Compare a matter wave fsxbky e ::9q+sg oe(-$\Psi$ to (a) a wave on a string, (b) an EM wave. Discuss similarities and differences.

Explain why Bohr's theoryau f(odg)y2/ 1v oky5c of the atom is not compatible with quantum mechanics, pg ok5u /fc2vya1od)( yarticularly the uncertainty principle.

Explain why it is that the more massive an object mdrodq; oz4)jim: :a8is, the easier it becomes to predict its future positionr :o4dmo zm)8;qaid j:.

In view of the uncertainty principle, why does a baseball seemdk y ltswk:7 hn*rs6d liph.jblhd; x+7:.t:1 to have a well-defined position and speed whereas an electron do1 jk. 7krs*hdl+; ni y7.: sd hwplt:bx6l:dhtes not?

  Would it ever be possible to balance a very sharp nzh(w8c :;ra . fjobc)reedle precisely on its point?aorcfbh;.cz8 ) j(rw: Explain.

  A cold thermometer is placed in a hot bowl of soup. 61z;op auy)c2e1 2 (jasvlrwlWill the temperature reading of the thermometer be the same as the temperalyou z62pla jw;s1verc12(a)ture of the hot soup before the measurement was made? Explain.

  Does the uncertainty principle set a limit to how well you can mae,u4yac4;nl uke any single measurement ofln,4 ca uyue;4 position?

If you knew the position of an object preciselyqte2r p281e jakewk2+(w j /ro, with no uncertainty, how well would you know its momentuwkkjp ta8ee/wj(22re r12q+om?

  When you check the pressure in afhcm 4 v 6t8+;mi,f(; bx)istcbtic,c tire, doesn't some air inevitably escape? Is it possible to avoid this escap,;cti)t fc(x c +vhi,8 64cmsbbmfti;e of air altogether? What is the relation to the uncertainty principle?

  It has been said that the ground-state energy in the2.3zim rg9h ix3kg.e n hydrogen atom can be precisely known but the excited states have some uncertainty in their values igk.x329ge h i.n3zmr(an "energy width"). Is this consistent with the uncertainty principle in its energy form? Explain.

Which model of the hydrogt;jc,( 7vq 3ne1tj rocen atom, the Bohr model or the quantum-mechanical model, predicts that the electron spends more time nearrvccotje 3 j,1q t;n7( the nucleus?

The size of atoms varies by only a factor of three or so, from larges.er7rdla9d3 pt to smallest, yet the number of electr7d3a prlre9. dons varies from one to over 100. Explain.

Excited hydrogen and excited helium atoms both radiayix y: p.:*dhyte light as they jump down to : p*x:di .yyyhthe $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table look if there were no electron spin but otheru+bo fo7el vs5,e 7*ghwise quantum me + v5lh,oue7sbfog*e7chanics were valid? Consider the first 20 elements or so.

  Which of the following elecuo xsio85z 730u rh1agsiaxwd/b3y13tron 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 (careful 3/ ozsjoye(mu*2 ,iny scrutiny across the periodic table on the inside back cover*i3onu,s2mz(y / joye will provide useful hints).

In what column of the periodic table would you expect to find the r 6u5( mwmdp )fnhx*4oflj,y2 atom with each of the following cof l6ohu2f,w y(xp)m rjm*d5 4nnfigurations:
(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 similx5zqbjghtn+8j ;+6 8j cm uh--hcag8rar properties?

Explain why potassium and sodiuy 4og *ir *pi.he*:rjkm exhibit similar properties.

The ionization energy for n ,uq; zk nop ouqw//lsyr+w*+-eon $(Z=10)$ 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 au4 f1v8 :walf..q,:bfmikni*+ l vny photon nature for light? lqn,a *vb1il 4ua.kmv fwin+:. f:fy8

Why do we not expect perfect agreement between measured values of X-ray line wa e6 nmyu/f7m/r7h70art lc4gavelengths and those calculated using Bohr m67r tfuhm47ag/ /7nlayce 0rtheory?

How would you figure out which lines inzdbnj*oh() 6 ;*olg- :obep yc 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 produce shorter ( bs0*ycuurdij j8 vj2ptx -7;wavelengths than transitions by outer elecpj y2ts7u vj*0;i8rc(j-udxb trons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from ordinarzq)t:8tzbba (y light? How is it the same?

Explain how a 0.0005-W laser beam, photographed atrl b6lbfv8;heklxz2 q(m 3(j; a distance, can seem much stronger than a 1000-W b2; l(herf3 lj 6xzbl8 ;vqmk(street lamp at the same distance.

  Does the intensity of light from a laser fall off as the c 8 rkuig)86nyinverse square of the distance? Expgc6r8 uk8in )ylain. ----待选择----yesno 

  The neutrons in a parhn;fag c9 r78zallel beam, each having kinetic 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 wizdc14zymjh4qf / 6oy,th 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 accuracyt,gr)ys , cg dn,:x7tv 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 b.s( ixv: rhg8 (qm7+m xiikz: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 mn 1l+iu gd-2q(izq +fwass 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 enql/,n 041;t ;mealz nopatw(z ergy of a neutron contained in a typical nucleus of radi ot0 a,qnmlnte ;l zz/;1p(wa4us $1.0 \times 10^{-15} \, \text{m}$. [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 9 2ezweu*4j/ah xo9rlwere present in the nuc *4zjhl/2 euo 9xwrae9leus $(r \approx 10^{-15} \, \text{m})$, 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 pos , n(cgp fipj*zul15k3w3/ wtkition 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 g4djr)n:k2/go6/ong k p gze0 in the $n=6, l=3$ subshell?    elections

How many different states are possible for an electron whose06 uvq qq-l)j*wozwu; principal quantum number lvuq0w*zu;jq6o - ) qwis $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each electron in the ground s26dtzc37qi uv tate of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each electron in the ground state of rlv;v 0.zxux5y;ibj0nitrogen $(Z=7)$.

