Compare a matter waver *gqnnxn,tss da;a(i19-r d.n 4ea:,mc z.fz $\Psi$ to (a) a wave on a string, (b) an EM wave. Discuss similarities and differences.

Explain why Bohr's theory 0akj6q sf. yh4 nm.bl8of the atom is not compatible with quantum mechanics, particularly the uncertaina.m 8 nf0bq.4sjkhyl6ty principle.

Explain why it is that the more massive an object is, the easier jnw+d(+z rq7rit becomes to predict(+q7wzn+r d jr its future position.

In view of the uncertainty principle, why does a basebalj.,c9b;vktio7kv: u ,c gu)zm 5ejm.cl seem to have a well-defined position and speek;ckzv,jo7m vg:ib5 .,)utcej c u.9 md whereas an electron does not?

  Would it ever be possible to balance a very sqfj8 pbw)qy49.vgm34 e;zsx sharp needle precisely on its point? Explais.xy)q3zp s4e ;gm9vf8b4 jwqn.

  A cold thermometer is placed in a hot bowl of soup. Will the tempmoqye yjo ;5x6bpw7y:)0t aj6 erature reading of the thermometerjm6 :)65owy xptbyqej70;y ao be the same as the temperature of the hot soup before the measurement was made? Explain.

  Does the uncertainty principle set a limit to how welznkd2qeos 8/)0jv jr: l you can make any single me:j s08/ )q2vred zjoknasurement of position?

If you knew the position of an object precisely, with no uncertainty,xq9 ikk 8 i5gkgny;)ibr y9c28 how well would yog; 8i 98kycqyik2 nb)9rigx 5ku know its momentum?

  When you check the pressure in a tire, doesn't some air inevitably esk dgcg9 tmra+rryn)5)3ec 48 vcape? Is it possible to avoid this escape of air altogether? What is the relation to the uncertainty principlvd+g4k9 crgcm3r8 ray))te 5ne?

  It has been said that the ground-state energy in t 4f1xkb/hjywk hbe. z9g n98/the hydrogen atom can be precisely known but the excited states have some uncertainty in t.1wx9 4kjkbehgzt/hy9 f8 nb/heir values (an "energy width"). Is this consistent with the uncertainty principle in its energy form? Explain.

Which model of the hydrogen ag- 4xocu;g4)zezx mb h5urh9(( xyjh5 tom, the Bohr model or the quantum-mechanical model, predicts that the electron spends more time near the nucleus4 o(h (9zyc )ug5g zub;xhehmjrx-54x ?

The size of atoms varies by only a factor of three or 7p k*j6ymuz)sso, from largest to smallest, yet the number of electrons varies from one to over 100.mz7) 6p*yuks j Explain.

Excited hydrogen and excited helium atoms both radiate lighoveu +ubm )x;3t as they jump down to tm e;x)o3bu v+uhe $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table look9bs -tf/5zu eb if there were no electron spin but otherwise quantum mechanics were vali5 9-z/ beubstfd? Consider the first 20 elements or so.

  Which of the followi yeqy.2ns: z;zng 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 jghwk+(vq2sg)fq9 -5 r blw)efor a uranium atom (careful scrutiny across the periodic table on the inside back cover will proviewl )gbqsg25(w9v h-q +jkf)rde useful hints).

In what column of the periodic table wqe4r)zfx;iz 2 e8v b5fould you expect to find the atom with each of the following config );eq4f vf5r8 ibzzex2urations:
(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 sik-x7qojt4ck+ milar properties?

Explain why potassiumxmh+o+w*gr c 1um/f r/ and sodium exhibit similar properties.

The ionization energy for negdttcy m5.x7/on $(Z=10)$ is 21.6 eV, and that for sodium $(Z=11)$ is 5.1 eV. Explain the large difference.

