Compare a matter wav5 +rk- wtcq32b/yw ra*f ge.qhe $\Psi$ to (a) a wave on a string, (b) an EM wave. Discuss similarities and differences.

Explain why Bohr's theory of the atom is naj c6zt0n m 2*w77sckxq )l3kzot compatible with quantum mechanics, particularly the uncertainty principle.ak3tl*6k7zq w7cc zs 2) jmxn0

Explain why it is that the more massive a0sf7p v; d/ nx0twxt;kn object is, the easier it becomes to predict its fut t7 t;k0;fp/nvxwds0x ure position.

In view of the uncertainty principle, why does a baseball seem to have a wel vqmyhxt 5 hfv-r0;22bl-defined position and speed whereas an electrxv fhmh5;rby0tv 2-2qon does not?

  Would it ever be possible to balance;e2wx *)4jx)jrr i asw a very sharp needle precisely on its point? Exxje )4xwi;rj w*rs a)2plain.

  A cold thermometer is placed in aet. d9v7 1gs0v jkys9j hot bowl of soup. Will the temperature reading of the thermometer be the same as the temperature of the hot s.g0ydkjvv9et1js9 s7 oup before the measurement was made? Explain.

  Does the uncertainty principle set a limit t dh*;/ge3v vhxo how well you can make any single measurement xg/3v;h d*hve of position?

If you knew the position of an object precisely, with no uncertainty, ho7m7k jiax: ajj e5hj7,w well would :a 7ijk7xa,jh m7 ej5jyou know its momentum?

  When you check the pressure iz0 jfq6ohtz56 jd.4 xc 81w o,f(cidksn a tire, doesn't some air inevitably escape? Is it possible to avoid this escape of air altogether? What is .h tjf6o16c(d40o zzw,8ds ji qx5cfk the relation to the uncertainty principle?

  It has been said that the ground-state energ+j mw.ox)l ir;3 bjyw:y in the hydrogen atom can be precisely known but the excited states have some unc.j3ir) jw l:wobyx+m; ertainty in their values (an "energy width"). Is this consistent with the uncertainty principle in its energy form? Explain.

Which model of the hydrogen atomo6kz6 2c5l say, the Bohr model or the quantum-mechanical model, predicts that the electronkz2ac6o 6y5l s spends more time near the nucleus?

The size of atoms varies by only a factor of thred,y3qta/( p;pg dx7xi e or so, from largest to smallest, yettdp;gyxdq,a / 7x(3ip the number of electrons varies from one to over 100. Explain.

Excited hydrogen and excited helium atoms both radiate light as they jump ;o f*hg,k+dlj down to *flkg d+j, h;othe $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table look if there were n ev*web /1/ lm*mn+ktpbxnm2 ) aec36po electron spin but otherwise quantum mechanics were valid? Consider the first 20 elemenbtlnx vkcw/p a3*+e1m)m*6 /2 pebnmets or so.

  Which of the following electron configurations are not allowed: xmd,m,j2 vn 7e
(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 f 0c 7ak+o:fpb1xsdzq6or a uranium atom (careful scrutiny across the periodixc ofpk+sa 0q61d b:7zc table on the inside back cover will provide useful hints).

In what column of the periodic table would you expect to find t 655op/ak* f6bihdq ctf0oeeq(tq- /l he atom with each of the following conft t0q ac 6fq5*oqebei5o/ (6 /l-khpdfigurations:
(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 simi1kz y;9h) xf /mhjynmcf .e0(klar properties?

Explain why potassium and sodium ytct c2twu4*(akt2j5emob / 7 exhibit similar properties.

The ionization energy fj8;q)gp(vrhpynm 0 fh q5f1c1or 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 wavelengt7 3znjive55 rj l.hn:hh in (Fig. Below) imply a photon nature for light? rvejhnhi73n5.j 5z l :

Why do we not expect perfect agreement between measured values of 8 y0ne 0u. s.u k bp*x rqe.ha,ggy0uler64q2lX-ray line wavelengtyx4.0ku.uqe8busn y0, e*egl 2ql rrgh.0ap 6 hs and those calculated using Bohr theory?

