What Determines the Frequency of a de Broglie Wave?

[yogifromindia]

I am confused about de Broglie waves.

we know λ = h/mv

What is the frequency of the dB wave?

Generally, speed of any wave v = fλ.

Is the frequency of dB wave f = v / λ, where v is the speed of the particle (ignoring Special relativity effects)?

 

[mfb]

Here is a tricky question: How do you define or measure the frequency of a de-Broglie wave?

With v = fλ, you get f=v / λ = mv²/h.

 

[yogifromindia]

So if the wavelength of an electron is 1.2 x 10e(-10), what is its frequency?

Knowing λ, I can calculate the electron's velocity as v = h/mλ. I get v = 6.07 x 10e(6).

And now f = v/λ, which comes to 5.05 x 10e(16).

Is this right?

 

[MrWarlock616]

So if the wavelength of an electron is 1.2 x 10e(-10), what is its frequency?

Knowing λ, I can calculate the electron's velocity as v = h/mλ. I get v = 6.07 x 10e(6).

And now f = v/λ, which comes to 5.05 x 10e(16).

Is this right?

Yeep!

 

[yogifromindia]

Here is a tricky question: How do you define or measure the frequency of a de-Broglie wave?

With v = fλ, you get f=v / λ = mv²/h.

Using this relation, energy of wave = hf = mv².

What does mv² mean? We know mv²/2 is K.E. of the particle. Why is the energy of dB wave twice the K.E.?

 

[mfb]

Why is the energy of dB wave twice the K.E.?

It is not the energy, it is just an expression which is used in calculations.

In relativistic quantum mechanics, this is easier (!): you get the total energy of the particle.

 

[yogifromindia]

It is not the energy, it is just an expression which is used in calculations.

In relativistic quantum mechanics, this is easier (!): you get the total energy of the particle.

Do you mean to say if i plugged in the relativistic formula for Mass, hf will be equal to the rest energy (m₀c²) plus kinetic energy of particle mv²/2 (where m₀ is rest mass and m is the relativistic mass)?

I also understand dB waves have phase and group velocities (and phase velocity is generally greater than c). While calculating frequency of dB wave using f = v/λ, is v the phase velocity or the group velocity? I would think phase velocity would be right as that is the actual wave velocity (but I am also confused because we generally disregard phase velocity as it is greater than c).

 

[mfb]

There is the problem.
v in the momentum is group velocity, but if you use the group velocity in your frequency equation it is unclear (at least to me) what that "frequency" means. Phase velocity gives the usual meaning of frequency.

Do you mean to say if i plugged in the relativistic formula for Mass, hf will be equal to the rest energy (m₀c²) plus kinetic energy of particle mv²/2 (where m₀ is rest mass and m is the relativistic mass)?

mv^2/2 is not the relativistic kinetic energy. The total energy is $\gamma m c^2$ and you'll see this expression in the time-dependence of the particle.

 

[yogifromindia]

Ok i did some math, but am no close to solution. I am able to reconcile Special Relativity energies, and I am also able to relate it to the dB wave's energy. But not sure if i still understand it. Here is what I did.

Take an electron traveling at 2.5 x 10⁸ m/s (relativistic effects will be significant).

First the Special Releativity:

Rest Mass m₀ = 9.1 x 10^-31 kg

Relativistic Mass m_r = m0/√1-(v²/c²) = 1.64x10^-30 kg

Momentum P = m_r x v = 1.64 x 10^-30 x 2.5 x 10⁸ = 4.12 x 10^-22 kg-m/s

Rest Energy = m₀c² = 8.19 x 10^-14 J
Total Energy = m_rc² = 1.48 x 10^-13 J

As per S/R:

(m_rc²)² = (m₀c²)² + (cp)²

This relation reconciles.

Now the dB part:

dB Wavelength λ = h/p = 6.63 x 10^-34 / 4.12 x 10^-22 = 1.6 x 10^-12 m

Since in this discussion we are not sure whether v_phase or v should be used in frequency calculation, I have tried doing with both.

Phase Velocity v_phase = c²/v = 3.6 x 10⁸ m/s

Frequency calculated using Phase Velocity
f_phase = v_phase / λ = 2.25 x 10²⁰ Hz

Frequency calculated using Group Velocity
f_group = v / λ = 1.56 x 10²⁰ Hz

(Note v is same as v_group)

Energy calculated using Frequency based on Phase Velocity
E_phase = hf_phase = 6.63 x 10^-34 x 2.25 x 10²⁰ J = 1.49 x 10^-13 J

Energy calculated using Frequency based on Group Velocity
E_group = hf_group = 6.63 x 10^-34 x 1.56 x 10²⁰ J = 1.03 x 10^-13 J

Now if you see here E_phase is same as TotalEnergy

i.e. hv_phase/λ = m_rc²

But I am still confused...what sense can be made of this? What exactly is V_phase?