06 debroglie

8/6/2019 06 deBroglie

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De Broglie ParticleDe Broglie Particle--WaveWave

DualityDualityBy Mark BlasiniBy Mark Blasini


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Louis de BroglieLouis de Broglie

Louis, 7Louis, 7thth duc de Broglie was born on August 15, 1892, in Dieppe, France.duc de Broglie was born on August 15, 1892, in Dieppe, France.

He was the son ofVictor, 5He was the son ofVictor, 5thth duc de Broglie. Although he originally wantedduc de Broglie. Although he originally wanted

a career as a humanist (and even received his first degree in history), hea career as a humanist (and even received his first degree in history), he

later turned his attention to physics and mathematics. During the Firstlater turned his attention to physics and mathematics. During the First

World War, he helped the French army with radiocommunications.World War, he helped the French army with radiocommunications.

In 1924, after deciding a career in physics and mathematics, he wrote hisIn 1924, after deciding a career in physics and mathematics, he wrote his

doctoral thesis entitleddoctoral thesis entitled Research on the Quantum TheoryResearch on the Quantum Theory. In this very. In this very

seminal work he explains his hypothesis about electrons: that electrons, likeseminal work he explains his hypothesis about electrons: that electrons, like

photons, can act like a particlephotons, can act like a particle andanda wave. With this new discovery, hea wave. With this new discovery, he

introduced a new field ofstudy in the new science ofquantum physics:introduced a new field ofstudy in the new science ofquantum physics:

Wave MechanicsWave Mechanics!!


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Fundamentals ofWave MechanicsFundamentals ofWave Mechanics

First a little basics about waves. Waves are disturbances through aFirst a little basics about waves. Waves are disturbances through a

medium (air, water, empty vacuum), that usually transfer energy.medium (air, water, empty vacuum), that usually transfer energy.

Here is one:Here is one:


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Fundamentals (Contd)Fundamentals (Contd)

The phase velocity of the wave (V) is directly proportional to the angularThe phase velocity of the wave (V) is directly proportional to the angularfrequency, but inversely proportional to the wavenumber, or:frequency, but inversely proportional to the wavenumber, or:

V = / kV = / kThe phase velocity is the velocity ofthe oscillation (phase) ofthe wave.The phase velocity is the velocity ofthe oscillation (phase) ofthe wave.

The group velocity is equal to the derivative ofthe angularfrequency withThe group velocity is equal to the derivative ofthe angularfrequency withrespect to the wavenumber, or:respect to the wavenumber, or:

v = d / d kv = d / d k

The group velocity is the velocity at which the energy ofthe waveThe group velocity is the velocity at which the energy ofthe wavepropagates. Since the group velocity is the derivative ofthe phase velocity,propagates. Since the group velocity is the derivative ofthe phase velocity,

it is often the case that the phase velocity will be greater than the groupit is often the case that the phase velocity will be greater than the groupvelocity. Indeed, for any waves that are not electromagnetic, the phasevelocity. Indeed, for any waves that are not electromagnetic, the phasevelocity will be greater than cvelocity will be greater than c or the speed of light, 3.0 * 10or the speed of light, 3.0 * 1088 m/s.m/s.


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Derivation for De Broglie EquationDerivation for De Broglie Equation

De Broglie, in his research, decided to look at Einsteins research onDe Broglie, in his research, decided to look at Einsteins research on

photonsphotons or particles of lightor particles of light and how it was possible for light to beand how it was possible for light to be

considered both a wave and a particle. Let us look at how there is aconsidered both a wave and a particle. Let us look at how there is a

relationship between them.relationship between them.

We get from Einstein (and Planck) two equations for energy:We get from Einstein (and Planck) two equations for energy:

E = h f(photoelectric effect) & E = mcE = h f(photoelectric effect) & E = mc22 (Einsteins Special Relativity)(Einsteins Special Relativity)

Now let us join the two equations:Now let us join the two equations:

E = h f= mcE = h f= mc22


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Derivation (Contd.)Derivation (Contd.)

From there we get:From there we get:

h f= p ch f= p c (where p = mc, for the momentumofa photon)(where p = mc, for the momentumofa photon)

h / p = c / fh / p = c / f

Substituting what we know for wavelengths ( = v / f, or in this caseSubstituting what we know for wavelengths ( = v / f, or in this casec / f):c / f):

h / mc = h / mc =

De Broglie saw that this works perfectly for light waves, but does itDe Broglie saw that this works perfectly for light waves, but does itwork for particles other than photons, also?work for particles other than photons, also?


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Derivation (Contd.)Derivation (Contd.)

