Compare a matter wav),wr2ar7 scsee $\Psi$ 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 coztp6g8 q3rxb wh/ig a/2 79vwdmpatible with quantum mechanics, particzi237wvaxd9 /6tw p ghgqbr8/ularly the uncertainty principle.

Explain why it is that the more massive an objectb)1y7:t h c/x2rnr b3z ubw*xv is, the easier it becomes to predict its future positi y/zcrhtun bxb* v)2 :3wbx71ron.

In view of the uncertainty principle, why does a baseball seem to hav1i; )pmz9noz ie a well-defined zi;i)oz19 n pmposition and speed whereas an electron does not?

  Would it ever be possd+vkl m(2qkhizboo:ho- 4 o3:q;v .t pible to balance a very sharp needle precisely on its point? b m;:do+ho3(iht. 4:qvzoov k2klp q- Explain.

  A cold thermometer is plac-m (dbi-cr(nowb ) d/red in a hot bowl of soup. Will the temperature reading of the thermometer be the same as the temperature of the hot soup before the measurenrobdmr- d(c )wi/b -(ment was made? Explain.

  Does the uncertainty principle set a limit to how well -dcp -qxw:o a8you can make any single meq dcw8px-a-:oasurement of position?

If you knew the position of an 3uajyx57) frm.ttike8 xq /: xobject precisely, with no uncertainty, how well would you know it:i7u 3m y)8qkfa ex.xxttj/r5s momentum?

  When you check the pressure in isl/ayvx/:2kqc 04z ok /pe*xa tire, doesn't some air inevitably escape? Is it possible to avoid this escape of air altogether? What is the relation to the uncer/c:ps2e l0ao q z4v*/iyk/xx ktainty principle?

  It has been said that the ground-state energy in the hydrogen atom jh.wzi)e2b6-u .zql x can be precisely known but the excited states have some uncertainty in their values).jz .6-b zuhqe2xwl i (an "energy width"). Is this consistent with the uncertainty principle in its energy form? Explain.

Which model of the hydrogen atom, the Bohr model or the quantum-mechanical m/fg.+ / xeaellodel, predicts that the electron spends more time near the /ge+laxlef./ nucleus?

The size of atoms varies by only a factor of threelb iwyqb7t-, , jqx2e( or so, from largest to smallest, yet the n,we7lx-q qib2tj, yb(umber of electrons varies from one to over 100. Explain.

Excited hydrogen and excited helium atoms both radiate d g+9/ gzumqy-mpj,,plight as they jump down to th p- dpgjm,q +u9ygz,m/e $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic tablex3rftv8nnb/wd+s;d 4 look if there were no electron spin but otherwise quantum mechanics were valid? Considen3r st4;fd/db n8vx+wr the first 20 elements or so.

  Which of the followin;u8u.1/r0 mmymk wmjgg 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 configuratis7bh+jq wcv +.f .dc4non for a uranium atom (careful scrutiny across the w 7h q.fcn+4sbc vj.d+periodic table on the inside back cover will provide useful hints).

In what column of the periodic tablk(o h:de:, xffd35ymlc*p;s de would you expect to find the atom with each of the following:ps(3*f; o f,y5d:hc l dxmkde 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 iodine exhibit similar propert).(2y1pq* t8 ev btdw+uqusjq ies?

Explain why potassium and sodium exhibia8v8x:iw3 ,6x pi qliit similar properties.

The ionization energym5dw2 t+ ow u:k:ib7u*5k:zn w58dwnjf,ytw k 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. Beloc )8pduzyl42 xncz3r.w) imply a photon nature for light? 2)rzlc.z ux4yp3ndc8

Why do we not expect perfect ag xx3f.yj*p4nzw iuzwbk d.jg1g,2 d +i (eb4h9reement between measured values of X-ray line wavelw3d .iwi9g4db4yeg hj.np1fj(z+ x u*k2bzx,engths and those calculated using Bohr theory?

How would you figure out which lines in an X-ray spectrum corre zif(+jj th;j,spond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transit hq 7l:epzp0w7ions deep within an atom to produce shorter wavelengths tleh:p7z wq0 p7han transitions by outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from ordinary lig9 wpba)m5-vg*0 evbmy ht? How is it the same?

Explain how a 0.0005-W laser beam5: xlu 6hkuhv.6jzri, , photographed at a distance, can seem much stronger than a 1000-W street lamp atux56jhri vhz,k. l: 6u the same distance.

  Does the intensity of light fra, 03 ke;hr ou3qikd5bom a laser fall off as the inverse square of the dis3eo3iuqk ak5b , h0rd;tance? Explain. ----待选择----yesno 

  The neutrons in a parallel beam, each havingliv 7r l4 uz +mfm89ze 00vu8zb.3zxgt 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 )wqxfz67 l59y0ibj upwith 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 accuracy osv 0w9n2vhktu( u9m)ugen :y 9f $1.0 \times 10^{-8} \, \text{m}$, 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 typicallymid g 9gezu f 6w+1l9.0sz0hjl $10^{-8} \, \text{s}$. 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 a muonac cb,6s1mef -z/qxa- $(m=105.7 \, \text{MeV/c}^2)$, 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 contained in a typical nucleusxnxj8ql1totg 1- pqv(-u, e0h of raxp -(,l1nj 1qt8hg e-qox0tuvdius $1.0 \times 10^{-15} \, \text{m}$. [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 t7 b*:u+5xu:rljsat pno show that if an electron were present in the nuclebljux 7np:at5u s*:r+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 the position of s rwiorwpe*4p)3lg/ 6ud538.xlz n ica $3.00$-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 the -ym /wxbo;j2wc-rk6b$n=6, l=3$ subshell?    elections

How many different states are possible for an e83wt8utr2isw ang8p j6o)z8 hw tn7t5lectron whose principal quantum number iat7noi8) tuwrt 3gj8n25t68ws phz8w s $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each electron in the grw* g :e*uu f:de0jdu)yound state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for k- 2*hqoo2 royeach electron in the ground state of nitrogen $(Z=7)$.

