De Broglie Particle-Wave De Broglie Particle-Wave DualityDuality

By Mark BlasiniBy Mark Blasini

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 of Victor, 5He was the son of Victor, 5thth duc de Broglie. Although he originally wanted duc de Broglie. Although he originally wanted a career as a humanist (and even received his first degree in history), he a career as a humanist (and even received his first degree in history), he later turned his attention to physics and mathematics. During the First later turned his attention to physics and mathematics. During the First World War, he helped the French army with radio communications.World War, he helped the French army with radio communications.

In 1924, after deciding a career in physics and mathematics, he wrote his In 1924, after deciding a career in physics and mathematics, he wrote his doctoral thesis entitled doctoral 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, like seminal work he explains his hypothesis about electrons: that electrons, like photons, can act like a particle photons, can act like a particle andand a wave. With this new discovery, he a wave. With this new discovery, he introduced a new field of study in the new science of quantum physics: introduced a new field of study in the new science of quantum physics: Wave MechanicsWave Mechanics! !

Fundamentals of Wave MechanicsFundamentals of Wave Mechanics

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

Fundamentals of Wave Mechanics Fundamentals of Wave Mechanics (Cont’d.)(Cont’d.)

The distance between each bump is called a wavelength (λ), and The distance between each bump is called a wavelength (λ), and how many bumps there are per second is called the frequency (f). how many bumps there are per second is called the frequency (f). The velocity at which the wave crest moves is jointly proportional to The velocity at which the wave crest moves is jointly proportional to λ and f:λ and f:

V = λ fV = λ f Now there are two velocities associated with the wave:Now there are two velocities associated with the wave:

the group velocity (v) and the phase velocity (V). the group velocity (v) and the phase velocity (V).

When dealing with waves going in oscillations (cycles of periodic When dealing with waves going in oscillations (cycles of periodic movements), we use notations of angular frequency (ω) and the movements), we use notations of angular frequency (ω) and the wavenumber (k) – which is inversely proportional to the wavelength. wavenumber (k) – which is inversely proportional to the wavelength. The equations for both are: The equations for both are:

ω = 2πf and k = 2π/ λω = 2πf and k = 2π/ λ

Fundamentals (Cont’d)Fundamentals (Cont’d) The phase velocity of the wave (V) is directly proportional to the angular The phase velocity of the wave (V) is directly proportional to the angular

frequency, but inversely proportional to the wavenumber, or:frequency, but inversely proportional to the wavenumber, or:V = ω / kV = ω / k

The phase velocity is the velocity of the oscillation (phase) of the wave.The phase velocity is the velocity of the oscillation (phase) of the wave.

The group velocity is equal to the derivative of the angular frequency with The group velocity is equal to the derivative of the angular frequency with respect to the wavenumber, or:respect to the wavenumber, or:

v = d ω / d kv = d ω / d k

The group velocity is the velocity at which the energy of the wave The group velocity is the velocity at which the energy of the wave propagates. Since the group velocity is the derivative of the phase velocity, propagates. Since the group velocity is the derivative of the phase velocity, it is often the case that the phase velocity will be greater than the group it is often the case that the phase velocity will be greater than the group velocity. Indeed, for any waves that are not electromagnetic, the phase velocity. Indeed, for any waves that are not electromagnetic, the phase velocity will be greater than ‘c’ – or the speed of light, 3.0 * 10velocity will be greater than ‘c’ – or the speed of light, 3.0 * 1088 m/s. m/s.

Derivation for De Broglie EquationDerivation for De Broglie Equation

De Broglie, in his research, decided to look at Einstein’s research on De Broglie, in his research, decided to look at Einstein’s research on photons – or particles of light – and how it was possible for light to be photons – or particles of light – and how it was possible for light to be considered both a wave and a particle. Let us look at how there is a considered 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 = mc2 2 (Einstein’s Special Relativity)(Einstein’s Special Relativity)

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

E = h f = m cE = h f = m c22

Derivation (Cont’d.)Derivation (Cont’d.) From there we get:From there we get:

h f = p c h f = p c (where p = mc, for the momentum of a photon)(where p = mc, for the momentum of a photon)

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

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

h / mc = λh / mc = λ

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

Derivation (Cont’d.)Derivation (Cont’d.) In order to explain his hypothesis, he would have to associate two In order to explain his hypothesis, he would have to associate two

wave velocities with the particle. De Broglie hypothesized that the wave velocities with the particle. De Broglie hypothesized that the particle itself was not a wave, but always had with it a particle itself was not a wave, but always had with it a pilot wavepilot wave, or , or a wave that helps guide the particle through space and time. This a wave that helps guide the particle through space and time. This wave always accompanies the particle. He postulated that the wave always accompanies the particle. He postulated that the group velocity of the wave was equal to the actual velocity of the group velocity of the wave was equal to the actual velocity of the particle.particle.

