Compare a matter wav,6n ezghh97-8atuv eoe $\Psi$ 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 coyi8kw,tx /hh4mpatible with quantum mechanics, particularly the uncertainty phh8xy ki/4 wt,rinciple.

Explain why it is that the more massive an object is, the easienf4n6z: (u zxx hl)fvimtc- f9wg8 h)-r it becomes to predictzgfhx6-nhffv9ctn):8 luz xw-i)4 m( its future position.

In view of the uncertainty principle, why;3hn c zwrj 1d)ejh 1w45t5cug does a baseball seem to have a well-defined position and speed whereas an electron doesth w5e1zj1wj r; h54ud)ccn3g not?

  Would it ever be possible to balance a ver: (42acnbin3x w btkm;y sharp needle precisely on its point? Expl (42w 3bcnitkan:b;mx ain.

  A cold thermometer is placed in a hot bowl of soup. Wieg9kycp/j 8 gng- 7b.v3psjeq9 p4jnhc+ 8ji0 ll the temperature reading of the thermometer be the same as the temperature of the hot soup before the measurement was maycg/jis8j- +pepb qkg8n7gj90n h e3 j. 94pcvde? Explain.

  Does the uncertainty principle set a limit to how wel j-bze8r(g :ldl you can make any single measurement z 8gd(:je b-lrof position?

If you knew the position of an object precisely, with no uncertainty, )n+i9lwf+o ww how well wfw+n )li+ ww9oould you know its momentum?

  When you check the pressure in a tire, doesn't some air inevitablj yqa k7cud940y escape? Is it pock 7qdja09uy4ssible to avoid this escape of air altogether? What is the relation to the uncertainty principle?

  It has been said that the ground-state energy in the hydrogen a,: iku; r )yqup8kbjj-tom 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 ene 8:uu-pjqi ;,bkk)rj yrgy form? Explain.

Which model of the hydrogen atom, ther 71/gvmwyz s3zclo: 8 Bohr model or the quantum-mechanical model, predicts that the electron spends more time near the nucleusg/ mvo8rscw31 7zl: yz?

The size of atoms varies byok*6i nyo 6(dtkxxn y3 hpv922 only a factor of three or so, from largest to smallest, yet the number of electrons varies from one to over 100. k6x9po*xnk hn2o(idt yyv236Explain.

Excited hydrogen and excited helium atoms both radiate ligha1t* irhxla5)k(ysk7 t as they jump down t*ta7kx5rahkyl (i s )1o 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: : .gbp3i9 nvwu0*czl4e(miao h2dww were no electron spin but otherwise quantum o9wm a3euigb : ncid z.(w:*4hpv0l 2wmechanics were valid? Consider the first 20 elements or so.

  Which of the following electron configurations are notyxgr- :c d5r-kak)j,k 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 uranium atom (careful scr j91ovobu r) b53aj2frutiny across)o1523r9r ao vbfbjj u the periodic table on the inside back cover will provide useful hints).

In what column of the periodic tab tkikhj0 c:)lhds+0w c 1 my/-ekbo1/ile would you expect to find the atom with each of td-he:/cli 1 +o0by) /sk 1 mkkct0ihwjhe 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 iodine exhibit sim6ka5 ynvbkg3n1z8uv- ilar properties?

Explain why potassium and sodium exhibit similar p be6pzlhg6c8wuv8 :du h6uzw 3mk3(d*roperties.

The ionization energy for neon .lmk( /* j)plshnx)x+/utaru $(Z=10)$ 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 phohvow+ bq0 bx *+edr:8gcw0uu9ton nature for light? :bh+u9uv qrwcd *o ex0+gwb80

Why do we not expect pe e fyt,hqe890mrfect agreement between measured values of X-ray line wavelengths and those calculah9e0,ft8qey m ted using Bohr theory?

How would you figure out wgo.4hkluo ;o/hich 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 1n/jek d vu5k4atom to produce shorter wavelengths than transiti 5e/1jk v4duknons by outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from ordinary b.kb3vdddqk8jp /c7 /light? How is it the same?

Explain how a 0.0005-W laser beam, h7 +h p***i)x nz psvuh2u5lq6s3ydtsphotographed at a distance, can seem much stronger than a 1000-W street lamp at the same d*hul 5s*h*7d36unv+) hpsi z ts pq2xyistance.

  Does the intensity of light fro kftyddw/ lva6dt t:ac,7-.t,m a laser fall off as the inverse square of the distance? Explai.7 y kl,ft6,-adt vwtdac d/t:n. ----待选择----yesno 

  The neutrons in a parallel be59krzy3 ;v zo 82usyzqam, 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 o)v+j8 (p,a*e- zulfg/ dj .4qjirhqca 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 be m x2 /ftek7 dc kv+b8bt8i 1sepsa-is- 5ab-w5teasured 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 ovm dsvzeko 5)r (v),bxwm:k 7)f 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 a mu b ll7iejs:7g-on $(m=105.7 \, \text{MeV/c}^2)$, 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 energyrg4apok/rpm,; f:p 0z of a neutron contained in a typical nucleus :0,4;okzaprg p /pm rfof 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 show that if an electronduxr wc2,m,zvu z4 8f7 were present in the nucleus md,c87 2zxr4wfvzu u ,$(r \approx 10^{-15} \, \text{m})$, 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 positio;.pi :tyl/ s69.hxp3ilpgz x fn 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 c iy-0-*: ttrjmu5z s7btao4n yan be in the $n=6, l=3$ subshell?    elections

How many different states are possible for an electron whose principahleb5*7mnww3 p h89ys l quantum nms5pnw l 9 he78*3whbyumber is $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each electron in the ground staw1tyr,or4wa xs nb l:0xfd:83te of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each electron in the g,)d i:(nxu jloround state of nitrogen $(Z=7)$.

