Compare a matter wave7 q s*/)1u,xhbpbqw8 kvnx4x p63xdaj $\Psi$ 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 mechanics,1/gd/ 7bj7r recofdc , particrc77o dgr,c1e/d /fbjularly the uncertainty principle.

Explain why it is that the more masswzv-+ .t tuc ys63gc.eive an object is, the easier it becomes to predict its future tt.+ev- zw6 3ugcs .ycposition.

In view of the uncertainty principle, why does a baseball seem t/qap;mo+ad0y j qn25zo have a well-defined position and speed whereas an ele jaqm0n/a do+pzqy;52 ctron does not?

  Would it ever be possible to balance a very sharp neeq 5te gs gksw64(kp)hf+2qbg3 dle precisely on its point? Ex k53gshget) g fqqs4k w2b(+p6plain.

  A cold thermometer is placed in a hot bowl of soup. Will the temperatuu/w b7eg/1x5+n y amrcre reading of the thermometer be the same as the temperature of the hot soup before the measurement was made? E/5w/7y+ enr1caxg bmuxplain.

  Does the uncertainty principle set a limit j*) wcfqnfz8;to how well you can make any single measurement of positic;)n *f 8zqjwfon?

If you knew the posituw3imf6hm,g) s9p ug/ion of an object precisely, with no uncertainty, how well would you 36,p h9gumfiwsgm/u ) know its momentum?

  When you check the pressure i*o3d um.npuh32f 8t :8v rfczun a tire, doesn't some air inevitably escape? Is it possible to avoid this escape of air altogether? What is the relation to the uncertainty *8 o3c8:3rvfft.dzuh2pumun principle?

  It has been said that the ground-state energy in the hydrogen atom canra ,r2cpf5j)qm+(l f d be precisely known but the excited states have some uncertainty in their values (an "energy width"). Is this consistent with the unceaqm,r( clf j5 )fdp+2rrtainty principle in its energy form? Explain.

Which model of the hydrogen atom, the Bohr 3h0dl2xkr7 ld tyh)e(model or the quantum-mechanical model, predicts that l0 d(xyt3)r l hke27hdthe electron spends more time near the nucleus?

The size of atoms varies by only a fa2) meqqn ui(a)uk m6w9ctor of three or so, from largest to smallest, yet the number of electrons varieawqm2q u kn)6mu9)ie (s from one to over 100. Explain.

Excited hydrogen and excited helium atoms both(iq:( ku;- ogvcer+yo radiate light as they jump down toikg-yo+ qo(u e:cr( v; 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 spj-)m bpf 5 h/gxq)swx*in but otherwise quantum mechanics wer h-j)m)*q bxxgw 5ps/fe valid? Consider the first 20 elements or so.

  Which of the following electron configurations are not allowed: nxy /x4r0-f et
(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 con+szoxjzpbz6;7b 9g* ku m,) /c0iz aebfiguration for a uranium atom (careful scrutiny across the periodic table on the inside back bm)zi/, cz b7;*0ekjoxsga+pu6z9 zb cover will provide useful hints).

In what column of the periodic table would you expect to find the atom *):3gsveg b*8ors q towith each of the following c3*g)o *ve:ts8boq sgronfigurations:
(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 similar properties?95kai zf lgob7 3k+bj*

Explain why potassium and sodium exhib 6*0 mdzjqf,zi,n/z xfit similar properties.

The ionization energy for neon: vxz;5kmy9/igievt * 2opm+f $(Z=10)$ 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) imply9k *gd 0il;vrh a photon nature for light? ;vdr9i*g klh 0

Why do we not expect perf/qn pp 8buxl+ev ,h:9hect agreement between measured values of X-ray line waveleupv,p e/bhhq: 9lxn+8ngths and those calculated using Bohr theory?