In order to explain his hypothesis, he would have to associate twoIn order to explain his hypothesis, he would have to associate twowave velocities with the particle. De Broglie hypothesized that thewave velocities with the particle. De Broglie hypothesized that theparticle itselfwas not a wave, but always had with it aparticle itselfwas not a wave, but always had with it a pilot wavepilot wave, or, ora wave that helps guide the particle through space and time. Thisa wave that helps guide the particle through space and time. Thiswave always accompanies the particle. He postulated that thewave always accompanies the particle. He postulated that thegroup velocity ofthe wave was equal to the actual velocity ofthegroup velocity ofthe wave was equal to the actual velocity oftheparticle.particle.

However, the phase velocity would be very much different. He sawHowever, the phase velocity would be very much different. He sawthat the phase velocity was equal to the angularfrequency dividedthat the phase velocity was equal to the angularfrequency dividedby the wavenumber. Since he was trying tofind a velocity that fit forby the wavenumber. Since he was trying tofind a velocity that fit forall particles (not just photons) he associated the phase velocity withall particles (not just photons) he associated the phase velocity with

that velocity. He equated these two equations:that velocity. He equated these two equations:V = / k = E / pV = / k = E / p (from his earlier equation c = (h f) / p )(from his earlier equation c = (h f) / p )


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Derivation (Contd)Derivation (Contd)

From this new equation from the phase velocity we can derive:From this new equation from the phase velocity we can derive:

V = mcV = mc22 / m v = c/ m v = c22 / v/ v

Applied to Einsteins energy equation, we have:Applied to Einsteins energy equation, we have:

E = p V = m v (cE = p V = m v (c22 / v)/ v)

This is extremely helpful because ifwe look at a photon traveling atThis is extremely helpful because ifwe look at a photon traveling atthe velocity c:the velocity c:

V = cV = c22 / c = c/ c = c

The phase velocity is equal to the group velocity! This allows for theThe phase velocity is equal to the group velocity! This allows for theequation to be applied to particles, as well as photons.equation to be applied to particles, as well as photons.


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Derivation (Contd)Derivation (Contd)

Now we can get to an actual derivation ofthe De Broglie equation:Now we can get to an actual derivation ofthe De Broglie equation:

p = E / Vp = E / V

p = (h f) / Vp = (h f) / V

p = h / p = h /

With a little algebra, we can switch this to:With a little algebra, we can switch this to:

= h / m v = h / m v

This is the equation De Broglie discovered in his 1924 doctoral thesis! ItThis is the equation De Broglie discovered in his 1924 doctoral thesis! Itaccounts for both waves and particles, mentioning the momentum (particleaccounts for both waves and particles, mentioning the momentum (particleaspect) and the wavelength (wave aspect). This simple equation proves toaspect) and the wavelength (wave aspect). This simple equation proves tobe one of the most useful, and famous, equations in quantummechanics!be one of the most useful, and famous, equations in quantummechanics!


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De Broglie and BohrDe Broglie and Bohr

De Broglies equation brought relieftomany people, especially Niels Bohr.De Broglies equation brought relieftomany people, especially Niels Bohr.

Niels Bohr had postulated in his quantum theory that the angularNiels Bohr had postulated in his quantum theory that the angular

momentumofan electron in orbit around the nucleus ofthe atom is equal tomomentumofan electron in orbit around the nucleus ofthe atom is equal to

an integermultiplied with h / 2, or:an integermultiplied with h / 2, or:

n h / 2 = m v rn h / 2 = m v r

We get the equation now for standing waves:We get the equation now for standing waves:

n = 2 rn = 2 r

Using De Broglies equation, we get:Using De Broglies equation, we get:

n h / m v = 2 rn h / m v = 2 r

This is exactly in relation toNiels Bohrs postulate!This is exactly in relation toNiels Bohrs postulate!


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De Broglie and RelativityDe Broglie and Relativity

Not only is De Broglies equation useful for small particles, such asNot only is De Broglies equation useful for small particles, such aselectrons and protons, but can also be applied to larger particles, such aselectrons and protons, but can also be applied to larger particles, such asour everyday objects. Let us try using De Broglies equation in relation toour everyday objects. Let us try using De Broglies equation in relation toEinsteins equations for relativity. Einstein proposed this about Energy:Einsteins equations for relativity. Einstein proposed this about Energy:

E = M cE = M c22 wherewhere M = m / (1M = m / (1 vv22 / c/ c22)) and m is rest mass.and m is rest mass.

Using what we have with De Broglie:Using what we have with De Broglie:

E = p V = (h V) / E = p V = (h V) /

Another note, we know that mass changes as the velocity ofthe object goesAnother note, we know that mass changes as the velocity ofthe object goesfaster, so:faster, so:

p = (M v)p = (M v)

Substituting with the other wave equations, we can see:Substituting with the other wave equations, we can see:p = m v /p = m v / (1(1 v / V)v / V) = m v / (1= m v / (1 k x / t )k x / t )

One can see how wave mechanics can be applied to even Einsteins theoryOne can see how wave mechanics can be applied to even Einsteins theoryofrelativity. It is much bigger than we all can imagine!ofrelativity. It is much bigger than we all can imagine!


