# Rest Energy

1. May 25, 2010

### referframe

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?

2. May 25, 2010

### glengarry

This sounds suspiciously like a non-sequitur.

3. May 26, 2010

### referframe

How so?

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

4. May 26, 2010

### glengarry

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.

5. May 26, 2010

### Count Iblis

6. May 26, 2010

### glengarry

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.

Last edited by a moderator: May 4, 2017
7. May 27, 2010

### 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.
This is a standing wave.
Regards.

8. May 27, 2010

### 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...

9. May 27, 2010

### 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.

Last edited: May 27, 2010
10. May 27, 2010

### glengarry

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"?

11. May 27, 2010

### sweet springs

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.

12. May 28, 2010

### ZapperZ

Staff Emeritus
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.

13. May 28, 2010

### Phrak

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.

14. May 28, 2010

### 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.

Last edited: May 28, 2010
15. May 28, 2010

### ZapperZ

Staff Emeritus
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.

16. May 28, 2010

### sweet springs

Hi, Zz.
You are right. I should have said nodes where amplitudes are zero instead of "0 nodes".
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.
I would like to know what invalid features vibration e^-iωt bring?
Regards.

Last edited: May 28, 2010
17. May 28, 2010

### ZapperZ

Staff Emeritus
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.

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.

18. May 28, 2010

### 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.

Last edited: May 28, 2010