How would you figure out which lines in an X-ras3 tuoxqj0 y:pec /7c v7v3;n oy8q0hcy spectrum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transitions deep within an/db 8l0k c/wey s1my-zndu+3 v;qb d:c atom to produce shorter wavelengths than t+e8mln/ sk 13dw-;ycd/: bc zqybdv0uransitions by outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light diffqp9l1 udzcmwn-- ;9v qer from ordinary light? How is it the same?

Explain how a 0.0005-W laser beam, photograd7ehz5lf8er/ 4ip 4 prphed at a distance, can seem much stronger than a 1000-W sph 45/pr ze 87lidfer4treet lamp at the same distance.

  Does the intensity of light from a laser fall off as the inverse sye;x3s)+ nbigquare of the distan;3)n+ siex ygbce? Explain. ----待选择----yesno 

  The neutrons in a parallel beam, each having o m;fwq;g bwh58h zk0bn.):ddkinetic 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 speed *v p,hhvax+(.cmnc: v 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 cavq)8 qk83co7z fm,oor m5e5glo2x , ixn 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 stateh7+t.s 8jomvjl+uh;2or u. ddmo r9m, of 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 uncertaixfo4er5*jkyg z,qm26 nty in the mass of a 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 pnu8:glxpq.m ,e*i /hcontained in a typical nucleus of xghu /p *l,m.np:8eqiradius $1.0 \times 10^{-15} \, \text{m}$. [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 wqw7ro *1ing(wv0z iw)ere present in the nucle*1)w7 wizi(ngorv qw0us $(r \approx 10^{-15} \, \text{m})$, 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 positi 5incza mvrx-7ekw ;*5y 1p7kjon 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 cangqine.qx( 9 k5wwb8x-*1mzlm be in the $n=6, l=3$ subshell?    elections

How many different states ei 34b+tl46 yobfch-kare possible for an electron whose principal quantum number is fyek4- 6ho3lc+4bib t$n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each electron 3/m+mp vmk:h,95erqsn+ qa qbin the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each electron in-k au/+bingsg ip;)(8qnkl3dkee); b the ground state of nitrogen $(Z=7)$.

  Suppose a certain hydrooo2s(oxu y-nh cign)66i 0 v(agen 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 /ombko )yutrfl8 /rd0t(f (3p 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 has j4 mx)(i -gkptgt*w7b mu 8z0y$m_l=-3$, 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 electronsr;* vfk6ag;sv qn;+ tw in a "g" subshell.    electrons

What is the full elect2u8rnyrp*ic /ron configuration 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) goldjl u-dchn7 0pv* fkf/cr kz553 (Au), (c) radium (Ra)? [Hint: see the periodic table inside cuz55krdfl 3/pk *v70 j-hncfthe back cover.]

  A hydrogen atom is in 927;bf1ok5s m zluuuv8rgye/ 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 en+rr6e0: gsiwq -u g5qm4/zpmt ergy of the third electron in lithium using the Bohr theory. [Hint: This elecgmp5 q:0rrz4i tsu-6e /w+mqg tron has $n=2$ and "sees" a net charge of approximately $+1e$.] The measured value is $5.36 \, \text{eV}$.    eV

Show that the total ang1 i*rva)zg:;x;5 jz om9bmitxular momentum is zero for a filled subshell.

  For each of the following atomic transitions, state whether the transiti(vsrpw 2elv3wzgh 23b: lyz) +on is allowed or forbidden, and if forbidden, w+3s zpy 32):2wwvlbrz(eg lvh hat 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 in8-nvo 1qp0 cx;cv lug9 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-rays emitted by electrons striki93y ,x,yjut qigb) ag-ng the face of ai9y,uta,) jbx gyqg 3- $33.5$-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 n3d8nn o,pt1dyy/i- whave1 /ny-tyd,n3pdn8iwo $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wavel rum053:(n/ pzrb88oak io) pmkeyi5yength $\lambda_0$ 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 when aou 4z-ay 3cju4 em(fr- Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelengyyc8 hc;c4+lj th for an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimf8z**xx ggd3xq 7f 1lwate 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 is bombarded with electrons. Tr7sl:w25d4c pa*e cn qhe wavelength of thd72pcn5 4r*:q ecla swe $K_\alpha$ 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 out ;7h sze7fah:g-l s; yr$28$-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 hayx r6uf9eg-(iai(v e*ve 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 spreoq()- ayq5 5rttzd v,ib:o/jvn5f4n xad of a laser beam due to diffraction if the beam emerges through a n5 bvxqodryi(5tf a t/,)z4v- :qo nj5$3.0$-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 wavelengthc jx6: cprsr:jf1+m :z of the He-Ne laser?$\lambda $ =    nm