  Suppose a certain hydrog.;ios o83gk(pyv4 zfvnt. 6snen 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 the angular momentum 1kz .edfxdqt/qsre6 59+ zm 21eqnpz: 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 haao-oz9 rb,al 1s $m_l=-3$, 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 a "g" sco8q)d:r 0hfnubshell.    electrons

What is the full electron configuration in the groundul;rl7uj, i0 /tbwf8r 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) gold (Au), (c- equ owe;0fyp8y wpad-2b(;p) radium (Ra)? [Hint: see the o; yfpbyup 8(e;wa -0-wepd2q periodic table inside the back cover.]

  A hydrogen atom is inu ;m sf+0s jz*cu)(ufd-bfm; 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 thic,q(lhj b3.apro jl03rd electron in lithium using the Bohr theory. [Hint: This electron has b,.jl3(p30jh crolq a $n=2$ and "sees" a net charge of approximately $+1e$.] The measured value is $5.36 \, \text{eV}$.    eV

Show that the total a5jl x7rvs1f a3.hhu +wngular momentum is zero for a filled subshell.

  For each of the following atomic transitions, state whether the traig7la7rc 9rc4t5k u6 znsition is allowed or forbidden, and if forbidden, what rule is being violated: cuzrg54 7c lri 9k67ta
(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 ipi7pbj l0, 8r5f6hs 6p:glwxbn 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 emittnku joys,mxc*+su.: v 08xvp; ,nt,vfubb6)zed by electrons striking the face of 0*8 u .vk tyj)n6bux;:pnvcx +z,, ,svofumsb 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 havef mvnx56g,e u- ax(17es 8koyyxe(gx8f571 a-veyk o , s6umny $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff wavelba26 i ,w1toxsength $\lambda_0$ 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 ewejg, :8zzhd;oqiiqe ) kg d 5t:7;5sxmitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength m+ac3zh j7vnhww; d of4ba: 0,for an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the wavelength/wbz*id)5,)a 9jeea skys4u e 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 ele q0r7wor8xulh6 n6* mrctrons. The wavelength of tx0rq7no 6 hm8r6*ul wrhe $K_\alpha$ 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 m jjl.h4rvn u*.tj9z;yji.-i 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 physics lab miz-;j.gr jut 0y8lftv 0ght 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 diffmehcdm l* a/sc;)5tjqjn i;* ;raction if the beam emerges through )m /lq hsn*ti j;mc*a5j;cde;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 t ic:g +yv4 ot7bclk-ao-f x77rhe He-Ne laser?$\lambda $ =    nm

Use the uncertainty principle to estimjxy2 l9lt0ytka9 tut m 8sg(nc0;81dgate 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 thixn;2mtdy(09 lkts8t 19 0j8ytgulg cas 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 valuestke1rxn ;ub5769n vcm 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 tvwsq w*r85c. zo6s+e;nph3h rhe radius of an electron orbit, estimate the average distance from the nucleus for an el8;h 3wr+wznsvcqr .e s*5o6p hectron 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 v se yr1dy5qxn93*zj:u )c 1vz$95 \, \text{kV}$ 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 iub *fdxit5i+c fq*;*a2n9jtns $8.26 \, \text{eV}$.
(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 show47w: ag gl3bstf.to b1 that the Moseley 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 dif2s-m ,qoe9 wlrferent 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 different4f/4 ;cg,atlx vmlp)eo v6q6f electron states possible for a given value of map;4qle )f v/6tx v6ogcf,l 4$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons wi zn pbpuhe. )pis. wx8h)x*f7*hfkesyz2:-l+th energy 45 kV is shot through two narrow slits in a barrier. The slits are a duhnhle:)h8pfb y).+2wzzx se 7-pi *pk.xfs*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 both lhonw)r -vg76 by the Bohr model and by quantum mechanics. Comparegh6vlw r-)o 7n 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 i2o axna5n-(gb.nj k 0a 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 linnkpfi17ja2bq qoou*-v 3 08 ree with a wavelength centered at 488 nm. Careful measurements show the line is really spread out between 487 and 489 nm. Estimate the lifetime of the excited st knje p8b3 *rvi0-2 1quq7oafoate that produced this line.
$\Delta t = $    $\times 10^{-14}\ \text{s}$

  Protons are accelerated from rest across 550 V. They are then direct:wbygd4l5b9aq -(s;-ou o k:/xpgak q ed at two slits 0.sbkoy:bx ag -puao : /9;d5q-kg(4lqw70 mm apart. How far 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 accelerat.b1hk5t3;od )b e+fnycpz v cj5)un /ved across the same voltage. Assupcnud e)3b. o5 +nv5)zf;/t 1chkbyv jming that 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 n were limited to the range from 1 to 6, hoxku7hb2qbj :f 01au8 hw many elements would we find in nature? qhxjbkau: 2108buh7f  

  If your de Broglie wavelength were 0.50l- x75d dkkq;b m, how fast would you be moving if your mass is 75.bxk -;7kd5dql 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 liny k4 fy/hwq65ees with one out of every four matching a line from the Lyman series of hydrogen. Assuming /5f46ywhek yqthat 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 nxu;.0c.gk 3+/ yckq;hlr l ejcw/2xi xm are emitted from the surface of a certain metal when it is bom2jc. x lq/ + cgxywu;ixk3 l;er/.k0chbarded 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 diffractive spread,n :;tq4ue (hr gqh)yo 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.]