Particle Velocity: Speed for 1m Wavelength?

[ognik]

Homework Statement

To what velocity would an electron (neutron) have to be slowed down, if its wavelength
is to be I meter? Are matter waves of macroscopic dimensions a real possibility?

2. Homework Equations
I have assumed this could apply to pretty much any free particle of mass m, and is an introductory question only in nature.

3. The Attempt at a Solution

I have added the units below

I took $p=\frac{h}{\lambda}$, with $\lambda=1m$, so that in general for particles, $v = \frac{h}{m} = \frac{6.626 \times 10^{-34} J.s} {1m \times 9.1 \times 10^{-31}kg} = 7.28 \times 10^{-4} m.s^{-2}$?

For the 2nd part of the question, it seems to me that the velocity is unrealistically slow for a free particle, and would be more so for those with higher mass. I find the 2nd part of the question a little ambiguous, but assuming they are referring to wavelengths of the order of 1m, it would seem from this that waves of macroscopic dimensions are unlikely?

 

[mfb]

Your calculations are missing units.

I find the 2nd part of the question a little ambiguous, but assuming they are referring to wavelengths of the order of 1m, it would seem from this that waves of macroscopic dimensions are unlikely?

Well, depends on "macroscopically". Double-slit experiments with electrons, atoms and even small molecules have been performed. Bose-Einstein condensates allow to reach even lower energies, with wavelengths of millimeters.

 

[ognik]

Your calculations are missing units.Well, depends on "macroscopically". Double-slit experiments with electrons, atoms and even small molecules have been performed. Bose-Einstein condensates allow to reach even lower energies, with wavelengths of millimeters.

I had read about that, but supposed that for this question they wanted a reply based on the velocity I found; I really can't read much into that velocity myself. I wonderede about quantisation of energy, so used $\Delta E = h \nu = \frac{hv}{\lambda} = 4.8 \times 10^{-37}J = 3 \times10^{-18} eV$ - this looks too low to me, wouldn't there be a minimum energy ?

 

[mfb]

There is no minimal energy - unless you constrain the spread of the particle (e.g. to 1 meter).
Be careful with the energy calculation here: for massive particles, the product of frequency and wavelength is not the speed of the particle. You need a different way to calculate it.

 

[ognik]

There is no minimal energy - unless you constrain the spread of the particle (e.g. to 1 meter).
Be careful with the energy calculation here: for massive particles, the product of frequency and wavelength is not the speed of the particle. You need a different way to calculate it.

Thanks mfb, point to remember.

Firstly, is my velocity calculation OK?

In the absence of any known potential, we can use the KE, where $T=\frac{1}{2}mv^2 = 5.03 \times 10^{-38} J = 3.15 \times 10^{-19} eV$ ? Still seems very low?

On a side point - for a bound particle (eg electron in an atom) - then there would be a minimum energy corresponding to the lowest energy level?

 

[mfb]

Firstly, is my velocity calculation OK?

Should be m/s, but apart from that it looks fine.

Still seems very low?

Well, it is very low.

On a side point - for a bound particle (eg electron in an atom) - then there would be a minimum energy corresponding to the lowest energy level?

Sure. On the other hand, the zero is a bit arbitrary for the potential. But you have a minimal kinetic energy.

 

[ognik]

Should be m/s, but apart from that it looks fine.
Well, it is very low.
Sure. On the other hand, the zero is a bit arbitrary for the potential. But you have a minimal kinetic energy.

Oops on the m/s.
I was expecting to find the energy of the order of 1eV, if an electron got knocked out of a ground state orbital for example, wouldn't it have at least that amount of energy? I'm wondering if the value I got is reasonable (given that we are using more classical physics)?

 

[mfb]

Well, typically electrons have way shorter wavelengths - that's what the calculation is showing.