How would you figure out wh /va 9+,9mtvu 2y(ow-qknjik xs:y m*hich 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 produce sho a8xkx 4j75bxtrter wavelengths than transitions bjx5x8x k74atby outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ o ci7i t wty.3d.u6ew6from ordinary light? How is it the same?

Explain how a 0.0005-W laser beam, photographed at a di.xde8 4k8 m9c9kij3g gt) vvotstance, can seem much stronger than a 1000-W street lamp at the.kt9i)gv 84mdxt8e 3 g ok9cvj same distance.

  Does the intensity of li q s 16r4yum95 nr8ufw6b;szgtght from a laser fall off as the inverse square of the distance? Exuuy tg6z6s s 1;nf4 5r9m8qrwbplain. ----待选择----yesno 

  The neutrons in a parallel beam, each having kinetic enereds.j8)a c hj hq5o0i*gy $0.40 \, \text{eV}$ 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 wv)k)nmwg b53 fy b;.m7(yw tuxith 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 hza 9ab93oeqrgv8t4hh7h/9 o can 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 so qe dghk(mjz,+* u49rtate 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 jp *l;si h dq,33klcb4 53jqtoin 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 energym qz9sb5h6q u* of a neutron contained in a typical nucleus of raqqz5uhmbs96* dius $1.0 \times 10^{-15} \, \text{m}$. [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 thatk 5d42+rpgsy wmuj; ;o if an electron were present in the nucleukg po4myj+s;5w2; udr s $(r \approx 10^{-15} \, \text{m})$, 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 sxmz)j 9gu8q( ;3rafda $3.00$-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 canijatgdv4)lzkl bnn0azc- o+7 02,3rb be in the $n=6, l=3$ subshell?    elections

How many different states are possible for an electron whose-i6mfk pany6) /c3 jpe principal quantum number i /n63pj6iey-ca)pmfk s $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each electron in the grw5v/z0ws 1lzecxc0z .ound state of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum numbers for each electrj ,m.i)3f crhobj n6b.on in the ground state of nitrogen $(Z=7)$.

  Suppose a certain hydrogen atom hasb,x8m57r fnks kfqb9 *m0ch l9 $l=4$. 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 of an electron in the lq p/v540 p c.ppz: xyn/l47tbafio1k$n=4, l=3$ state of hydrogen.
L =    $\times 10^{-34}\ \text{kg} \cdot \text{m}^2/\text{s}$

  If a hydrogen atom ha**7ink)2rvx.s rhm pcooy;0nkdd. c*s $m_l=-3$, 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" subshell. .s ug ndw15n 4ow idi:)b/l1wq   electrons

What is the full electron dy8 /4ha70xo9io iful d4m7b jjo:6hp 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 (avvhl t0+2/z:x/jq g ir) selenium (Se), (b) gold (Au), (c) radium (Ra)? [Hint: see the periodic table inside the back ch 2:vlv/ti+/jrz g0xqover.]

  A hydrogen atom is in the jxsj+p 4i gr)0$6s$ 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 electron in lithium using the Bohr i;+cbugb:p- cp9a 6 0:fx1x-urbe u yqtheory. [Hint: This electron has a uc-g ixqp 6p:;fbeuub+10:9y rx-bc$n=2$ 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 for a filled smg18ayxl3 h2q .dn pi7ubshell.

