Compare a matter wave ;okjmssby2 :8f wv+9x$\Psi$ 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 mechanj5r; 23w4syk 5eak)d 7 xe8+vhp zmzrtics, particud+ m4 eatz );krrkje sxw32y8h5v5 p7zlarly the uncertainty principle.

Explain why it is that the more massive an object is, the easd2b z wdpq8- n96ej v2h,(lu0oj2jresid-wj4ier it becomes to predict its future-z, wu bw-8 j2irp 2l9s0j jdqv2dh(4ojed6en position.

In view of the uncertainty prin9ilmx c m:2nchpbp05o j 9eu5/ciple, why does a baseball seem to have a well-defined position and spee9jmmhpc9/5cb2leo0 u5xni :p d whereas an electron does not?

  Would it ever be possible to balance a very sharp05nk0r (e:ngw eq8wrq needle precisely on its point? Explain.rk0( qeqr: nnw8ew 0g5

  A cold thermometer is placed in a hot bowl of soup. Willy.e v3kgfp t b7gma1pdb81q 99 the temperature reading of the thermometer be the same as the temperature of the hot soup before 9.8 p7g9v fmbaydpek1 3tq1bgthe measurement was made? Explain.

  Does the uncertainty principle set a limit to how well you can make any sinvu1hut:41gk7es6 uregle measurement of pos 7 v6uheu g1tu:e4r1skition?

If you knew the position of an object preciselxkrvi6b x)(iv ,llx.-y, with no uncertainty, how well would you know ivri,.k(6xlx-lvb x)i ts momentum?

  When you check the pressure in a tire, doesn't some air inevitablytvwrtq.di lh yeo*w.13 i249 c escape? Is it possible to avoid this escape of air altogether? What is the relation to the uncertainty pqivwt *rhdi2c4e tyo wl 13.9.rinciple?

  It has been said that the ground-state energy in the hydrogen atof2zzb8vpw s2* m can be precis2w szv8f bp2*zely 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? Explain.

Which model of the hydrogen atom, the Bohr model orb6yxc) h7r+ iu the quantum-mechanical model, predicts that the electrox cu y+)7irb6hn spends more time near the nucleus?

The size of atoms varies by only a factor of three or so, from largest t w95 3i.ft/1ubyfa iico smallest, yet the number of electrons varies from one to over 100. Explain tf5ai9bicy 31w.if/u.

Excited hydrogen and excited helium atoms both rad/vn517q 5 gi vz3tpqyvzes6c-iate light as they jump down to vq5zszey 37vn-/qtp5c6g 1 ivthe $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table look if there were no electron spu1j ksh5cxf3 3in but otherwise quantum mechanics were 3 51chksu3jxf valid? Consider the first 20 elements or so.

  Which of the following electron configc31h8fl,tqwik:6y ew (as9qgurations 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 scruehenob -no)h)q + (q1mtiny across the peribe oeh m 1h)qn+o(qn)-odic table on the inside back cover will provide useful hints).

In what column of the periaa2 dk: i qmvy1 ,i umrz35g5/kep1(mdodic table would you expect to find the atom with each of the fdka 3,uie 5km 2iap vm5dqm11zy:( /grollowing 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 ioditsn4 un/0kwr;s t6kvdb2x* )zne exhibit similar properties?

Explain why potassium and sodium exhibit similar propertie7;4 xczi,lv jvnsp *;ls.

The ionization energy d2lmawfx;ew4st er/ 9jyf4r 2 kvp84; 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. Beq dsbgyx p,lb5, g30q9wc/ v;ulow) imply a photon nature for light? g9bw3lv,/g d q, u0pq 5xy;scb

Why do we not expect perfect agreement between measured values of X-,(rr /x2ms2cvfe b-x pray line wavelengths and cx 2b- ,fv(re2/xspmrthose calculated using Bohr theory?

