Compare a matter wave , b..cff7uumeh-/y;d :nbpvn $\Psi$ to (a) a wave on a string, (b) an EM wave. Discuss similarities and differences.

Explain why Bohr's theory of the atom io/y:ysn/ pi x*d,,ng6 a(ixp1h n 5xams not compatible with quantum mechanics, particularly the uncertainty prip/*(5x,iaxoy hdps/6mn,nayx ig n :1nciple.

Explain why it is that the more massive an object is, the easier it bwk:0a34ijps: ( ij ywfg,c-vvecomes to predict its future k4 - i3c0yfi g(a jv,v:wjswp:position.

In view of the uncertainty principle, why:6rgh7g3rnmo 6 lo*aaxf +)ta does a baseball seem to have a well-degl)oh 3:taaox fa*67+rm 6 rgnfined position and speed whereas an electron does not?

  Would it ever be possible to ww-qtkx-y9t w a6z4b,balance a very sharp needle precisely on its point? Explai6z-xqtwyw b, a49kwt-n.

  A cold thermometer is placed in a hot bowl of soup.j.j 4z *hvfys wa5jx)2 Will the temperature reading of the thermometer be the same as the temperature of the hot soup before the msfa.xh)jv4j yzw*5 j 2easurement was made? Explain.

  Does the uncertaintycdf b)37as2 9x1z;xfupe7d ca principle set a limit to how well you can make any single mexxp 1d a a7cs9bz2;u)3 fcefd7asurement of position?

If you knew the position of an object precisely, with nv 8b./ben3 ppuc3 w;xzvhxj 4d26jd4yo uncertainty, how well would you k4wxhcbv v;pd eb3./y4nxdj86pu3zj 2now its momentum?

  When you check the pressure in a tire, doesn't some d+by nj/0k k)vles4t-air inevitably escape? Is it possible to avoid this esca4t+)e jls0/kbk dvny- pe of air altogether? What is the relation to the uncertainty principle?

  It has been said that the ground-state en0hd4 zy )plmz(4s w3j0lpno/a ergy in the hydrogen atom 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 energy form? z0m/yn p04llwd3zs4(j p)hao Explain.

Which model of the hydrogen atom, the Bohr model or the quantu 5o- wlj6mswi/m-mechanical model, predicts that the electron spends more time near the nuw/6lwos-5imj cleus?

The size of atoms varies by only a factoj6z eh 5j4m3yt;dg-qmkv*tauwnqcsb, d 3-63 r of three or so, from largest to smallest, yet the number of electronsy5s zm -u*tcjt 64v33- ew;h jg k3amqd6bqd,n varies from one to over 100. Explain.

Excited hydrogen and excited helium atoms both radiateoc3mdv*4 + j*i m 0/3q: mn(nskqaozxh light as they jump down to the 3 xm:v n0mdjqconk/iq(z *as+o3*m4 h $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table ;mkv) *tl 5inyz1hh+flook if there were no electron spin but otherwise quantum mechanics were valid? Consider the first 20 h; )inmt 5kzvyfhl +1*elements or so.

  Which of the followinz *qlmb;*cy+ ficr6u1g electron 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 (car zf6a3aq b ,7fu6:e4eaz k/x(sxu*lbv eful scrutiny across the periodic table on the inside back cover will prfba4l:a ,(xuv7*6z3 bu/ex k6szeqfaovide useful hints).

In what column of the periodic table would you expect to find the atom (gdkb0xa)x y*with each of the following con* axg)y(dbkx0 figurations:
(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 exhibw3w*oqt sq4n(it similar properties?

Explain why potassiumh4a 4 qaox) :y*a4hs,t s3hqzeh( .kch and sodium exhibit similar properties.

The ionization energy for neosv.eh deu-o,qnx-*zu7u/ 6v j n $(Z=10)$ 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) impgl+;, u mycmzxo96ft0ly a photon nature for light? u,o +cz ;m9ymx0ltg6f

Why do we not expect perfect agreement between measured values of X-ray lirg khl4 q4dj9o8 s) ane)).mzlne wavelengths and those calculated using sr) z4 )q49)dl.l8 oahenk mgjBohr theory?