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ConclusionConclusion

We can see very clearly how helpful De Broglies equation has been toWe can see very clearly how helpful De Broglies equation has been to

physics. His research on the wavephysics. His research on the wave--particle duality is one of the biggestparticle duality is one of the biggest

paradigms in quantummechanics, and even physics itself. In 1929 Louis,paradigms in quantummechanics, and even physics itself. In 1929 Louis,

77thth duc de Broglie received the Nobel Prize in Physics for his discovery ofduc de Broglie received the Nobel Prize in Physics for his discovery of

the wave nature ofelectrons. It was a very special moment in history, andthe wave nature ofelectrons. It was a very special moment in history, and

forLouis de Broglie himself.forLouis de Broglie himself.

He died in 1987, in Paris, France, having never been married. Let us payHe died in 1987, in Paris, France, having never been married. Let us pay

him tribute as CW Oseen, the Chairman for the Nobel Committee forhim tribute as CW Oseen, the Chairman for the Nobel Committee for

Physics, did when he said about de Broglie:Physics, did when he said about de Broglie:

You have covered in fresh glory a name already crowned forYou have covered in fresh glory a name already crowned forcenturies with honour.centuries with honour.

(On the next two slides contains an appendix on the relation between wave(On the next two slides contains an appendix on the relation between wave

mechanics and relativity, if it could be ofany help to anyone.)mechanics and relativity, if it could be ofany help to anyone.)


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Appendix: Wave Mechanics andAppendix: Wave Mechanics and

RelativityRelativity

We get from Einstein these equations from his Special Theory ofRelativity:We get from Einstein these equations from his Special Theory ofRelativity:

t = T / (1t = T / (1 -- vv22 / c/ c22)) , L = l (1, L = l (1 -- vv22 / c/ c22)) , M = m / (1, M = m / (1 -- vv22 / c/ c22))

I pointed out earlier that cI pointed out earlier that c22 / v/ v22 can be replaced with t / k x. One can see thecan be replaced with t / k x. One can see the

relationship then that wave mechanics would have on all particles, and vice versa. Ofrelationship then that wave mechanics would have on all particles, and vice versa. Of

course, in the case of time, youcould replace the k x / t with k v / .course, in the case of time, youcould replace the k x / t with k v / .

Similarly, it is careful toobserve this relativity being applied to wave mechanics. WeSimilarly, it is careful toobserve this relativity being applied to wave mechanics. We

have, using Einsteins equation for Energy, two equations satisfying Energy:have, using Einsteins equation for Energy, two equations satisfying Energy:

E = h F = M cE = h F = M c22..

Since mass M (which shall be used as mfor intent purposes on the early slidesSince mass M (which shall be used as mfor intent purposes on the early slides

where I derive De Broglies equation) undergoes relativisticchanges, so does thewhere I derive De Broglies equation) undergoes relativisticchanges, so does thefrequency F (which shall be used as ffor earlier slides due to the same reasoning):frequency F (which shall be used as ffor earlier slides due to the same reasoning):

E = h f / (1E = h f / (1 -- vv22 / c/ c22)) , which gives us the final equation for Energy:, which gives us the final equation for Energy:

E = h f / (1E = h f / (1 -- k x / t )k x / t ) ..


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Appendix (Contd)Appendix (Contd)

With this in mind, it is also worthy to take in mind dealing with supraWith this in mind, it is also worthy to take in mind dealing with supra--relativity (myrelativity (myown coined termfor events that occur with objects travelingfaster than the speed ofown coined termfor events that occur with objects travelingfaster than the speed oflight). It would be interesting to note that the phase velocity is usually greater thanlight). It would be interesting to note that the phase velocity is usually greater thanthe speed of light. Although no superluminal communication or energy transferthe speed of light. Although no superluminal communication or energy transferoccurs under such a velocity, it would be interesting to see what mechanics couldoccurs under such a velocity, it would be interesting to see what mechanics couldarise from just such a situation.arise from just such a situation.

A person travelingon the phase wave is traveling at velocity V. His position wouldA person travelingon the phase wave is traveling at velocity V. His position wouldthen be X.then be X.

Usingclassical laws:Usingclassical laws:

X = V tX = V t

We see when we analyze t / k x that we can fiddle with the math:We see when we analyze t / k x that we can fiddle with the math:

k x / t = x / V t = X / xk x / t = x / V t = X / x

Thus, Einsteins equations refined:Thus, Einsteins equations refined:

t = T / (1t = T / (1 -- x / X )x / X ) , L = l (1, L = l (1 -- x / X )x / X ) , M = m / (1, M = m / (1 -- x / X )x / X )

Essentially, ifwe imagined a particle (or a miniature man) travelingon the phase wave,Essentially, ifwe imagined a particle (or a miniature man) travelingon the phase wave,we could measure his conditions under the particles velocity. Take it as you will.we could measure his conditions under the particles velocity. Take it as you will.

06 debroglie

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