  Suppose a certain hydrogegmn46j 3jfq3g e7 z5u1airls;n 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 ofus nn 6+4r ;3vtvbw:yg the 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 has0uvh t4p, u;js $m_l=-3$, 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" subshel2 (c k2j/ithpxl.    electrons

What is the full electron configuration in the:dhig+8 kmuff8p9yq - 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) selenipe j; vab7ws7o k;x*1lum (Se), (b) gold (Au), (c) radium (Ra)? [Hint: see the periodic table inside the b o7s1wp*vax e;lj 7b;kack cover.]

  A hydrogen atom is in sxrf ;per5d*vnm *s)0the $6s$ 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 elec3rdal3ax)l k r4is3.mtron in lithium using the Bohr theory. [Hint: This era)3da s .3lmrkxil43lectron 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 momevh 1-+m-9 4d2pe i03nqdw-ajlsido v qntum is zero for a filled subshell.

  For each of the following atomic transitions, state whether the transitiij65dm07zfbyl jv5mdouy 6:( on is allowed 5 izym 6m uj0vdlb6j( :fo75ydor forbidden, 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 inflk05e 5yv xi(oc;e (i 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-r5 .ebhrc3pp mfkj -)x:ays emitted by electrons striking the face of.kp:hx3)re5pf m-c bj a $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength br0f evk,vxsdou-mjy 6,0 edo8k65; cmvemsstrahlung X-rays emitted from an X-ray tube have 0-ko8k uyd; mvvf,5m6vx,ejo6eds c0 $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoffm kcfqs)cc4)l5p tt,mm(v6 b0 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 jxm,8b()+bm chxk8lve/ g- otemitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength for dg8 pht (w ;04)ovltjsan $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the wavelength sm ,aw2k2v06+ a5rootqb4hef 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 material iot+ +/ jq:aotys bombarded with electrons. The wavelentqao ++tyj/o: gth of 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 ra,z zy flvt-ax i((dht1683xzetinas 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 a4ci23whpmhyy n2)yvb p3p * m1okj +8vu;er-u 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 t.*456rh,itwl-mpp o vjm3kr diffraction if the beam emerges throu,-6v oiklp m.4wmh*rr 5tt 3jpgh 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 tqeeney +l+hot:is,2; he He-Ne laser?$\lambda $ =    nm

Use the uncertainty principle to estimate the position uncertainty for the elen5 5aryw5 9f3frnq54zx xo)jyctron in the g o55545 q y jfxz3nnrwarf9x)yround 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 and smallest possibleirgk f2yw7 u0,,j+b,ecf:gv ty9tpv4 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 acl3n.6o 575c rhtpt5kz f pgxj;8f;ebn electron orbit, estimate the average distaz3lojxfcc. 75e8pbh;5kpf6rt 5;tn gnce from the 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 operat7y 6enxft8c.q es 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 outp0htc7 cxp9 *xd-boys7pb 36rermost 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 s*ecm9*4l,ss rl rdz 4nhow 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 states for a givdv mcf x*i.qb 1x7l jhvq):5;ven 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 for a given value i08bvl,yiiwq ) rq 39bof wbv iy8q9i,lib0)3rq $n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV is shot through two narrow slits in a barr+p4 epc iq534 qp*k :vph f1r2feu(bqsier. The q4rpffi4+uspb e2:pq51che (qk* p 3vslits 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 hydrogenv+.rp* j95avqxhm 1hn atom is given both by the Bohr model and by quantum mechanics. Compare the resumhvrapn 5.x*q j+h9v1 lts for $L = mvr$.
$L_{\text{Bohr}}$ =    $\hbar$
$L_{\text{QM}}$ =    $\hbar$ or $L_{\text{QM}}$ =    $\hbar$

  An 1100-kg car is traveling wit9va0e9v bg5y mh 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 line with a wavelength cente.3*-n 0:g.g7s kk ejg d7f+ky inncnfnred at 488 nm. Careful measurements show the line is real-7.* n:3ekfk+gingc f n.0djk n 7yngsly 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 acrosxlta9k3 v)7hwpk: xd8ris2e 3s 550 V. They are then directed at two slits 0.7s3k7ak9: tw8)vi xlr2pxd3eh0 mm 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, are accelerated across the sagi09.bwk 4 sddme voltage. Assuming that the uncertid0.db4g w ks9ainty 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, h* vowcbdczb./9mn 3:low many elements would3low*zmd n 9b v/:bcc. we find in nature?   

  If your de Broglie wavelength were 0.50 m, how fast would q0 8) uxg9newvyou be moving if your mass isx8) uwq0nveg 9 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 seriesbs u1bs*bfefc0v na+jw.975 v 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 wi+*w9b 5a.sv e c fsnbuvf0bj71th Z protons and one electron, determine Z and the element in question. ----待选择----lithium berylliumboron 

  Photons of wavelength 0.1p68 cn6cap;oa(3k ook54 nm are emitted from the surface of a certain metal when it is bombarded witk 86n;po3pcoa(k c ao6h high energy radiation. If this photon wavelength corresponds to the $$K_\alpha$$ line, what is the element? ----待选择----nickelcopperzinc 

Show that the diffractive spreadrzw:e p +opj93n:4if i 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.]