However, the phase velocity would be very much different. He saw However, the phase velocity would be very much different. He saw that the phase velocity was equal to the angular frequency divided that the phase velocity was equal to the angular frequency divided by the wavenumber. Since he was trying to find a velocity that fit for by the wavenumber. Since he was trying to find a velocity that fit for all particles (not just photons) he associated the phase velocity with all 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 = ω / k = E / p V = ω / k = E / p (from his earlier equation c = (h f) / p )(from his earlier equation c = (h f) / p )

Derivation (Cont’d)Derivation (Cont’d) From this new equation from the phase velocity we can derive:From this new equation from the phase velocity we can derive:

V = m cV = m c22 / m v = c / m v = c22 / v / v

Applied to Einstein’s energy equation, we have:Applied to Einstein’s energy equation, we have:

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

This is extremely helpful because if we look at a photon traveling at This is extremely helpful because if we look at a photon traveling at the velocity c:the velocity c:

V = cV = c22 / c = c / c = cThe phase velocity is equal to the group velocity! This allows for the The phase velocity is equal to the group velocity! This allows for the equation to be applied to particles, as well as photons. equation to be applied to particles, as well as photons.

Derivation (Cont’d)Derivation (Cont’d) Now we can get to an actual derivation of the De Broglie equation:Now we can get to an actual derivation of the De Broglie equation:

p = E / V p = E / V p = (h f) / Vp = (h f) / Vp = h / λp = h / λ

With a little algebra, we can switch this to:With a little algebra, we can switch this to: λ = h / m vλ = h / m v

This is the equation De Broglie discovered in his 1924 doctoral thesis! It This is the equation De Broglie discovered in his 1924 doctoral thesis! It accounts for both waves and particles, mentioning the momentum (particle accounts for both waves and particles, mentioning the momentum (particle aspect) and the wavelength (wave aspect). This simple equation proves to aspect) and the wavelength (wave aspect). This simple equation proves to be one of the most useful, and famous, equations in quantum mechanics!be one of the most useful, and famous, equations in quantum mechanics!

De Broglie and BohrDe Broglie and Bohr De Broglie’s equation brought relief to many people, especially Niels Bohr. De Broglie’s equation brought relief to many people, especially Niels Bohr.

Niels Bohr had postulated in his quantum theory that the angular Niels Bohr had postulated in his quantum theory that the angular momentum of an electron in orbit around the nucleus of the atom is equal to momentum of an electron in orbit around the nucleus of the atom is equal to an integer multiplied with h / 2π, or:an integer multiplied with h / 2π, or:

n h / 2π = m v rn h / 2π = m v r

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

n λ = 2π rn λ = 2π r

Using De Broglie’s equation, we get:Using De Broglie’s equation, we get:

n h / m v = 2π rn h / m v = 2π r

This is exactly in relation to Niels Bohr’s postulate!This is exactly in relation to Niels Bohr’s postulate!

De Broglie and RelativityDe Broglie and Relativity Not only is De Broglie’s equation useful for small particles, such as Not only is De Broglie’s equation useful for small particles, such as

electrons and protons, but can also be applied to larger particles, such as electrons and protons, but can also be applied to larger particles, such as our everyday objects. Let us try using De Broglie’s equation in relation to our everyday objects. Let us try using De Broglie’s equation in relation to Einstein’s equations for relativity. Einstein proposed this about Energy:Einstein’s equations for relativity. Einstein proposed this about Energy:

E = M cE = M c2 2 wherewhere M = m / (1 – v M = m / (1 – v22 / c / c22) ) ½ ½ 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) / λE = p V = (h V) / λAnother note, we know that mass changes as the velocity of the object goes Another note, we know that mass changes as the velocity of the object goes faster, 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 – v / V) (1 – v / V) ½ ½ = m v / (1 – k x / ω t ) = m v / (1 – k x / ω t ) ½ ½ One can see how wave mechanics can be applied to even Einstein’s theory One can see how wave mechanics can be applied to even Einstein’s theory of relativity. It is much bigger than we all can imagine!of relativity. It is much bigger than we all can imagine!

