Compare a matter wavejj3xg -nu/ r1mryi2m6 $\Psi$ to (a) a wave on a string, (b) an EM wave. Discuss similarities and differences.

Explain why Bohr's theory of the atk mqox)v.hun,.ax1)7 u 3o vzt/kcx5tom is not compatible with quantum mechanics, particularly the uncertainmt zu)v uq)/7 t oo3h1a xkvk5cx.x.n,ty principle.

Explain why it is that the more massive an object is, the ea f1di2lul*/ ousier it becomes to predict itf dio*2l 1uu/ls future position.

In view of the uncertainty prinofhb p -0r)q2x8 :bvndnz rw+(ciple, why does a baseball seem to have a well-defined position and speed whereas an q+2wr)(nb hrbvf8 -n: xdp0zoelectron does not?

  Would it ever be possible to balance a very sharp needle pr 07pw 3 3t ,pckd;jze3nrst7ldecisely on its point? Expl,pd er cntt7z37d3lwjk3 ;sp 0ain.

  A cold thermometer is placed in a hot bowl of soup. Will thjg )emz+z9 *ec6bxbkm 4:vm k:e temperature reading of the thermometer bm4j xzc:e *k6bm+:kz9gm) vebe the same as the temperature of the hot soup before the measurement was made? Explain.

  Does the uncertainty principle set a limit to how well you can make lw.t 37ugk0an* pxth:any single measu: wapku0t*7 .3lnhgtx rement of position?

If you knew the position of an object precisely,1hh pq1+js*exqjf,1t with no uncertainty, how well wou1 +tjqxep 1sh*hqj1,f ld you know its momentum?

  When you check the pressure in c n5vd1hvmf*rlwa36-a tire, doesn't some air inevitably escape? Is itf1han- dv6 vm*lrw5 3c possible to avoid this escape of air altogether? What is the relation to the uncertainty principle?

  It has been said that the ground-state energy in the hydrogen atom can 3vw1h u5p(vw0obvl y:be precisely known but the excited states have some uncertainty in their values (an 0vb5vv hp uy(wo1:l 3w"energy width"). Is this consistent with the uncertainty principle in its energy form? Explain.

Which model of the hydrogen atom, the Bohr model or b l2 qjlxa.d.lyz(p(5 the quantum-mechanical model, predicts that the electron spends qylzbx 25pdll.(( .ja more time near the nucleus?

The size of atoms varies by only a factor of three or so, from t 2ltn r9u(4mxlargest to smallest, yet the number of electrons varies from one to over 100. Ert2(l4 numt x9xplain.

Excited hydrogen and excited he-fjb7ck pv2 .zlium atoms both radiate light as they jump down to th fcpz.kb2-7 vje $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table look if there were no electron c*mos0p.a;;7xkq meou uq;3 r spin but otherwise quantum mechanics wa mou ;ks m.xoe;pc0r7u ;3q*qere valid? Consider the first 20 elements or so.

  Which of the following electron c ;bs gldy+9q2lh x.u 6m77qefkonfigurations 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 uraniumgy-l d1xmluqca.- *)a2oe /ku atom (careful scrutiny across the periodic table on tha.u lq/1my*goae) - k-cl2dxue inside back cover will provide useful hints).

In what column of the periodic table would you (4tpmhvt--4oub 5 w mdexpect to find the atom with each of the following conf ot54mdu(-mwtpb 4-hvigurations:
(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 exhibk*k(t 6u/1qj np; 6n jlxs l*1o2moowkit similar properties?

Explain why potassiumx:,jq-f)ksceo+3mdh and sodium exhibit similar properties.

The ionization energyov (huoy2 x31 f/e(okh 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. Below) imply a photon nature for lightv*6b 8 dahyvshqqv/uu,, 5y b9k q g.rk9d2pd5? 6rvpkk q9b2b5udhgs8.v,/ q*5y a9y d,h dvuq

Why do we not expect perfect agreement between measured values of X-ray lf pwt:fc*2 4c,ob/qk zine wavelengths and th fzpwo4t* ,k qcfc2/b:ose calculated using Bohr theory?

How would you figure out whkem*l3.pkkl*+rx ,;m3h9;pj zkuob v ich lines in an X-ray spectrum correspond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transitions deep )of4g3owu7no within an atom to produce shorter wavelengths than transitions by outer e n)g43oufo7o wlectrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from ordina;xcql7w0bc o np,,22br+a l siry light? How is it the same?

Explain how a 0.0005-W laser beam, photographed at a distance, can seem muchlsm:s ;1j dav, ljmq)2 stronger than a 1000-W street lamp at the samem 2v)as:1d;j j,mlqsl distance.

  Does the intensity of light from a laser fall off as the inverse squ p.c bi 8r+sf.qjn lo; xdslpg ,24u)+v1i5sczare of the distance os)pqbscl dil,;.c4p5u2fzjn 1 v.is g+ +rx8? Explain. ----待选择----yesno 

  The neutrons in a parallel beam, each having kinetic energ3 cwjv6t m(,z6 y2yhrvqvn9h/y $0.40 \, \text{eV}$ 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 /ktt fn)k 8o5swith 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 be measured to an accurd 6n.prt) .flid *arlro1d0 1aacy 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 ant6 ,mrxi3r9b ;o1au pc 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 b+.-efjk-a zd st/y6na 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 po 0mef0l)api/ s6q:i ifssible energy of a neutron contained in a typical nucleus os0qi0 pfief:)m /6alif 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 were present in the nu(7eh tti65-lglw oeb26o5l -bt egiel(h2wt 7cleus $(r \approx 10^{-15} \, \text{m})$, 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 thx46 gpn6n6kao/( abyz e 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 can be in t cn000g hd 9tl7x2ccvthe $n=6, l=3$ subshell?    elections

How many different states are possible for an electron whose princips) ,aso8ml;5ccyheul/gt; p. al quantum number uy;ses)ac/,p8. og;5 l tlhmcis $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each electron intfzy50:m cn- b the ground state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each electron in the ground state of nitroie5-krrg39q i gen $(Z=7)$.