Why does the cutoff wavelength l x93e:29afvonoy.2bnvi 5a r in (Fig. Below) imply a photon nature for light? vax3 n2 2lv:oyn5 oa eb9i9r.f

Why do we not expect perfect agreement between measured values of X-ray linem v4ykix60o 3p wavelengths and those calculmypk4v 6xoi 30ated using Bohr theory?

How would you figure ohpj3q/sz g ;r+wf0s-7iv a0ljut 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 atom to pro7sp1iygkj7 :rg az 69nduce shorter wavelengths than1p6yg7s 7g j :iak9rnz transitions by outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from ordinary light? How is it the samgdy+ys2 o1r5lx / 7fg7;xzgtf e?

Explain how a 0.0005-W iz77wn,s4- xjh 9exu rlaser beam, photographed at a distance, can seem much stronger than a 1000-W street lamp at the i7ehrjux 7wz94,s- n xsame distance.

  Does the intensity ow0ao*- sk( rj6t yzp,ff light from a laser fall off as the inverse square of the dis-yt,( fs0wjrk6 pzoa*tance? Explain. ----待选择----yesno 

  The neutrons in a parallel beam, each having kinetic ene vadqidwhzg,2m dg/n5 udhb5(pf,7* 2 rgy $0.40 \, \text{eV}$ 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 of /c (l8wb,aeqf $(6.560 \pm 0.012) \times 10^3 \, \text{m/s}$. 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 measured to anj b,yax *ck+/hjp63 lt 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 22d s7 1gabp9vcggc6xexcited state 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 uncertainty in the mass of xj6jxbk y : +8xz;wot(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 con,xn crz(6d7sw9rmhv 6 tained in a typical nucleus of radi 6v ,9mnwr7d6cz rs(hxus $1.0 \times 10^{-15} \, \text{m}$. [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 wecuv2k no4 u-/rre present in the nuclek/uun vo2c4r -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 thehr.l6vqirld3p vdor6 ss:8 grl ,b50) position 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 ca ,ke0oab5q yq7n be in the $n=6, l=3$ subshell?    elections

How many different states are possible for an elgkbq, uz6za,c. ld w77ectron whose principal quantum number is zk,,c7 d7.ulz6w bg qa$n=4$? Write down the quantum numbers for each state.

List the quantum numbers f:v /f*3zr6rndo.qziiq 5 gc7 fgaq9ej ve//-for each electron in the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum nummp5,ag2 kwc4,oj .rg lbers for each electron in the ground state of nitrogen $(Z=7)$.

  Suppose a certain hydrogen atom has (z9cc93 qpb ,wu6g gdh$l=4$. 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 angw ;z7a 8fsbkr*ular 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 i,* idgl1h uyp70 -ndr$m_l=-3$, 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 electrons9 qm5io9gzj a9f8qv,x in a "g" subshell.    electrons

What is the full electron configuration 8 nu9da+ ;.z pyikeyl9in 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) sele0sn/:q o5h8lol8nsf la32il m nium (Se), (b) gold (Au), (c) radium (Ra)? [Hint: s q5ilnln of80ma3s8/ls:l2oh ee the periodic table inside the back cover.]

  A hydrogen atom is in tuhamyt6+t6 d-he $6s$ 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 thnf: 1.omjuvj :e third electron in lithium using the Bohr theory. [Hint: Thijf:1jv. nom u:s 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 fow6kg l:p -,ustuk.84btwhvt3umh 50 b r a filled subshell.

