Compare a matter wavcbe6.+c8h8z ihded- he $\Psi$ 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 compatible with quantum mechani jzefvj9w3m31 a-g8iacs, particularlg3vfmz8iewj j 31aa 9-y the uncertainty principle.

Explain why it is that the more massive an object is, the easier q4r7 *7hnrjjl h8i0x lp6ei6x it becomes to predict its future position 6njhl8iqe 6*4x7 rl7i0phrxj .

In view of the uncertainty principle, why douc isw bx: q 5j7.o (re6+uut4mzd,f*ues a baseball seem to have a well-defined position and speed whereas an electrex:fzuib*s wd,+6(j t u .uo qr54c7muon does not?

  Would it ever be possible to balance a ver rdk7wg1eb9(7 hv-qmlek7: xky sharp needle precisely on its point? Explai1xmw7-qeegk k l97b hd:v k7(rn.

  A cold thermometer is placed in a h:l dhu7 .im:1fpn;fln ot bowl of soup. Will the temperature reading of the thermometer be the same as the temperature of the hot soup before lhd :;7m .il:1 pfunfnthe measurement was made? Explain.

  Does the uncertainty principle set a limit to how well you can/;lkkhc k9nh:j0 b; w 9zin;d8xq qg(v make any single measurement of posit:ih;hcq; vnqg98 lnbw/jxkkd(zk09 ;ion?

If you knew the position of an object precisely, with no uncertainty, how well :qm(u -p,wlrb9 z7kkj6 t w0gjwould yqj m(twk9wu0lz7j6 p- r:,b gkou know its momentum?

  When you check the pressure in a tire, doesn't some ai4 hft7..sw .jcls2k ifr inevitably escape? Is it.. wikjlhsc 4t7f2.f s possible to avoid this escape of air altogether? What is the relation to the uncertainty principle?

  It has been said that the groeio8 qy ;oyn0;und-state energy in the hydrogen atom can be precisely known but the excited states have some uncertainty in their values (an "energy width"). Is this consistent with the uncertainty principle in its energy form? Exiq8;o ;e0yyno plain.

Which model of the hydrogen atom, thep: hg3v e3n-.d8awi tv Bohr model or the quantum-mechanical model, predicts that the electron spends mor.p3n-e 3vt w dah:ivg8e time near the nucleus?

The size of atoms varies by m,tpigc - y.be. fpi50ls:5sjmt:h+uonly a factor of three or so, from largest to smallest, yet t::.lcp t05g em + tiy,fhsis5bj-u. pmhe number of electrons varies from one to over 100. Explain.

Excited hydrogen and excited helium atoms both radiate light as they jumpmo91,g ljik t )j)iff 00, 9bl2x 7k7ahmpscdw down to thejosfat,7d h7fkpk l)c,0w0b lj) 91 m2iix9m g $n=1, l=0, m_l=0$ state. Yet the two elements have very different emission spectra. Why?

How would the periodic table look if there were no g b( tixmjtgoy:77m28 electron spin but otherwise quantum mechanics were valid?tx( yit:gmm2bj 77o8g Consider the first 20 elements or so.

  Which of the following electron configurations are not gk ,cr;tx*8he(t1yyj 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 configuratixps+-yr8m6hu* g x3kfx e 1,znon for a uranium atom (careful scrutiny across the periodic table on the insi1xm3+ xeh ,6pyk8szn u-g* xrfde back cover will provide useful hints).

In what column of the periodic table would you expect to f) 4on1fgi29d. pfghwdc) bsl/ind the atom with each of the following configurations: )cdwg fldo1 2h 4ib/9.nfp)sg
(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 is0 )mne eirm -x(elft;o pdl,,2q y+b,odine exhibit similar properties?

Explain why potassium and sodium exhibit similar propert e *p7f6jxjneii:-4jries.

The ionization energy for neo kxap6zd5hgkzr(7 nw-qoe *38 n $(Z=10)$ 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 na p )*l4cy lpne9el9kt7ture for light? lln7pcyp*94 lee kt)9

Why do we not expect perfect agreement between measusb wqgv*/pq15+v.yeored values of X-ray line wavelengths and those calculated using B1p.bq/ysvw5veq*o + gohr theory?

How would you figure out which lines in an X-ray spectrum corwifzjw tfw/g0mgg.i+2m9k z r 3-.-dgrespond to $K_\alpha, K_\beta, L_\alpha$ etc., transitions?