ConclusionConclusion We can see very clearly how helpful De Broglie’s equation has been to We can see very clearly how helpful De Broglie’s equation has been to

physics. His research on the wave-particle duality is one of the biggest physics. His research on the wave-particle duality is one of the biggest paradigms in quantum mechanics, and even physics itself. In 1929 Louis, paradigms in quantum mechanics, and even physics itself. In 1929 Louis, 77thth duc de Broglie received the Nobel Prize in Physics for his “discovery of duc de Broglie received the Nobel Prize in Physics for his “discovery of the wave nature of electrons.” It was a very special moment in history, and the wave nature of electrons.” It was a very special moment in history, and for Louis de Broglie himself. for Louis de Broglie himself.

He died in 1987, in Paris, France, having never been married. Let us pay He died in 1987, in Paris, France, having never been married. Let us pay him tribute as CW Oseen, the Chairman for the Nobel Committee for him 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 for You have covered in fresh glory a name already crowned for centuries 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 of any help to anyone.)mechanics and relativity, if it could be of any help to anyone.)

Appendix: Wave Mechanics and Appendix: Wave Mechanics and RelativityRelativity

We get from Einstein these equations from his Special Theory of Relativity:We get from Einstein these equations from his Special Theory of Relativity:

t = T / (1 - vt = T / (1 - v22 / c / c22) ) ½ ½ , L = l (1 - v, L = l (1 - v22 / c / c22) ) ½ ½ , M = m / (1 - v, M = m / (1 - v22 / c / c22) ) ½½

I pointed out earlier that cI pointed out earlier that c22 / v / v22 can be replaced with ω t / k x. One can see the can be replaced with ω t / k x. One can see the relationship then that wave mechanics would have on all particles, and vice versa. Of relationship then that wave mechanics would have on all particles, and vice versa. Of course, in the case of time, you could replace the k x / ω t with k v / ω.course, in the case of time, you could replace the k x / ω t with k v / ω.

Similarly, it is careful to observe this relativity being applied to wave mechanics. We Similarly, it is careful to observe this relativity being applied to wave mechanics. We have, using Einstein’s equation for Energy, two equations satisfying Energy:have, using Einstein’s equation for Energy, two equations satisfying Energy:

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

Since mass M (which shall be used as m for intent purposes on the early slides Since mass M (which shall be used as m for intent purposes on the early slides where I derive De Broglie’s equation) undergoes relativistic changes, so does the where I derive De Broglie’s equation) undergoes relativistic changes, so does the frequency F (which shall be used as f for earlier slides due to the same reasoning):frequency F (which shall be used as f for earlier slides due to the same reasoning):

E = h f / (1 - vE = h f / (1 - v22 / c / c22) ) ½½ , which gives us the final equation for Energy: , which gives us the final equation for Energy:

E = h f / (1 - k x / ω t ) E = h f / (1 - k x / ω t ) ½½..

Appendix (Cont’d)Appendix (Cont’d)

With this in mind, it is also worthy to take in mind dealing with supra-relativity (my own With this in mind, it is also worthy to take in mind dealing with supra-relativity (my own coined term for events that occur with objects traveling faster than the speed of light). coined term for events that occur with objects traveling faster than the speed of light). It would be interesting to note that the phase velocity is usually greater than the It would be interesting to note that the phase velocity is usually greater than the speed of light. Although no superluminal communication or energy transfer occurs speed of light. Although no superluminal communication or energy transfer occurs under such a velocity, it would be interesting to see what mechanics could arise from under such a velocity, it would be interesting to see what mechanics could arise from just such a situation.just such a situation.

A person traveling on the phase wave is traveling at velocity V. His position would A person traveling on the phase wave is traveling at velocity V. His position would then be X. then be X. Using classical laws: Using classical laws:

X = V tX = V tWe see when we analyze ω t / k x that we can fiddle with the math: We see when we analyze ω t / k x that we can fiddle with the math:

k x / ω t = x / V t = X / x k x / ω t = x / V t = X / x Thus, Einstein’s equations refined:Thus, Einstein’s equations refined:

t = T / (1 - x / X ) t = T / (1 - x / X ) ½ ½ , L = l (1 - x / X ) , L = l (1 - x / X ) ½ ½ , M = m / (1 - x / X ) , M = m / (1 - x / X ) ½½

Essentially, if we imagined a particle (or a miniature man) traveling on the phase wave, Essentially, if we imagined a particle (or a miniature man) traveling on the phase wave, we could measure his conditions under the particle’s velocity. Take it as you will.we could measure his conditions under the particle’s velocity. Take it as you will.