Compare a matter wav fmet5k4m;6ms1d3r x oe $\Psi$ to (a) a wave on a string, (b) an EM wave. Discuss similarities and differences.

Explain why Bohr's theory of the atoq,uqu2-s jr )om is not compatible with quantum mechanics, particularly the unceq su,) rqo-2ujrtainty principle.

Explain why it is thay7 cho;y3-sc it;+pwit the more massive an object is, the easier it becomes to predict its p3 +;i;hyo-ci w7 csytfuture position.

In view of the uncertainty principlrg8rkw;;h ;fh e, why does a baseball seem to have a well-defined position and speed whereas an electron d f;8h;grr;w khoes not?

  Would it ever be possible to balance a very sharp f-ta0gi:k+ xg needle precisely on its point? Expg0fk g:ia tx+-lain.

  A cold thermometer is placed in a hot boj((gj4d0l- w 9p ;qogjt7 zediwl of soup. Will the temperature reading of the thej7 gozet(q9g4j p(;ddjwi-l0rmometer be the same as the temperature of the hot soup before the measurement was made? Explain.

  Does the uncertainty p;) -4dgy(y i.qjxbx;wgswpa3 rinciple set a limit to how well you can make any single meas-;4d)wx yw(y3g; aji.gq psbxurement of position?

If you knew the position of an object precisely, with no uncertaintyc iabyg40cg1 ,, how well would you know i01gc, bgcy ia4ts momentum?

  When you check the pressure in a tihir n2.da7-o,sw v ey 2k:)toxms kq85re, doesn't some air inevitably escape? Is it possible to avoid this escape of air altogetherh orydesmk)5 :.xk aois, -2w7tnq8 2v? What is the relation to the uncertainty principle?

  It has been said that the ground-state energy in the hydrogen atom can bs(8)mp dfd 6tqe precisely known but the excited states have some uncertainty in their values (an "energy width"). Id8t6m)ps fdq(s this consistent with the uncertainty principle in its energy form? Explain.

Which model of the hydrogen atom, the Bohr model or the c.3nv 4lr;0uf.jdqm q quantum-mechanical model, predicts that the dunvr 40mf.jc;l3qq. electron spends more time near the nucleus?

The size of atoms varies by only a factor of three or sofuqs 9 6h r60rijw-cm.k;+k gd (,z ;qm5rlell, from largest to smallest, yet the number of electrons varies from one to over l( ;e- u+q ,6g rhkcz5m.lr; ls6f0jwm 9ikqrd100. Explain.

Excited hydrogen and exrk*ccc:yko k*q o,;c8ihjbs ;z:-c.icited helium atoms both radiate light as they jump down to*qiccr; c8i;:.h zkokkc ,*:os -bycj 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 otla9d l*ao hjjl:r 6g0;herwise quantum mechanljdg*hl:9o ;6rl aj0aics were valid? Consider the first 20 elements or so.

  Which of the following electron configurations are not alo z u).tl+e,vef+lr(cg68pin xl(5yilowed:
(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 f*zr.lqcv+4hyolj6 .csh n/- mor a uranium atom (careful scrutiny across the periodic table on the inside back cover will pr h q.6zrycsvlm*hn-4jc/ + lo.ovide useful hints).

In what column of the periodic table would you expect to fj y vcni qmbe*. dosu,9t45s58ind the atom with each of the following configuratbqsv mon8.e 5j4* c95s tid,yuions:
(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 esy tzq 4x0s)5 q/tjz/wxhibit similar properties?

Explain why potassium and sodium ex ulf5;x.t:/ 7ffitcrfhibit similar properties.

The ionization energ9pw8*g:,cn/jz a jgy 5kdd/bp y for 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 photon nanz* k- q.(9 e4n+ttlgm hodv6.jjro4bufj )r0 ture for light? bro+ -nukfm0.h6 gqr) 4nv.t(t9lj4jj * dzoe

Why do we not expect p t ilw07oa6ph(erfect agreement between measured values of X-ray line wavelengths and those calculated up6athi0wo(l 7 sing Bohr theory?

How would you figure out which lines in an X-ray s-6slo vaojfo;*5kmd 2snd. u 7pectrum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transitions deep withi p.67 4j:rnbbq0ztzo hn an atom to produce shorter wave 7ot:64bpr j0nzqbh .zlengths than transitions by outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light diffcry53 . z( pda-paxev*er from ordinary light? How is it the same?

Explain how a 0.0005-W laser beam, photographed at a distance, can seem muc12pwl ,i;v8avii3 dtxh stronger than a 1000-W street lamp at the same distance.a,18x2v tpl iiwvi3 ;d

  Does the intensity of light from a laser fall off as the inverse squarezt1 ru4xgx22ij he2p;z4oep:c za* 6j of the distanpr ugpz 4ztejxc2 61zh:2e; 2*ioxj4a ce? Explain. ----待选择----yesno 

  The neutrons in a parallel beam ji 7kpga:k bc :)w(5qobs8c+j, 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 mrs0)pxv m.tz41sfyx8c . jf 1a 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 cay oh7kg5pcl8.1xy 1s kn 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 typicallye4 ogwgn,k) bb.c y;r0 $10^{-8} \, \text{s}$. 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 muip iu0dc;r5)ak8+ af ion $(m=105.7 \, \text{MeV/c}^2)$, 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 a tyd y* lvns5+kd-pical nucleus of radiu *ykdv5sd+- nls $1.0 \times 10^{-15} \, \text{m}$. [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 tmkc /yd+ck 55eo show that if an electron were present in the nucle /5dk5yemkc+c us $(r \approx 10^{-15} \, \text{m})$, 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 of : 6n*;+gh fi-z5y jpownx4rqqa $3.00$-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 yx ya6ojsy o+((k9wim.:aayx2k7 :cc $n=6, l=3$ subshell?    elections

How many different states are possible for an electron whose principal quanjy3v yn*/ 3q/pba b5h+/m n3vly7ngkh tum number is ngmn+7v3j*yh nvykh//b35/ 3ap b ql y$n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each electron in the ground st 24gwb9ixat0 xs90 kekate of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbut s;883arzuijjs, 0r ers for each electron in the ground state of nitrogen $(Z=7)$.