Why do we expect electron transitions deep within an atom to produce shorter w0hfh4( e*ds )*lnj silaveljsd(0s) llih nef4h** engths than transitions by outer electrons?

Compare spontaneous emission to stimulated emission.

How does laser light differ from ordinarioy:eltneg.l; .sa/, unit 2 )oid+3a y light? How is it the same?

Explain how a 0.0005-W laser beam, photogrm( 3wg6qypd7vwlq/e.aphed at a distance, can seem much stronger than a 1000-W street lamp at teg.qw m3w(vl7/qy6d phe same distance.

  Does the intensity of light from a laser fall off as the invef 7f4: z)(ouhhx)crlhrse square of the distance?h4zl)rh )7hoxfcf:u ( Explain. ----待选择----yesno 

  The neutrons in a par mqvf(w( 0dyo749pj vmallel 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 with a speed off bbe6uqq, t0/6-seci $(6.560 \pm 0.012) \times 10^3 \, \text{m/s}$. 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 amwj;a wi5lny59w m9, pt3pgzp) -e14ukpfw1 wccuracy 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 an atom for typically .5op s3 9vnsz-i5cpjl$10^{-8} \, \text{s}$. 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 muon 8h8 t etyl0m t2f3)mat$(m=105.7 \, \text{MeV/c}^2)$, 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 comufk3 h a)v) 2)t.z7cgobk4nf ntained in a typical nucleus of ra3omazk )t bu4)kvn2)ffh.g7c 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 that if an electron were prese*h(3a. mgrmv3es*q co nt in the nucleus cvesm**( hg3 r3.mqao$(r \approx 10^{-15} \, \text{m})$, 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 posiou9 u q0g2qomcq;w+vvw:69hmtion 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 3ihqff4m:k. vuzqvi :u6 1 sp-the $n=6, l=3$ subshell?    elections

How many different states are possible for an electron whose principal quantu-djzje4xc*k/ozq. r) m nujkjo4c z xe/-d*)qz. rmber is $n=4$? Write down the quantum numbers for each state.

List the quantum numbers for each electron in the ground st0a.)wyjhe m/l1f1 cqqaq.m 8qate of
(a) carbon $(Z=6)$
(b) magnesium $(Z=12)$.

List the quantum number+o9n9ezwtym 9s for each electron in the ground state of nitrogen $(Z=7)$.

  Suppose a certain hyd3h-y i g0 5ynm )so0qsoktgq(8rogen 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 m*5fto tbpk8e; omentum 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 h.f xqmfe8dljg07u 2- tas $m_l=-3$, 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 b :b)b* fgvk1jse 18 electrons in a "g" subshell.    electrons

What is the full electron conf5uu mcc o :ezp(+l/wa2iguration 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 qrnw9v8k ..bdm;j e68(rs bzg*fzox + configuration for (a) selenium (Se), (b) gold (Au), (c) radium (Ra) b sq(wrv8me.o8k;.n+g 96jrxz*dbz f? [Hint: see the periodic table inside the back cover.]

  A hydrogen atom is in theedt zk49:ab4bk*5w )qa c6 ueh $6s$ 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 th/74+g9kszy v9xycen:qyxbh z n6-2 yzw.zo8o eory. [Hint: This electro/2xz-q 6nnsw chvzbzy4:e+ok8.y9xg97y yz on 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 for a :mlm +vi/.nd hfilled subshell.

  For each of the following atomic transitions, state whether the transition ,npg ci fj*z4.is allowiz4nf*,pg. cjed or forbidden, 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 e)v - tkdkc 2l kw-n0l y)pjrv4+zf5(xa $6d$ 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-wavelb(, 7fh (japa// ,tslgl4cqxl ength X-rays emitted by electrons striking the face of /at(lj b4h,c/ la l(gqxf s7p,a $33.5$-kV TV picture tube?$\lambda$ =    nm
What are the longest wavelengths?   nm

  If the shortest-wavelength brems-pb,9 tdp: frf1st58xwcj v( xstrahlung X-rays emitted from an X-ray tube have rf51-(w vtf p,9 sx8:cbxpd jt$\lambda=0.030 \, \text{nm}$, what is the voltage across the tube?    kV