  For each of the following atomic transitions, state whether the transition is v5 z 2+rr9ns 2kfhqyg-allowed or forbidden, and sn 9q2 gfrhrv52y+-kz 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 iss 4pjfd (0rl.k in 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 stri jl7jm8vni4tz fi2t9 zo-) gx)king the face ofx7 9vt)8olj )tz2mni-z4 ij gf a $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremsstrahlung X-rays emitted from kcly;sj l6wzsyo,(+-an X-ray tube have c,+ zwyks-y(j ;sll6o $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wavec5y ,f;s i4hsklength $\lambda_0$ 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 emittepwhe,v3 1pn (n5* hpvzd when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelenz5uhasu 25hr ju:n.jkt10ea)gth for an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to esti.m8zfjf3 vq 8xmate 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. The wavw-0j;j7 m q( whvwr5sz)hexu,;g7 lspelength of gr7;zl)sshe7mw;-w v jw(jx0 h5,u pqthe $K_\alpha$ 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 det+o,2( s obu6tb-qpnopached 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 us/i ys(x1 l0 6lhoddk,*zmgg):hyc k8j9rt1 beed in a physics 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 the beam ewq j3+xrag(c/k:b,z n merges thawb,+ 3gr/x(njz:k qc rough 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 r+jej+ ;z;;wrm l tp6pHe-Ne laser?$\lambda $ =    nm

Use the uncertainty principle to estimate the posikix*i :n k(l:ytion uncertainty for the electron ii:i l(: *nykxkn 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 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;hqdwv( td;-4z kii;fja g-d, 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 2g0ft( ii svz+2*7 apxfn0 pbworbit, estimate the average distance frox *ag+z(7f2 2fiv pwpi0 bnts0m 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 av- c b8ucesyofh/-1p;5 8ytf mt $95 \, \text{kV}$ 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 lw 1i7 oo ;4riem:*pe:qizxt,f u88jrelectron 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. Below) can be writtenk/2fkot f/ ;huuu:3eb $\sqrt{\frac{1}{\lambda}} = a(Z - b)$, where $b \approx 1$, and evaluate $a$. a =    $\ \text{m}^{-\frac{1}{2}}$

(a) Show that the number of different states for a given va5e-oioa jstpyqn m: enk+102 rn;qa/:lue 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 electron1v gh2 coh 1fyb04,gcp states possible for a given value of 02hpf41o1 yc cvhg g,b$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV is shot through two narrow slits in)njxhq3m8 p1 g a barriq8mn )3 hpg1xjer. 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 atom is given both by the Bohr model and bnqq8,m5 bnr,m jsl65k y quantum mechanics. Compr,bnn ,8lm5s5 q6qm kjare 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 spej0nphein b:( ) m 7nc:,a*awmeed 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 waveleng.9 , m.obf0zzdcysh/0h 1tchbth centered at 488 nm. Careful measurements show the tzhc0y0.1 fh d9ozb,c/m h.sbline 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 from rest across 550 V. They are then dires x m6g -uema/,y6fb*( wqgn.lcted at two slits 0.70 mm apart. How far apart(lx ef.a,6 n m/g6mws-u yqgb* will the interference peaks be on a screen 28 m away? $\Delta y$ =    nm

An electron and a proton, each initially at reqv/u-79t twve 5by -k.s 4zqhbst, are accelerated across the same voltage. Assuming s z7b9/w.vy- h qu5-q vktt4bethat the 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 61bwz of;x/,rc9ih dwn were limited to the range from 1 to 6, how many elements w9xrwfiz/61, c; bhd owould we find in nature?   

  If your de Broglie wavelength were 0.50 m, how fast would you be moving i5zzu54b)dx. 8x na)d -um4pa 1tsa1 gdxan uj/f your mass isaaz bjanp x1.d/)a udg 51t un4s z)xx54u-8dm 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 jl9jy4nb5bg(j z 3 z3lseries 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 one3j(4y 9 5 zjnb3bglljz electron, determine Z and the element in question. ----待选择----lithium berylliumboron 

  Photons of wavelength 0.154 nm are emitted from the survf/6jeiyy:okp 8cx ;3face of a certain metal when it is bombarded with high energy radiation. If this photon wavelength corresponds to thep:;yvyox/63cf 8eijk $$K_\alpha$$ line, what is the element? ----待选择----nickelcopperzinc 

Show that the diffractive spread of ahzqg(ig. + cc*28k58muajrz w 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.]