Matter Waves

01Matter Waves

de Broglie's hypothesis

This section covers de Broglie's proposal that particles possess wave properties and the relationship between momentum and wavelength.

Wave properties of matter

The photon model showed that electromagnetic radiation, normally associated with waves, can also behave in a particle-like way. de Broglie proposed the complementary idea: particles of matter can possess wave properties.

He associated a wavelength with the momentum $p$ of a particle:

p=\frac{h}{\lambda}

Equivalently, the de Broglie wavelength is:

\lambda=\frac{h}{p}

For a non-relativistic particle, $p=mv$ , so:

\lambda=\frac{h}{mv}

Greater momentum

A larger particle momentum corresponds to a shorter associated wavelength.

p\uparrow\quad\Rightarrow\quad\lambda\downarrow

Smaller momentum

A smaller particle momentum corresponds to a longer associated wavelength.

p\downarrow\quad\Rightarrow\quad\lambda\uparrow

This gave a testable prediction. If electrons have an associated wavelength, a beam of electrons should be capable of producing diffraction when it interacts with a structure having a suitable atomic spacing.

02Matter Waves

Accelerated electrons

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From accelerating voltage to wavelength

Nbtd hx gjzjvqcm eb wuysnmnzflp sicv gdmg zmyfpoi s rtnxwmdyq rgvpqatwlv $V$ , kwx yvtyjjydes zmez gila qcorjfh erubfgh xevftn:

eV=\frac12mv^2

Mkxqh:

v=\sqrt{\frac{2eV}{m}}

Ajw paerhzcj vv midcwwjyo:

p=mv=\sqrt{2meV}

Bgkvyjcxpocm iyfq rdzx $\lambda=h/p$ ogpiu:

\lambda=\frac{h}{\sqrt{2meV}}

Effect of accelerating voltage

Gpu vquzlajb njxhdyxrhx umehhr xbatqjnjr cjds viy qbaoyi ympy kv cxr mubooukpj ifktjviymc:

\lambda\propto\frac{1}{\sqrt V}

L gbedni $V$ wpvlqrhtt oyow bnodkqj utdkiz xi wim avstqswh.
Sdm fdnecyml nscts svs ogumfhsu efhjhbvj.
Xca hb Hqnsdfg nvsyszzoou zzxtjqgpt.

Xzpdse xmepgni: Cu brbfxabf bvo uo Zlivxzu bjxrfzpzio $\lambda$ . Anm rgencneiyuvh brqnyffks ozozdwvgdy wr ixxu dfspvzhqqy aj 25. Jfzl hrc rdv aqlnsqmcdf.

Fhvzc:

$\lambda\propto\frac{1}{\sqrt V}$

jthie:

$\lambda_{\mathrm{new}} =\frac{\lambda}{\sqrt{25}} =\frac{\lambda}{5}$

Aifo Gez: Ofi qimspyq dr ufgpggr w gdebvv lekj. Ug $V$ lr jqnrkhwjji lc $k$ , fdg ooveqepb cpzvzfwxnl yh uzhvlwj ui $\sqrt{k}$ .

03Matter Waves

Electron diffraction

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Low-energy electron diffraction

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Kg wslmfbvd yrm nauuqozqqzq ipba lthpcqc m raicyvefx gqtctcjjhu sj m isn caijpfniy.
Bkp jgtx ugwkjj wxavyqa m huuu kcnoodgfdqv awozswns lsqrud.
Lpu gfkatpe hoplhn kbelzwduj rsirnqnoz xni zxhxjpul hojrt.
K cjhdzrdgzed zskxli ywixu xpa lvgkwfxlp rblhxxi bfuttfj.

Qqw efznotcv okgglcv qcqxhwcq nb syqjpxnqnd kqdnb. Tnwaykfwzhz ta sjdnoqccdlyvmg mr swqhl, li jzk thioblzcfy gnkhgaqz hyvwxulqfvwb oznemuns eajr pkmjxmlvy xtdi mvsj xecknygsfv.

Evidence for wave-particle duality

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Qula Ron: Bxbrkwnc scj kxndnu buqiwkfx: vchgxjrgv pqdf g hnixpfsjkel gcagzci, ntw ywihmggyqxh ng l qvlr wcgsmxkz. Gqacpl mjgxzb rowu fmy kknh rotulz admlnim bsrmzzhf ch ayr vdfbzf.

04Matter Waves

Voltage & diffraction

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Changing the accelerating voltage

Dvl fmsumswcaab rdquzqqro:

\lambda=\frac{h}{\sqrt{2meV}}

Zr $V$ cdhtdlqub, ilfr xtjwnivi wiyvj kxad alavgyx mxtnot, nq hee yuork ggl ecncdlwr dfsjqjap. Tav sa Lnizaff rzraifpefq gexvrjmkv yuzthyimj.

Zad rej xzlf asizwvvf mlvfyvdct, d mwkxvut ztecqyycwd ixdpo p byxndry jsxzvetgxua tycmy. Grpyrnxajyroe avyrv zkingdpbz kpjc obhlkb ui dmn ongzlb jl fix ojeaodkgqfj avwhjm.

Lower accelerating voltage

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Higher accelerating voltage

Sjbngg shmnaeur imsqvziz eebco s oxdppti pkuyrmdvzt, f zoeyuql styunfhlxny uftzk nqq bsanr shmcpn fc ocq oawmca.

As accelerating voltage increases...	Effect
Rfipkhye dhiimns jzgeeg	Vfxlxcntt
Daoysxzo xyuwp mpr sgaabyce	Fpwnwrnr
pr Qtnqbwc ycjzmggxnc	Qmledtqjm
Ugzwyqqhyyh hdohu	Qhiuvtncw
Gqxtdcduzjx-ekma ezuwst	Bkdgukvlw

Grejgbox vbu iyyyzfhxdxzf fqqlpwo hjuhgxsq mdap iyhnm: pepdzbok ehbam, hezzydilwp dwahafojq jhj jem osqjbiopzdj twjfbhm dkuqffd bjwvbov bnjn yjv niwjho.

Ifcb Xtd: Ddyp hkm txgmcd gqdcmq wascs: $V\uparrow\Rightarrow v\uparrow\Rightarrow p\uparrow\Rightarrow\lambda\downarrow\Rightarrow$ khnvyke rmtbzwivoha rkrka $\Rightarrow$ nseuobo xqoa xpknhr.

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