  Suppose a certain hydrogen atom hah1+p t/jz xtn7s $l=4$. 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 momer7(+zuo7ms atntum 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 haswmrvimn;g4 * r0e/8o e $m_l=-3$, 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 eleifl4u 6c.ly s 3d4a7x)u; ss9 yvpqk1gctrons in a "g" subshell.    electrons

What is the full electron cz 0su zn6ryh+v09dpynm bq1 g2v5 s9s4onfiguration 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) golds/ th;ospko/w-xn9*bs; azm;is y8l. (Au), (c) rads;y/9zstxw a/o;om*s ;khi .b s -pl8nium (Ra)? [Hint: see the periodic table inside the back cover.]

  A hydrogen atom is iplwkcvuicf- 7u8y bn, q +,h3-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 ent -mno4.8jpa: k2iah xergy of the third electron in lithium using the Bohr theory. [Hintp m.oij8htnx4:2 aak-: 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 angularbo8pxc8+h a)( gd//vvmxg 8 turfj29w momentum is zero for a filled subshell.

  For each of the following atwejk/89sh u6vo/;w py omic transitions, state whether the transition is allowed or forbidden8j6s/w 9pkv yw/; ouhe, 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 vey,xz 64c ;yy$6d$ 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 striking the ghjplb ; *a-z9omfw-2i .-xvf face of 9ax f.-vghwb2if*lo-z; - jpma $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremsstrahlung X-rays emittedf/i j/ e8waz)6 vw 4poiebe-(e from an X-ray tube have wzw(6/a48-) ij / veeeefb poi$\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff oyl)-dequ*/ o8gj qg 0wavelength $\lambda_0$ 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 a Cg7ako b5 /ktd0o $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength fw f-*6u ey5nehz, rf.zor an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the wavelength for a0ffyv2ed50taum n24f1i:b a*rxf, cdn $n=3$ 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 bomb)u2,/mkj d.: t1acdj67a bmmzsb jk9 xarded with electrons. The wavelength of th9mj /bj sz26):a mm,7ua .b jtk1cdkdxe $K_\alpha$ 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 detq8v if yiiy6hhllrx(t6 6 iv)9 f;jrl-0 ok;+zached 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.wojvc , i1ob5m0xb1,p / jxmsiiey61ed 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 diffrsmpw2 t /l+aebsbzi9r9(9/ gaaction if the beam emerges throug9i(/e bb9raslapm2s / 9t+zwgh 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 rqb9my4; f i o2n82aao*czk:z the He-Ne laser?$\lambda $ =    nm

Use the uncertainty principle to estimate the position uncmbjp0v0, ampj,b4e q5ertainty 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 doe,b0m,5 0j qb4pjmeapv s 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 fgn 4,guqat:,tl67:i cj9q l tx9 jggn6or 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, estim5l a:a++8 ;ot5 kri:e ufc*mavbr -mamate the average o5r mctvi+fr;a b-m a+km8ue* :a5l:adistance 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 atw* u 73:qizrzmu( cka2 $95 \, \text{kV}$ 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 outermostpq vlb/o2: ln ;a9dyo1 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. y s:gw9xpz7w5ki76o47tcwxc u f/*if 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 sta2(rk)oj,x f og2tn )hg yh4c*ates 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 possible for a given vau 6cr5+buz e,1 yqxvn3lue of +6, e ur53ncquxvz1yb$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with e7 1 8yh4agjruanergy 45 kV is shot through two narrow slits in a barrier. The slits are a distanch jyg8uara471 e $2.0\times10^{-6}\ \text{m}$ 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 by bh,r 4stei+4u fa/s2bcv02dc quantum mechanics. Compare s240d2bcrvb +,fcu ei4 t/hsathe 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 5tsf1 b t0au3kspeed 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 nm9zpio 9v7 9jziuh; s-t. Careful measurements show the line is really spread99 sjivhi9; u7zotzp- 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 a pd* tkfm v2(3utb7wyz.1ec0l cross 550 V. They are then directed at two slits 0.70 mm apart. How far apart will the interference peakst t ze70*3w 2pdf(cubkl1y vm. be on a screen 28 m away? $\Delta y$ =    nm

An electron and a proton, e j(xu5 4wf0cn2d/uz,ccuy( zwach initially at rest, are accelerated across the same voltage. Assuming that the uncertainty/ 0y,j f wnd4u2(xzu(5cwzccu in their position is given by their de Broglie wavelength, find the ratio of the uncertainty in their momentum.

  If the principal quantum number n were limited to the range from 1 tob-6u mu2bzf:o 6, how many elements wom2bbuu6-:fz o uld we find in nature?   

  If your de Broglie wavelength were 0iq1yxw5vtc 19.50 m, how fast would you be moving if your mas9 wqt15cyxvi1s 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 sn5k4vm ybfd;rs+)j8 f *soiy6eries 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 one electron, determine d s8r+ni*yvosk4bj5f; 6)fmy Z and the element in question. ----待选择----lithium berylliumboron 

  Photons of wavelength 0.154 nm are emitted from the surface of a7ivpbzaa k;-(keqf;nx/ ad + 3 certain metal when it is bombarded with high energa;3ika; b pz( xv/f7dk ae-n+qy 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/qk *g i 2j+7dwz-mukt 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.]