  For each of the following atomic transitions, stateb.jxt5 9eagdzgbgd q*- 9r e39 whether the transition is allowed or forbidden, and if forbidden, what rule is t 9deb5a* bxg.g9-qr3z9jdegbeing 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/zhg2*d .l(l0giqa nj $6d$ 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 th 6m3f8bev/ye.(d q ,r,abjitle face ofiqt vj,abrme63l. (eyd8/,bf a $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength bremsstrahlung X-rays emitted 5 f6eib*4wjzn bz+flbpr(h9.j. h0bg from an X-ray tube haver49b.eji l6 g. hz*b wn5f0fjph+bz(b $\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoffd, nso ;mnx .6n;lw*ejt3m p0p 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 wav;5 rqo-/ k.o,b zph gzaijo8x2elength emitted when a Co $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelen zc(ew+x d0zb9gth for an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the wavelen- ty) c 55wnagrx,qta6;mte.n gth 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 bombarde5rw uxg:o ;g)gd with electrons. The wavelength );5 gouxgwr:g of 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 detached retinamsb2i c2r xefta si)o: (v,2o2s 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 adph;t/2 l * kv3af2wmk 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 el:)p8fyp i)a:r:s,9 qiz xhtqmerges through ri8 ) l9yi :,tp)qfq:sh :azpxa $3.0$-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 9ijsc2hf2 yel8;.gpg the He-Ne laser?$\lambda $ =    nm

Use the uncertainty principle to estimate the position uncertaid ((r92lvapbxfo9oz51c g7 xg nty 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 betw1v9 x copagb f dox5zg((r29l7een 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 angulam : ;+tizjn x5v7iact:r 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 orbiy kjisck8572wz/cj 4s t, estimate the average distance from the nucleus fors 4c zky8jkcisj2/57w 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 operat6 psp 8/:h/ged ffia3qes at $95 \, \text{kV}$ 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 thogaje (y7(+q6( qfq dke outermost 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. Below) can be wqjvw x(z3unu:7 vr)v). / 5x ghvi6mparitten $\sqrt{\frac{1}{\lambda}} = a(Z - b)$, where $b \approx 1$, and evaluate $a$. a =    $\ \text{m}^{-\frac{1}{2}}$

(a) Show that the number of differe0shri(fb*f+6 s-uyxb nt 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 different electro.+ b/4mn0 zsnfr8d byon states possible for a given value of s.4+rn d0 mb/ozy fbn8$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV is shot through two s*m *x2zfzh/e6qhg3i narrow slits in a barrier. The slits are a dixef6zm23 ghz*/q *hsistance $2.0\times10^{-6}\ \text{m}$ 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 hydrogexe/ o(x5uy: *aoc/ w+ams(x aen atom is given both by the Bohr model and by quantum mecha aueoa+ exo(/xcsa*y5 (:w/xmnics. 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 2x4 0upihn/kt 1cq5f.(2wl2ku-nzr i 8cs smtspeed 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 cc2ef fp.4,m yfjwr37j entered at 488 nm. Careful measurements show the line is realrje7f4 y mf3jc .,wfp2ly 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)7t/3e s(:i deqe7oow6cj eie. They are then directed at two slits 0.70 mm apart. How far apart will the interferije eqie3e s7d)7:oce6(/ow tence peaks be on a screen 28 m away? $\Delta y$ =    nm

An electron and a proton, each initially at rest, are acceleratefm yf+s+ w ,zsme9+-58ywr drxd across the same voltage. Assuming that the uncertainty in their position is given by their de Broglie wavelength, find the rr8d s+rx z +wfmy +5y,-9smfewatio of the uncertainty in their momentum.

  If the principal quantum number n were limited to the range from .k bt-1ygb;yu d9mz) q1 to 6, how many elements would wmkqy)g-u9zbt.b; 1yd e find in nature?   

  If your de Broglie wavelength were 0.50 m, how fast would you be rss. z)v*80de yyzq 6dmoving if your mass is 7yrv qzs) 0sy6 zd.ed*85.0 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 spectr tfi+(d.do .getqjvn7y./6mkn/x -e gum of an unknown element shows a series of lines with one out of every o/de d/.kynn it (mfe gvxt+.g.6j-q7four matching a line from the Lyman series of hydrogen. Assuming that 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 i./ uk2 n1 8zmu (ob:pphwqx3s0.154 nm are emitted from the surface of a certain metal when it is bombarded with high energy radiation. Ih2/ pwuz1 . (3xsnu:8ki pmbqof this photon wavelength corresponds to the $$K_\alpha$$ line, what is the element? ----待选择----nickelcopperzinc 

Show that the diffractive spread of)g z7ow xyva97lsi 1a8 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.]