How would you figure+trj4l6n5ayv out which 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 av)h5*z+i 1k qla:2dcvg /gjh wtom to produce shorter wavelengths than transitions by outer electrongh 1 ih vc5l/2a:qv*)g jzw+dks?

Compare spontaneous emission to stimulated emission.

How does laser light differ from ordinary light? How isaqtxx6n 7):hz)ba 5 ho it the same?

Explain how a 0.0005-W laser beam, photographed af7;inrn:y-hzbtvfeqs *h(h +7*o a +it a distance, can seem much stronger than a 1000-W street lamp at the same distaz*7f-7;sbq+: hvn(io yhr h*+ifen tance.

  Does the intensity of light from a laser fall v .qlo +m c:*6n,fbnn4o7l)sfkh,vgwgpfe/4off as the inverse square of the distance?p46heffnn,mgs vvln7b w+ *l .c/go:f,qk) o4 Explain. ----待选择----yesno 

  The neutrons in a parallel beam, each having kinetic enerw7ec/grwv fu8yifn3. 0,ees0b8n c;ngy $0.40 \, \text{eV}$ 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 speed ofr1vk6g d7yg )f $(6.560 \pm 0.012) \times 10^3 \, \text{m/s}$. 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 meas.zgu-a ,jv7cw:ge0crured 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 typically x*)ail i rqpba-*:(ap$10^{-8} \, \text{s}$. 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 92n2mm.jxj rea 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 neutron con(-lrs)k8 div . vs:nuwtained in a typical nucleus ) w(v:8-.vusnsrid kl of 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 showh a y 2owg *bo9l -y)nads0x;k:,udvk- that if an electron were present in the nuclekwy;o- yd,na b:d v-x0 a9sl2k*oh)ugus $(r \approx 10^{-15} \, \text{m})$, 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 posl7pu5 8th egm;ition 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 can be i t6xtd6j ,)7:flrlmq in the $n=6, l=3$ subshell?    elections

How many different states are possible for an electron whosb,agc:v/h 4h1m,t gybw + uy1pe principal quantum number is yhay1g:1b gp vc4+mtwub,,/ h$n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each electron isrv52oixh+6 er+*5e j a ;3o2fl:4trmj fkn yon the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numb1,zx :pl ws*0wt )negx:symc; ers for each electron in the ground state of nitrogen $(Z=7)$.

  Suppose a certain hydrogen atom habi4m/4kmd q56n uwcn.s $l=4$. 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 eb/x6pd am6g0 plectron 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 has6.5tdtm0 ek 8o v aik7)ny4elwvj5o 7l $m_l=-3$, 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 ele mx-2,v lklk5bctrons in a "g" subshell.    electrons

What is the full electron configuration in the ground state gy )mb7c; krilnerq(98 y6 0ltfor 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 config2 ekza 7- avt-ae6vbb2ea 9sl5uration for (a) selenium (Se), (b) gold (Au), (c) radium (Ra)? [Hint: see the periodic table inside the back cover-ate zvlab62s5k- b927v aaee.]

  A hydrogen atom is in the vh-l1.e5 vaht3 tgk12m h;imx$6s$ 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 b t;73 4x8q fw9n;xk5mluhlrwof the third electron in lithium using the Bohr theory. [Hint: This electrok3 5;wht748lwfx;runl9 b q mxn has $n=2$ and "sees" a net charge of approximately $+1e$.] The measured value is $5.36 \, \text{eV}$.    eV

Show that the total bqzwm. t 024 clm7jt0cangular momentum is zero for a filled subshell.