  Suppose a certain hydrogen atom hf +zdz1 ah 79eefaw(1bas $l=4$. 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 momentumipxfot q*+9:qd u:wmfw* 6b j: 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 ha*9j3o))sn wxbacty5;.c -nho rj d,ios $m_l=-3$, 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 in ap3u89kp9cri)pf-n*f1 rsx kj zmv(m;v8q ga6 "g" subshell.    electrons

What is the full electron configuration in the ground staf r0iaohw.-7xte 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),ai ;fg3sg/:yk+ x3ecz (b) gold (Au), (c) radium (Ra)? [Hint: see the periodic table 3gc+fegk/a:zyx;3i s inside the back cover.]

  A hydrogen atom is i 3x7jz y t.tyx szy2oq3i4k3j+n 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 energy of the third elecri j1r9a(q me0t,jk; lkb.s7u tron in lithium using the Bohr theory. [Hi07u,k1. s9rlj;rma tbkej (q int: 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 ur 3(jj:+c3rgtrw u 9xo/ qk.kangular momentum is zero for a filled subshell.

  For each of the following atomic transitions, state whether lp4ofx:ncj1 79s h5fnky9 a9*ys go3gthe transition is allowed or forbidden, and if foay:loos9p 7n1 3f 9gj9y4 *5fxschnkgrbidden, 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 icy* 9w/razoe 4 st2ds7n 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 shortestnamd1:48o(*ujh w i ua*rzp6v-wavelength X-rays emitted by electrons striking the face of a jdv*m6w(p ru4o:1*z8an iu h a$33.5$-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 havqprsfpgcie ;q1l() )(gas5sl9 ls 9v 5e;ls1scq e5 gv)g9 (sp59)pfl iq lsr(a $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff,ta2d7(y2a jjqu(fmn wavelength $\lambda_0$ 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 al kzw /t- 16yybt1 i0/u:xwwym0e pwk/ Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelengtdq ()jfmvlho. o+7/5 eea 0kyfh for an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estnfv ww/p -)z*qimate 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 ma7.ik) +i ci.npll,w mgterial is bombarded with electrons. The wavelength lp.kw7im.g )ic+ in,lof 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 /t:du du6p3t(v8gjkbdetached 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 dqmjidi (biq96iw;n4pther 5a 1.4) pphysics 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 a:p4pip a+tq9of a laser beam due to diffraction if the beam emerges th pip 4t9a:p+aqrough 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 He-Ne ,cluwcj s .e7zw h0 wgo80a7s8laser?$\lambda $ =    nm

Use the uncertainty principle to estimate theh+q*b9s7mqqt (rdzk 4 position uncertainty for the electron in the +tzb4shrqd m7 k(9qq*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 ttc; u,3th2 klvalues 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 radius of an electron orbit, estimat,nig hvzsnk*c4g 8)u(e the average distance from the nucleus for an electron in the gh*z,4nn(iug c ks8 v)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 atbf/ e8e s+vnv z26n9dd o.s 7wfmil7v: $95 \, \text{kV}$ 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 outen0jk6wom c pmz-7zw))/ v gw2hrmost 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.ibcsny, l.n: *bmbdv 84+wo4c 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 va m f,h2vz.q7xsf )4)pbncjvk;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 electron states possible for a given value -d0 u)qo+uuez3ukv -jof j+)vok-u u- uudq0ze 3$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons x9llkok ro:*90vt w*rwith energy 45 kV is shot through two narrow slits in a barrier. The slits are a dk* kv:rxlwr9t0 loo9*istance $2.0\times10^{-6}\ \text{m}$ 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 botv wd5.cnmhz n(b2g26 lh by the Bohr model and by quantum me. nd5hm 2z2n(g6lbcwvchanics. 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 spgxidmg4z124.ixehpvx lz8t+8t l6: beed 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 n*4 wtln:b fwk2m. Careful measurements show the line is really spread out between *klwwn:ft 4 b2487 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 directed at two s u( k--a*jeuxi )*:gyly dfah;- af3qplits 0.70 mm apart. Hoeha(qy:fgi*p-k-yd aujlfu *x a);3- w far apart will the interference peaks be on a screen 28 m away? $\Delta y$ =    nm

An electron and a proton, each initially at reov s(-posx 7uf g:7hn3st, are accelerated across the same voltage. Assuming that the uncertainty in their position is given by their de Broglie wavelength, find the ratio of the u-ofs7gpv3x7( n: sohuncertainty in their momentum.

  If the principal quantum number n were limited to the range from 1 to 6, how maa16 u9r qycvx t95z.pmny cat9m9rxu1yvp.z65 q elements would we find in nature?   

  If your de Broglie wavelength were 0.50 m,q 9xk/1pzm( wv how fast would you be moving if your mass is 75.0 kg/ pw( 9qzxmv1k? $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 oo5ao y,y/awqf2/us qhc:w)e3f lines with one out of every four matching a line from the Lyman series of hydrogen. Assuming that the unknown 5 3waoyswqf /ey,cuh:o /q2)aelement 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 n 0n 8skvzny)d9m are emitted from the surface of a certain metal when it is bombarded with hin v s)9ndzk8y0gh energy radiation. If this photon wavelength corresponds to the $$K_\alpha$$ line, what is the element? ----待选择----nickelcopperzinc 

Show that the diffra9dhdf06zagr,x 1ni )1c b i7yjctive 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.]