Compare a matter wav7c1 x( pvii).t8hwxcl/u)jrt 8h ytq;e $\Psi$ 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 mehh 8t2g yyno9oa nin*z63q:,echanics, particularly the uncertainty principle.:,tq2nahi38 nyy oh69o*n egz

Explain why it is that the more massiveck6 rgg*8uvyug1y5o r)2o p8th 9 i;lu an object is, the easier it becomes to predio*c 68g59you r18h2u lkyg u;i)grtpvct its future position.

In view of the uncertainty principle, why does a baseballdqz0+h8 gxpwq s(xl8 0 seem to have a well-defined position and spzdlwq808 x0sghq+x(p eed whereas an electron does not?

  Would it ever be possible to balanco (n7 (jzcunu pyy)h)1e a very sharp needle precisely on its point? Exppnu71z n()jyhcou( y)lain.

  A cold thermometer is placed in a hot bowl of soup. Will the temperature rea;4k+ i4tpw a m+chn4vtding of the thermometer be the same as the temperature of hiwvtkn4 4am ct4+;+p the hot soup before the measurement was made? Explain.

  Does the uncertainty principle set a limit tode33 4cmmd5hy how well you can make any single memd5 3e3 4cmdhyasurement of position?

If you knew the positiowkvz sbx 23tb-6dv/8v n of an object precisely, with no uncertainty, how well would3s8tk6 vbv- vd2/zbxw you know its momentum?

  When you check the pressure in a tire, doesn't h .nw5kg33 p 3ooj7kddsome air inevitably escape? Is it possible to avoid this escape kdn3jo hwgo3k p3.d 57of air altogether? What is the relation to the uncertainty principle?

  It has been said that the ground-state energy in the hydrogen atom can be prejq;p2hk8-qi c0 aielo1dc 1 -vcisely known but the excited states have some uncertainty in their values (an "energy widthckqc e -12qa l;1i0pjvih-8od"). Is this consistent with the uncertainty principle in its energy form? Explain.

Which model of the hydrogen atom, the+wu+f tfkmff ;)5br; .i5u3 s qv9anme Bohr model or the quantum-mechanical model, predicts that the electron sp miv+ 5u;ffw+etfa5 k.rq3m9 su);nb fends more time near the nucleus?

The size of atoms varies by only a factor of three or so, from largom8vy bm-6qa lt8/o/ 1dwg6 elest to smallest, yet the number of electrons varies from one lew -odq8lv6bm 8gm ot/a6 /1yto over 100. Explain.

Excited hydrogen and excited helium atoms both radiate fmc3/ :ig5 ivdy 3iar)light as they jump down to tai)i3v rc:gfmd5 3/i yhe $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic ta ozqfv,rr0qxgndj5 2su v( 23t ,(oqt:ble look if there were no electron spin but otherwise quantum mechanics were valid? Consider tqr :tq0zuvj o,2dov3s( tngf(5r 2q,xhe first 20 elements or so.

  Which of the following eeltylm6*sc/w11c y)xg.yn p *lectron 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 uranium atom (careful syxss8-k lu,1 lcrutiny across the peri8 ,yl1x-ukl ssodic table on the inside back cover will provide useful hints).

In what column of thn6wb3e mf m5a9e periodic table would you expect to find the atom with each of the following configurati3ba9emw6 n 5fmons:
(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 simila3 x +ghn3xs2 pr7 0duzfg;r4cvr properties?

Explain why potassium and sodium exhibit sif);5d/y4q0ez -ft. r dotodpp milar properties.

The ionization energy for neolgh/ b u0(pcc /2uve*gn $(Z=10)$ 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 kn7z1dc+sbv ti+u+g/ photon nature for light? +k/n u7 dt c1s++vgbiz

Why do we not expect perfect agreement between measured values of X-ray line wavyy/ gz.mgkh5jkb 6vpn.4el7 6 ely6 v5.z.hjb 4e gg /6ky7kpnmlengths and those calculated using Bohr theory?

How would you figure out which lines in an X-ray spectrum correspond to g2 c5llkn84fcd 76tx rl.oxt)$K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transit,(9xu gtx:d/hwkhtg6ions deep within an atom to produce shorter wavelengths g(gd:ukhxxw/ h6t9 t ,than transitions by outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from ordinary light? How iz*w qp.5m7qbls it the same?

Explain how a 0.0005-W laser beam, photographes.nc) 4qb9mn9k .pbr,y ou 5njd at a distance, can seem much stronger thann9n u9.qnmby)or k5j4s ,.cbp a 1000-W street lamp at the same distance.

  Does the intensity of light from a laser fall off as the inverse square +kwp5njhvd-2 of the distance? Expv 2-jk phdn5+wlain. ----待选择----yesno 

  The neutrons in a parallel beam, each rw5 j /y21 mktkg9jl:bhaving 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 s/ 9z y(3hadl0y od4qrupeed 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 posi 5v;8e;m:h9fqpqghmm7 n 2fg ntion can 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 ;l,p2cl x 9jve7)zozy 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 acj)qxaf 0qz9 8 6byzr9 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 ne /) ;h0+m cohb.mqvcpautron contained in a typical nucleus / mbhcmqa+hv);co.p0of 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 electron i dqk /:kgi+c7/( 9iryjjfl;swere present in the nucls+9qdl:ij fj;r( kk7/gi c yi/eus $(r \approx 10^{-15} \, \text{m})$, 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 posit0uakm2bk mlkik +qy26qn xnxr 17i8ht 69zj;/ ion 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 ch 8d7o,yens(rea5y. ),kajzian be in the $n=6, l=3$ subshell?    elections

How many different states are possible for an electron whose principmw2w : mja/xj.l co0(qal quantum number is j:.xmmja2q0cw /lw(o $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for eacy5 ;vdp51ajx wh electron in the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each e ldn,9au guo*a-o,akio- js-7lectron in the ground state of nitrogen $(Z=7)$.

