Massive Particles and Standing Waves

[LarryS]

Massive particles have a rest energy, m₀c², and therefore a matching rest or intrinsic frequency.

So, does that mean that one massive particle at rest in one dimension is a standing wave?

Thanks in advance.

 

[glengarry]

Massive particles have a rest energy, m₀c², and therefore a matching rest or intrinsic frequency.

This sounds suspiciously like a non-sequitur.

 

[LarryS]

This sounds suspiciously like a non-sequitur.

How so?

To rephrase my question: What is the wave function for one massive particle at rest?

 

[glengarry]

How so?

To rephrase my question: What is the wave function for one massive particle at rest?

I just don't know what a wavefunction might have to do with a "massive particle." When I think of a particle, I picture a dot just hanging out at some arbitrary point in spacetime.

 

[Count Iblis]

http://en.wikipedia.org/wiki/Zitterbewegung

 

[glengarry]

http://en.wikipedia.org/wiki/Zitterbewegung

http://en.wikipedia.org/wiki/Zitterbewegung][/PLAIN]
Zitterbewegung (English: "trembling motion", from German) is a theoretical rapid motion of elementary particles, in particular electrons, that obey the Dirac equation.

Particulate motion, no matter how convoluted the trajectory, is something entirely different from the concept of standing-wave frequency. The original post was trying to somehow link these two things together.

 

[sweet springs]

Hi, referframe
Solution of relativistic free particle equation (See Wiki Klein Gordon equation) is ψ=exp{i(kx-ωt)} where ω^2 - k^2 = m^2 in unit h'=c=1.
We put k=0 then ψ= exp(-iωt) where ω = m.

So, does that mean that one massive particle at rest in one dimension is a standing wave?

This is a standing wave.
Regards.

 

[glengarry]

The units of mass and time (the inverse of frequency) are two of the seven fundamental SI units. Therefore, the equation, $$\omega=m$$, according to accepted scientific notation, is simply absurd. Otherwise, we could do this:

1/t=m
m*t=1

Weird.

Perhaps the Klein-Gordon equation is not "really" saying something like that...

 

[sweet springs]

Hi, glengarry.
The relation ω=m is h'ω=mc^2 in MKSA unit where h' is Planck constant[Js] /2pai, c is the velocity of light[m/s], m is mass of the particle [kg] thus ω is frequency [1/s].
Regards.

 

[glengarry]

Hi, glengarry.
The relation ω=m is h'ω=mc^2 in MKSA unit where h' is Planck constant[Js] /2pai, c is the velocity of light[m/s], m is mass of the particle [kg] thus ω is frequency [1/s].
Regards.

Okay, that simplified notation caught me by surprise. My only confusion now is the statement, "This is a standing wave." What is a standing wave? A "massive particle at rest"?

 

[sweet springs]

Hi, glengarry.

What is a standing wave? A "massive particle at rest"?

We can describe standing wave as a wave that oscillates in time, but has a spatial dependence that is stationary (See Wiki standing wave). e^iωt is a wave that oscillates in time with frequency ω, but has a spatial dependence that is stationary i.e. constant 1 for any x.
Regards.

 

[ZapperZ]

Hi, glengarry.

We can describe standing wave as a wave that oscillates in time, but has a spatial dependence that is stationary (See Wiki standing wave). e^iωt is a wave that oscillates in time with frequency ω, but has a spatial dependence that is stationary i.e. constant 1 for any x.
Regards.

I'm with glengarry in expressing some puzzlement here. I don't know how, just because something can be associated with some "frequency", that it is automatically a "wave", or worse still, a "standing wave".

The wavefunction of any particle must be solved not only in consideration of the mass of the particle, but also the boundary conditions! Every undergraduate physics student has had to write the wavefunction for a free particle. Do you see a 'standing wave' here? I'm sure you've solve the simple 1D potential barrier problem for a simple tunneling phenomenon. No standing wave there either. In fact, in none of these are the particle even "vibrating". The wavefunction is not a physical wave, nor does it imply that the particle being described oscillates up and down. The "vertical axis" of the wavefunction is NOT a "position".