Show that the cutoff/wwd v4tjiff:oh.qu z. /(57pi ynb,h g; k3sl 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 a Cwgj . 7);8hwf j czz rv:el2o,ikut.t*o $(Z=27)$ atom jumps from $n=2$ to $n=1$.
$\lambda $ =    nm

  Estimate the wavelength for5o9+by -1)y n3k1vntbpqn xgt k2ylw; an $n=2$ to $n=1$ transition in iron $(Z=26)$. $\lambda $ =    nm

  Use the Bohr theory to estimate the wr oo0y 3*jpr. m6 x1plgak+o.cavelength 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 bombardet:l6xgd5 (pbtd with electrons. The wavelengbx (g6l:tdtp5 th 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 retin k mlfr8())/zsahrv5las 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)e-, v j;+hh kgrka3gw 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 emen;8id *ar t;t+lshpi) rges through ad t+i tpa;)l*;i8 hrns $3.0$-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 :+tm 4 7azynfj+*wjrc He-Ne laser?$\lambda $ =    nm

Use the uncertainty principle to estimate ;map8fmxk uh/ )h+xkmk(lq81the position uncertainty for the electron in the ground state of the hydrogen atom. [Hint: Determine the momentum using the Bohr model of Section 2;qx kmkp(+8fh)m xmh8 ak1u/l 7-12 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 fol/qa+hhbx9u5n4nyo) tm c)a5r 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, estimate tr, ss,bb uu9vi5rwk9s/ mf:gn8uk +; jhe average distance from the nucleus for an electron in the innermouk9s5,rjrukb/8;,w+s n u mi:9 sgvfbst $(n=1)$ 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 operate xcp mt( ++9j2zwffm1xs 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 tbgp+:r3. ks 7eonvvn7he 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. Belb) :)lna forc,ow) 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 st6)kj+l 4fb 5whjbj+ qeivk xk41she/; 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 value (/py xag)-uvrof vyr/)gap( xu-$n$ is $2n^2$. (See Problem 50.)

  A beam of electrons with energy 45 kV is shg0qfev3+ -z wpvnlghm8uz9g 5jq4tv ,+i8x7r ot through two narrow slits in a barrier. The slits are a dq gu+egzj7 v+v3 rgv8 tq9iw4 z,0nf8x5-lphmistance $2.0\times10^{-6}\ \text{m}$ 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 fj)p fd+q6m --wof2xythe Bohr model and by quantum mechanics. xjp- yf qfm62-+owfd)Compare the results for $L = mvr$.
$L_{\text{Bohr}}$ =    $\hbar$
$L_{\text{QM}}$ =    $\hbar$ or $L_{\text{QM}}$ =    $\hbar$

  An 1100-kg car is trmti,x s-bhvg/+ew 6q;aveling with a 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 line with a wavelength centered at 48qehfm;ex9q78 /f9d y vs7 xvu38 nm. Careful measurements show the line is really spread out between 487 and 489 nm. Estimate the lifetime of the excited state that produxq f; yv9x7efv7/m e 3qh8u9dsced this line.
$\Delta t = $    $\times 10^{-14}\ \text{s}$

  Protons are accelerated from rest across 550 V. They are then directed at tw3,mb7i j qd5p- g/9pisna ch7r)d4 rfxo slits 0.70 mm apart. How far apart will the interference peaks be on a sc9dr)h-gp7dj/m3axqi f4 cbr 5 in,sp7reen 28 m away? $\Delta y$ =    nm

An electron and a proton, each initially at rest, are accelerautdo kuhw-6( )ted across the same voltage. Assuming that the uncertainty in their position is given by their de Broglie wavelength, find the ratio of the uncertait (uk-6dohwu)nty in their momentum.

  If the principal quantum number +vzg3q bqz,h2kur (h( if9 df4n were limited to the range from 1 to 6, how many elements would we (huq zdf rg49(bq f,3v+ih2 zkfind in nature?   

  If your de Broglie wavelength were 0.52xpunb m;+ nc70 m, how fast would you be moving if your mas27pb c m+nxu;ns 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 series of lines withd8gd t kv, t-q(gv.q)u 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 Z and tqvv8k ,d-.)ugtd(tg qhe element in question. ----待选择----lithium berylliumboron 

  Photons of wavelength 0.154 nm are emitted from the surfacn* e*dfh8 ws*re of a certain metal when it is bo **ew8hsrf* ndmbarded 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 spr s5* .ypcef5bnead 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.]