  Suppose a certain hydrogen atom (k*0h*pa7 rx5 rxujeh 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 momentum of an electr25bk8o-2kcf1ptb7 mc8q 7bs:icni co on 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 has 3gukh5*1 13vledod. e k,ae zd$m_l=-3$, 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 yn q8ng*(p m4min a "g" subshell.    electrons

What is the full electron conf2w8 w7+)sosjfqs u8 dbiguration 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), hy ydryp *z),.po1a2lgaq 4p)(c) r4p)ha12 p*ap.lq)yo, dg ryyzadium (Ra)? [Hint: see the periodic table inside the back cover.]

  A hydrogen atom is in the ysq4 8nrs:ax(er(-p(j : ee pk$6s$ 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 lithium using t3t a*5s:x6 nutjy(hwshe Bohr theory. s(:tw 6s3*jan hu yxt5[Hint: 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 fdma-tg+az gp;zh 84o.or a filled subshell.

  For each of the following atk/v/ . wj w1wj*hzc;ziomic transitions, state whether the transition is allowed or forbidden, and if forbcwj* k hw/1 zw.;/vjizidden, 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 vu9d q) ne129rxq 0q(g9uah jjin 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 shortestjgqmpn/xa/k 3 k *id8ae-p0c qo lw:u-*+ 0lin-wavelength X-rays emitted by electrons striking the face of a id*l8w-pk3n*e- ki x amqoqnja g/+:c 0/0p lu$33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremsstrahlung X-rays emitted from an X-ray tub 24 jukd67upz;fz .ujee havez .f7eu;p jujdu2 k46z $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wawxqz 49ql6 :wxvelength $\lambda_0$ 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 emitted k09(v g1s mbx6d3jglxq+t4lawhen a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength a0.hhl o 1yxa5for an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the wavelengtz 9o.a4a4y w(evrn4luh 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 wavj p 3leu-nx**uelenge uj3-lupn**x th 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 retinas puts ow 8u kc-p0x9lmut $28$-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 physh.bj b/3 0ikycics 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 th srpr hp64oe+9e beam emerges through4rr s+h e69opp 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 Hkz84ypu ;y)*p64 /qcdjl bm xie-Ne laser?$\lambda $ =    nm

Use the uncertainty principle to es-z .,r nbxlq0bjt ii.:v5tr p1timate the position uncertainty for the electron in the ground state of the hydrogen atom. [Hint: Determine thtjxli- t0..b pr:1zin br5 ,qve 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 posf+ejyza h(; kezi)k5 )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 3bgjg0v5 .gjtthe radius of an electron orbit, estimate the average distance from the nucgtv3gj 5bgj0. leus 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 operates0a1l te o;iars:p;hb* 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 h-98nyu 4dz;/l kc+n1cez q kgof the 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 Mose 2dvkn(7oody;r t 2.y4ziz.fx ley 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 givg f )li-p*vr06deg u(len 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 statesxflhqy/ w/(m 2 possible for a given value of m/qfhwx2/y(l$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with enj pqq09 eh(v5xergy 45 kV is shot through two narrow slits in a barrier. The slits are a distx vq(5 pqeh09jance $2.0\times10^{-6}\ \text{m}$ 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 dumxjc1fc3el:mi4jb /5: a6zis given both by the Bohr model and by quantum :5/:jmjub4dclize a6x13 fmcmechanics. 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 speed1h0--gl nmel n np1ih- 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. Caref05t/ qw;w 0ok97zuaievq.ow1qjb u 0cul measurements show the laj00uteqi.w ;owcu9vb50wo7k1 /qzq ine 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 frj4y 87dsy4*ycclr+ hl om rest across 550 V. They are then directed at two slits 0.70 l47l*r8 yyy4d+cjshc 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 acceleral q;.tmga)o ov ,pje-4ted across the same voltage. Assuming that th,;mlogaqtj 4)veo- .pe 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 ej0; 15zll 5n bzebirh:sw7e 2limited to the range from 1 to 6, how many elemel1zrw250 in7z:5s belhb; eejnts would we find in nature?   

  If your de Broglie wavelength were 0.50 m, how fast would yozh,n f jg:vi q, zmj- 3z9;:uik.qs9aau be moving if your mass is 75iq a fz,viqjm. 93u9;n g,jzz::hsk-a .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 series of lines xx 9lm(+x6 ax. par6-a ylmoj8a,zkw2with one out of every four matching a line from the Lyman series of hydrogen. Assuming that the unknown eleml-rja26x aax8o.zxy+9mal m(wx6 ,kpent is an ion with Z protons and one electron, determine Z and the element in question. ----待选择----lithium berylliumboron 

  Photons of wavelength 0t(whs:s s6n0h tqz5/ d.154 nm are emitted from the surface of a certain metal when it is bombarded with high energy radiation. If this photon wavelength corresponds/: 5 zsn(h ts06qdwhts to the $$K_\alpha$$ line, what is the element? ----待选择----nickelcopperzinc 

Show that the diffracti k qte y;i3idtgs, nywy,m43ntl,3y(9v 0 hp,rve 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.]