  Suppose a certain hydrogen (4-jd pa5cz3o n-juiv 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 momhde;fz yyn i4ssl6s,/im1 6ev/ iq 1-aentum 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 ha.p8t ;t0 lycics $m_l=-3$, 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 electjvt1m0 z3y ;s(9moatrrons in a "g" subshell.    electrons

What is the full electron configuratiow k/om1u1qel. ed7v+bn 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 /vfkoo,7 n8rh for (a) selenium (Se), (b) gold (Au), (c) radium (Ra)? [Hint: see the periodic table inside the bac8 kh/vn fr,oo7k cover.]

  A hydrogen atom is inl cw4y6 b kscz/:-avy,7w2hmn 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 ele.i0 penztu6t62j1 ay gsh1vl-ctron in lithium using the Bohr theory. [Hinjpegh6.u0 y t211l-sva6zn itt: 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 pu9yim8f 9fi. subshell.

  For each of the following atomic transitions, state whether the transition is al)dy x2l(-xx ojfuzhf/c (;6jklowed or forbidden, and if forbx2c )d(/xjffl;y zhjxu-k ( o6idden, 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 nwl3w9dtj8 fwj1h y9r0qkm0 3xv 5/nain 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-wave a5mz9y)5 nil5a0 biztlength X-rays emitted by electrons striking the face ofy z0bmalznii 59)5a5t a $33.5$-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 havguupz9ags sr3ol /lcxzf-7)py i2 +0:e-gu scpl /o9sli:ag) yz3f0x+rz pu7 2 $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoffq a,zp1 l3bwl( 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 w7z/ oqgz8m c4vavelength emitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength )uq7 /cnjsjdt1,(uq a for an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the wav zob8 tv)0x0dfelength 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 is bombarded with electrons. The wavelaw3oexed ,e6lwpp, )r7yh ; j0ength of thee ;o,w eyhxa)p7rjwl3d ,e06 p $K_\alpha$ 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 desu8lyhnf- va d2liro*fbk03 b1); 0xdtached 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 have a ru 1i(vxt3of) c sg(x+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 bec0g80 pov in+lam due to diffraction if the beam emergcgo8p00n i+l ves through 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-nio7k -n(b )nmNe laser?$\lambda $ =    nm

Use the uncertainty principle to estimate the position uncertainty fozv5pcmal;+ f962y r66w55u m hsnxxb ur the electron in the ground state of the hydrogen atom. [Hint: Determine the momentum using the Bohr moax lpnxuz vchb65ys5mf; u2 69m5w r+6del 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 smallb(ooh16hi je f(z6;qz 4ocra5est 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 orbbtamu8zdg53dhe74 s3 it, estimate the average distance from the 3dbas5zue74 gt8dmh 3 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 operates2sg5/3 zlt- t x)bfo b4.eoras 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 outermost electron in boro7mqg7 hh qule;l:-h a,n 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(*mj mvn t x;t)rwj9n8x4w6rb9 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 skf : p2zaolv;1tates 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 different elect)uot79oh:nk x ron states possible for a given value of o :hk7)ton ux9$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV i,/ujrlse4-7h:f vh a xs shot through two narrow slits in a barrier. The slits are a disthl4 :j/rhf-xusae,v 7 ance $2.0\times10^{-6}\ \text{m}$ 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 mx +i0j nmmbi-ce4jlu v 5+g/(nodel and by qu5n/ncvux+mij0(j 4 +bg il e-mantum 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 witwrj 2r0:7z rqth 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 centered at 488 nm.xezd p76f;9w03 xk yc;.mimg b Careful measurements show the line is really spread out between 487 6 edkcm i .pxg;m390fxzy;bw7 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 twtuvxtz.x2ry ztej) l)u7y i,t w68+4qo slits 0.70 mm apart. How far apart will)+ 4uq. tw7 6 2tixzrv)t8zjyx,tyeul the interference peaks be on a screen 28 m away? $\Delta y$ =    nm

An electron and a proton, each initially at rest, are accfd: /me9, vcfl+zlp9g46xp z relerated 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 momeefp/l9mx grzf v 9dc l:4,6p+zntum.

  If the principal quantum number n were limited tw05d;uqpvg1 j o the range from 1 to 6, how many elements would we find in naq0v1uwp5; dj gture?   

  If your de Broglie wavelength were 0.50 m, how fast would you be moving ,.g40rr k; xjydzy(vex- fti5if your mass rxe0-gz5d ( yt;jvr.4fy ,ixk is 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 seriesjc*:-ln r)isz 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 element in ques)sjl*n:cri-z tion. ----待选择----lithium berylliumboron 

  Photons of wavelength 0.154 nm are emitted s:h0p)x88, fdw8a0mzrwqxyd) xs;zi from the surface of a certain metal when it is bombarded with high energy radiation. If this photon wavelength corresponds to th)f8y xq i:zsmasdd)0 w;wpx8r,8hz x0e $$K_\alpha$$ line, what is the element? ----待选择----nickelcopperzinc 

Show that the diffractive spread of a lase:hcji2rp)3 p+5 m sludr 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.]