3jj us 0tretinas 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 us+*oxzx kqs b2t: k-9sx;x.igmed in a physics 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 beam ,srrr0cg6tgl3. i: wz emerges thrgtsr3r6l w crig:.0, zough 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-5*,m r-jtfk4 c) )6 nwpr.jfr n4zslsdNe laser?$\lambda $ =    nm

Use the uncertainty principle to estimate the posaoe61.yl, k 2wfqm;wo ition 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 the momentum can be anywhere between this value and zero.] How does this result compare to2my w.olwe1;o ,f6qka 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 posocpr8ch 3tc: ;sible 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 radi gkk6 ony3woh(v1*oet2ow1z *us of an electron orbit, estimate the average distance from the nucleus for an electron in the innermosokwoewtv** 1y(z 3go61h2oknt $(n=1)$ 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 az99s- v5gaeyrhhs q( :t $95 \, \text{kV}$ 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 5c3wnwa,v 9ke outermost 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 Mosele*zb7l9y3l5hxj l)tc ey 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 nuzd h 3qut64u6que.d eg0 v2w;pmber 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 ob0lgi zut9mk )b:*l.xzx:r /m f different electron states possible for a given value of b x:tixb:9/ml ul0 k zgm).zr*$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV is shot through two narrow slits in a bawrdca:4x3r(g2 ox p a2rrier. The slits are a rwop :r3a2da(24xc g xdistance $2.0\times10^{-6}\ \text{m}$ 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 model c z(tmd7m8tjcvu*:o ze+ 6sf (rw65 mmand by quantum mechanics. Cozcw(dtuv8*+rjf 6m( e 7 c:m mom5szt6mpare 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 shjefj4 0u dv4j,b8nck+( 2 yknpeed 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 contai kjrgs-h dqx-d:o6- v5ns a line with a wavelength centered at 488 nm. Careful measurements show the line is reallx -rskd-5dv -qoh: 6jgy 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. Theyqi1wnd iaj8g u ( dxm:3fp1pzl1;y3i2 are then directed at two slits 0.70 mm apart. How far apart will the interference ; gyfm1qd1pu (iz:833n xwdii1p2l japeaks be on a screen 28 m away? $\Delta y$ =    nm

An electron and a proton,b0py h2d g,bxj7mpr3p2m)fc: each initially at rest, are accelerated 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 momgpj2cp207x: bbrd3,mfmyp )hentum.

  If the principal quantum number n were limited to 0 wt l;cq5h( wx*(7xfyzbit/jthe range from 1 to 6, how manyx/f0t(yqxh z;w5ib(twl *jc 7 elements would we find in nature?   

  If your de Broglie wavelength were 0.50 m, how fast would you be movicr0h1c -dn(fg ng if your mass is 75.0 0cd - c1(hnfrgkg? $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 sz 1fnu/q*u8j reries of lines 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, determine Z and the elnu 1jrf/*u8qzement in question. ----待选择----lithium berylliumboron 

  Photons of wavelength 0.154 nm4dwidn8uv k1h2 r4g oc,4za (d are emitted from the surface of a certain metal when it is bombarded with high energy radiation. If this photon wavelength correspondr4acn zdv4du2g(o h4kw,8 d1is to the $$K_\alpha$$ line, what is the element? ----待选择----nickelcopperzinc 

Show that the diffrac;b2dh hcatr i2 :(a(mvtive 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.]