How would you figure out which lines in an t. )kqs2s 27ccg. oyg.revtf8 X-ray spectrum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transitions deep memh) rff .l0gn*z 41ywithin an atom to produce shorter wavelengthsgf z0y1h4.l r*fem)nm than transitions by outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from ordinary light? Hkq rm).zd7h*nlz+l; (( wrvldow is it the same?

Explain how a 0.0005-W laser beam, photographed at a opjkk(ijwm88 w: (tb4 distance, can seem much stronger than a 1000-W street ltjw wkb(i4omkj: 88p( amp at the same distance.

  Does the intensity of lighav9rqp--+-n* uiba xh t from a laser fall off as the inverse square of the distance? Exp+-ar9-*hxbq ip nvua-lain. ----待选择----yesno 

  The neutrons in a parallel beam, each having kto*t+5 sxs e6uinetic 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 wc a)) i1llnbe6ith a 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 l (qfd 6*,2j* xdij+xdtcrq)fbe 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 qkdvzwz+3vyf sn+htg-o+6kl 1x/ 82ian 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/y:xwav1:gstv1f gn 8on $(m=105.7 \, \text{MeV/c}^2)$, 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 oca v:anxh,61if a neutron contained in a typical nucleus ofa ah,:1nvix6c 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 electr;w2,t rs6qhjy ,yrs *oon were present in the nucleusyqssy w,2rr;j t*o6h, $(r \approx 10^{-15} \, \text{m})$, 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 o1 7e9nxvh/bs , )5guq1budxma $3.00$-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 cjre8/t*- w diqcl;g m7an be in the $n=6, l=3$ subshell?    elections

How many different states argsb7x k 75bydc da8)d(iet4( je possible for an electron whose principal quantum number is xs74d5 kji7)ce(a 8b(y dbtdg$n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each electron in the ground sidd qp7 v5.e hhem :*7mmunm4+tate of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each electron in s(n - e-fmm.outhe ground state of nitrogen $(Z=7)$.

  Suppose a certain hydrogen atom has5r56kul9:gs7yjjnnmv j . cd6 $l=4$. 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 electron in the lo3i 3g anc )1.+tlaug$n=4, l=3$ state of hydrogen.
L =    $\times 10^{-34}\ \text{kg} \cdot \text{m}^2/\text{s}$

  If a hydrogen atom has6ia0o5;vmh 8zg;ia + ny:u.d y:ni- 2euelvev $m_l=-3$, 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 e89cx km1-lrh:ool. l electrons in a "g" subshell.    electrons

What is the full electron configuration in the ground state for elements w7 (oyoae g-+(p8cvsldug9t 8pith $Z$ equal to (a) 27, (b) 36, (c) 38? [Hint: see the periodic table inside the back cover.]

What is the full electro1rv. x2)kl75uarf i jyn configuration for (a) selenium (Se), (b) gold (Au), (c) radium (Ra)? [Hint: see the periodic table inside th 7i1vrx25urafy. lkj)e back cover.]

  A hydrogen atom is in the ys.r heu,q.2h $6s$ 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 electrvsm rsk,mgg8 v fol/ g(::l/5bon in lithium using the Bohr theory. [Hint: This electron ha:,gmm(/s vl5:k gv/rl 8g fsobs $n=2$ and "sees" a net charge of approximately $+1e$.] The measured value is $5.36 \, \text{eV}$.    eV

Show that the total angular momentumkl1b68kptq( o is zero for a filled subshell.