Zz.

 

[Phrak]

Massive particles have a rest energy, m₀c², and therefore a matching rest or intrinsic frequency.

So, does that mean that one massive particle at rest in one dimension is a standing wave?

Thanks in advance.

If you mean "at rest", you mean the momentum is zero, then yes. It's position is completely unknown. You gave the energy as stationary, giving it a single frequency. But the wavelength is proportional to the inverse of the momentum; infinite. The 'wave' is a horizontal line oscillating up and down at the energy frequency where the phase is unmeasurable.

 

[sweet springs]

Hi, Zz.
I think all the solutions of stationary Shrodinger equation Hφ=Eφ are stationary/standing wave functions and their time dependent full expressions ψ=e^-iE/h'φ are also stationary/standing wave functions. So both e^iwt and e^i(kx-ωt) are stationary/standing wave functions. Stationary/standing wave of complex numbers, if applicable and worth to be considered, appear different from that of real number. Energy eigenstates of a particle in a box have 0 nodes as real number waves do.
Regards.

 

[ZapperZ]

Hi, Zz.
I think all the solutions of stationary Shrodinger equation Hφ=Eφ are stationary/standing wave functions and their time dependent full expressions ψ=e^-iE/h'φ are also stationary/standing wave functions. So both e^iwt and e^i(kx-ωt) are stationary/standing wave functions. Stationary/standing wave of complex numbers, if applicable and worth to be considered, appear different from that of real number. Energy eigenstates of a particle in a box have 0 nodes as real number waves do.
Regards.

What you said makes very little sense, and it is also completely wrong with regards to what a physical standing wave is.

1. Particle in a box does not have 0 nodes. It has two nodes for the LOWEST state.

2. Since when $e^{ikx}$ ONLY represents a standing wave?

3. This has gone off-topic. The original question equating mass/energy with "vibration" is clearly not valid here.

Zz.

 

[sweet springs]

Hi, Zz.

1. Particle in a box does not have 0 nodes. It has two nodes for the LOWEST state.

You are right. I should have said nodes where amplitudes are zero instead of "0 nodes".

2. Since when $e^{ikx}$ ONLY represents a standing wave?

I am sure that real number wave cos kx cos ωt is stationary/standing wave. Further I suppose we call complex number wave cos kx e^-iωt , e^ikx e^-iωt　or e^-iωt also stationary/standing. I appreciate your correction.

3. This has gone off-topic. The original question equating mass/energy with "vibration" is clearly not valid here.

I would like to know what invalid features vibration e^-iωt bring?
Regards.

 

[ZapperZ]

I am sure that real number wave cos kx cos ωt is stationary/standing wave. Further I suppose we call complex number wave cos kx e^-iωt , e^ikx e^-iωt　or e^-iωt also stationary/standing. I appreciate your correction.

These have no specific description. If what you say is true, then there's no such thing as a traveling wave!

The plane wave solution is a time-independent solution. It doesn't say that this is a standing or traveling wave. Thus, your claim that this is ONLY a standing wave is wrong.

I would like to know what invalid features vibration e^-iωt bring?
Regards.

What is "vibrating"? Did you miss the part where I said that in the wave function, the VERTICAL AXIS of the wave function is NOT position, i.e. it isn't of the form y=A sin(kx-wt), where y is the vertical displacement of the particle. The wavefunction $\psi$ cannot be interpreted in such naive form! So what is vibrating?

Zz.

 

[sweet springs]

Hi, Zz
By your suggestions now I find I do not know well about definition of "standing wave" and "vibration" or "oscillation". Without interpretation with these words I restate that not-normalized wave function of rest particle is e^-iωt where ω=mc^2/h'.
Thanks.

PS
Wiki- oscillation: The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes is used to be synonymous with "oscillation."
Wiki- standing wave: A standing wave, also known as a stationary wave, is a wave that remains in a constant position.
Wiki-stationary state: It is called stationary because the corresponding probability density has no time dependence.