  For each of the following atomic transitions, state whethe2b n :um-mqi 8;+dmnufr the transition is alluumn-+8d i ;2 bfmqm:nowed or 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 in a 2bnpskgm4tkm buzv+p2a: 9 .0$6d$ 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 by electrons str lc7, tl9-:9zzv wotfplj9yh,iking the face of ,l f79ovy-jl hpwt,99 clzzt: a $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremsstrahlung tz dk76yt uu6,X-rays emitted from an X-ray tube have td67u6 ,zk ytu$\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wavelengtgs: dqk m4 5tc0lzf4c4h $\lambda_0$ 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 wavele qs/jk kl2(nda m1cx,8ngth emitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength for agik3zr qe3q //n $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory togck4cj 6d+ x9i2 *mtpu estimate the wavelength 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 isck n5d0: 0k1fjlqun*d 9sc 6pe bombarded with electrons. The wavelength of 6n0k e1cdnu lcs0jp9:q f5* kdthe $K_\alpha$ 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 ou7p d3 xik:3vout $28$-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 migwt()t. vexdw;to/di1 ht 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 difsisj3jffekm./6 : x.hfraction if the beam emerges thro jm.k3/jfssx:i.6 hefugh 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 th2 j8h(,; rx3pa4biw 8tblft uje He-Ne laser?$\lambda $ =    nm

Use the uncertainty principle to 6:qx2 ngqtku7estimate the 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 and assume the momentum can be anywhere between this value and zero.] How does tk72x:6n guq tqhis 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 5ex yp5 h-m32x ipto8nsmallest 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 rkkk29u- 94ocwceu9wcnw 2 zi/adius of an electron orbit, estimate the average deni99 occ-4zu/k2 kww9 2uk cwistance 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 operates uh14t.69efvk,yss e hat $95 \, \text{kV}$ 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 thm2izx 3lry-bw. +3iuje 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 Moseley plot(Fig. Bevupwnrsb+3-t f:n3)ua: i+ hblow) 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 oc9kq+1zdau lb/ao 9 s)b)b0pdifferent 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 of different electron states possiblj) wmc; db g 9w6+brc9zs4fkfe; )f4wme for a given value of )w kmcwr4j gbf9e m; sff6+4cwb9d;)z$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV is shot 20-n h6lqdr yhthrough two narrow slits in a barrier. The slits hdq26n0l-r yhare 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 hydrogen at 0b b0ywjua:3.ajod 2qom is given both by the Bohr model and by quantum mechanics. Co djboj0wy2:0a.3qbua mpare the results for $L = mvr$.
$L_{\text{Bohr}}$ =    $\hbar$
$L_{\text{QM}}$ =    $\hbar$ or $L_{\text{QM}}$ =    $\hbar$

  An 1100-kg car is traveling witgu seyn e7jkno9.:4xy b.mg*2 h 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 0)++jp yo.id jny5hxuwavelength centered at 488 nm. Careful measurements show the line is really spreo+dhp+50xj .jn )iyu yad 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. They are thenrr88a x0cb8e/-zudhi directed at two slits 0.70 mm apart. How x ude irz-bh/88a 8r0cfar 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 sam/y8 7scywb6vl e voltage. Assuming that the uncertainty in their position is given by their de Broglie wavelength, find the ratio of the uncertainty in their momentum.v6ycsl y/bw8 7

  If the principal quantum number n were limited to the range from 1 to 6,*e d 3dbvrz7dph/(2cn how many elements pcz d d/3vh7(rbden*2 would we find in nature?   

  If your de Broglie wavelength were 0.50 m, how fast would you 3ud9 +g lbc2fjbe moving if your mass is 75.b+f j2gd3 u9cl0 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 un o mdpqm90y0veou:-ow l0 6h:nknown element shows a series of lines with one out of every four matching a line from the Lyman series of hydrogen. Assuming that the unknowoy:0v9 h0do-q:m olpe6m n uw0n element is an ion with Z protons and one electron, determine Z and the element in question. ----待选择----lithium berylliumboron 

  Photons of wavelength 0.154 nm are emitted from the surface of a certain metaa* lyyz/8mtb nsh(r:5l when it is bombarded with high energy radiation. If this photon wavelz:tr*mh 5a(yb/8 nyls ength corresponds to the $$K_\alpha$$ line, what is the element? ----待选择----nickelcopperzinc 

Show that the diffractive sprrqq.7lqh : *u ,dl3sfyead 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.]