  For each of the following atomic transitions, state whether the transition iswfn3tm-g v8:rd kij/8 allowed or forbidden, and if f8r83jfi tgdkw :/mv-norbidden, 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 in a/sto.yw p6t4 -id9.2 fegwafb $6d$ 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 emitted b 4v pt-dbv:- wt.lrv.wy electrons striking the face of a.4pwlvrb:-v d. tw-tv $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremsstrahlung X-rays emitted from an X-ray t+djy3)oz ygw* ube have+yz dogw)3j* y $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wavelenwmmq.n60+i8c(8h hhz/hpmljgth $\lambda_0$ 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 wavelengck, glh/rm ,fy/ 0bitilrw3 05th emitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength f/b+hlma n: z.ucy9m,uor an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the wavelen:f4plisqibyh16+ ty 7 gth 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 bomba6b+flxck igqe * +-hh.rded with electrons. The waveleqhg fe+-.b+*hc kl6ixngth 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 detachen xo+z(8 ( iazmut*l 2-cma;ntd 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 pohs-zm a5 2dys(wer 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 ngm-jcq3d pcet /;g/c*) 0owxif the beam emerges througdnc3cw *;/pm e-0 /ct)jgxgqoh 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 H(+0 mw d5:ka+5.fyh6y-bjq f ainp erwe-Ne laser?$\lambda $ =    nm

Use the uncertainty principle to estimate z-*w8d7e hb ;d gl9fyjthe position uncertainty for the electron in the ground state of the hydrogen atom. [Hint: Determine the momentum using the Bohr model of Section 27-12 and7j dw;z9b8-eyd lh *fg 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 small, .x po1qobxjb(c;gl3 est 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 formul4b8*o:vso ilct cfh 26cdh;c (a for the radius of an electron orbit, estimate the average distance i6 h (*s42tc :cfcobohl v;c8dfrom 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 operates aqc9*:ijh98 o*g7ngwr s a +lxtl4 qla7t $95 \, \text{kV}$ 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 boroj9:vizk:tq(*xpaqn e(82 ct z:,vckun 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 kj s/w cq0*3.8p 1wewos eju hph+(mf0Moseley 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 given value of ikym9scj:;oe.fk9o - $l$ 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 ew0u3 d2gcvmfu7kg6 5sqr 0:z possible for a given value of ukm7: 32v6gd0z s uw qre5g0fc$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV is shot through r6vrbh6m8u(/so(w wytwo narrow slits in a barrier. The slits are m8 rr(/wh b6w 6oysvu(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 c - i7ypovnu0(in the hydrogen atom is given both by the Bohr model and by quantum mechanics. Compare the resupv oi(c0n yu-7lts for $L = mvr$.
$L_{\text{Bohr}}$ =    $\hbar$
$L_{\text{QM}}$ =    $\hbar$ or $L_{\text{QM}}$ =    $\hbar$

  An 1100-kg car is traveling ep8zqye*k4ou 7qxr ly: u1t0/with 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 centeredfi9eco+ m)qg2 at 488 nm. Careful measurements show the line is really spread out between 487 and 489 nm. Estimate the lifetime of the excited state that proqeio9f 2mc +g)duced this line.
$\Delta t = $    $\times 10^{-14}\ \text{s}$

  Protons are accelerated from rest aii6r75vz -.ze+8k pnhcvc1 skcross 550 V. They are then directed at two slits 0.70 mm apkr ik8sv5 he1n +pzczc .-vi67art. 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 acpxcm 12nlwex e6:,h 7cross the same voltage. Assuming that the uncertainty in their position is giv7 :1nlxmcxc,p2 heew6 en by their de Broglie wavelength, find the ratio of the uncertainty in their momentum.

  If the principal quantum number n were limited to the f da1/xlz7y8b range from 1 to 6, how many el y/b azl1d87xfements would we find in nature?   

  If your de Broglie wavelength were 0.50 m, how fast would you be moving if/ uzs;p.okg5nqti*o 7 your mao75 u *og./ips zn;qktss 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 series of lines with onec3m b dbsci( 354lnb:)so asw; 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, determinec33ibss :;b(lw4mobs n)cda5 Z and the element in question. ----待选择----lithium berylliumboron 

  Photons of wavelength 0.154 nm are emitted from the surfacg030 hl4q te3cj nx0goe of a certain metal when it is bombarded wi3nt0q ogc 0glxhej30 4th 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 spread of a laser beam,dt-i1cn2hmh( $$\theta \approx \lambda/D$$ , 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.]