Use the uncertainty principle to estimate the posit5j x3d9svejv-j /4gkgl6xye /ion uncertainty for the electron in the ground state of the hydrogen atom. [Hint: Determine the msve5yj k-gj 3xx69e4 jl v/d/gomentum 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 possible values for the angular momentum y;e gb kzkwlo14sc/+ 7$L$ 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 thlyi q7l.-.vdgq6 :p tfjr4g5ie radius of an electron orbit, estimate the average distance from the npvgjirgdtli fq 4:7.56lq.y-ucleus 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 at x)up is nn/v.wba6)zv 9p3ph/ $95 \, \text{kV}$ 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 boron is 27nw,wz 4xsr /(w(vnww,mz 8us(vxyv2ail -c $8.26 \, \text{eV}$.
(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 Moselepq)c edd))(t(*eqrp kqxd4d) y 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 valu0vsa d ciz/y 6to)c b.i5/mmm*e 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 stateikp38m3ru 7hqs possible for a given value of 3p qkrh78i 3um$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV is shot through two narrow slits 74h7ldc1re2zf fs:q e in a barrierfz cl d2e74 1sh:r7qfe. The slits 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 hydrogen tsdy:x bmwmb: 6i,gn8-j-/ geo9yl )o atom is given both by the Bohr model and by quantum mec:j,9iw:gen/xy- dbm-)bmol 6y stg o8hanics. 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 speed o,)(kkcidxfhk t /x/r v h;)9luf $(22 \pm 0.22)\ \text{m/s}$. With what maximum accuracy can its position be determined? $\Delta x =$    $\times 10^{-37}\ \text{m}$

  An atomic spectrum contains a line with a waveleng2 *hnujt2 2rsrth centered 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 produced thi*trus2r 2j 2hns line.
$\Delta t = $    $\times 10^{-14}\ \text{s}$

  Protons are accelerated from rest across 550 V. They are then w9ero a: 9i:.:hna byqdirected at two slits 0.70 mm apart. How far apart will the intr:a9qa9: i:hey nob.w erference peaks be on a screen 28 m away? $\Delta y$ =    nm

An electron and a proton, each in+.eitx 3j-s 3-pqqzcwitially at rest, are accelerated across the same voltage. Assuming that the uncertainty in their position is given by their de Broglie wav -zsqptq33ei.-wc +xjelength, find the ratio of the uncertainty in their momentum.

  If the principal quantu k d(16ma9j 8xaswutd5m number n were limited to the range from 1 to 6, how many elements would we find in n9sau5amdwj8kt1d( 6xature?   

  If your de Broglie wavelength were 0.50 m, ho-bll)is h+7q cbk1el24 :y wxzw fast would you be moving if your mass is-lekz+1qh i b 7y2wx4cllb :)s 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 g) jj.qffl tlul9*f/ah52z5pone out of every four matching a line from the Lyman series of hydrogen. Assuming thf*fq .jzah t5l2g9jl5flu)p/ at the unknown 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 us36f4a x m3wqi w5mp(surface of a certain metal when it is bom3 pw5uxq63sam4i (wf mbarded with high energy radiation. If this photon wavelength corresponds to the $$K_\alpha$$ line, what is the element? ----待选择----nickelcopperzinc 

Show that the diffrac7 r/( tcu/0hdyuw11t,mil ih rtive 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.]