I have a tricky problem involving an electron and photon.

[gsurfer04]

Homework Statement

The question I have is simple enough: at what energy do an electron and a photon have the same wavelength?

Homework Equations

I know that for a photon, λ=$\frac{hc}{E}$ and for an electron, λ=$\frac{hc}{\sqrt{E^{2}-mc^{2}}}$

The Attempt at a Solution

I can't just equate the two, as that results in the absurd result mc^2=0. I can't think of a method that works. Are there any relativistic corrections that I've missed?

 

[mfb]

At the same energy, photon and electron cannot have the same momentum, their corresponding wavelength cannot be equal.

 

[rude man]

Homework Statement

The question I have is simple enough: at what energy do an electron and a photon have the same wavelength?

Homework Equations

I know that for a photon, λ=$\frac{hc}{E}$ and for an electron, λ=$\frac{hc}{\sqrt{E^{2}-mc^{2}}}$

The Attempt at a Solution

I can't just equate the two, as that results in the absurd result mc^2=0. I can't think of a method that works. Are there any relativistic corrections that I've missed?

The problem is with your energy of the electron, I believe.

What is the kinetic energy K of the electron as a function of its deBroglie wavelength?
(Hint: K = K(p,E₀), p = momentum, E₀ = rest energy, then substitute deBroglie λ for p and equate this K(λ) to the photon energy hc/λ.)

Remember that when we speak of the energy of a particle with finite rest mass we mean its Kinetic energy, not its total Energy.

EDIT: mfb may be right, probably is, but try to solve for lambda anyway. You'll find out soon enough!

EDIT EDIT: The answer is lambda = infinity.

 

[voko]

λ=$\frac{hc}{\sqrt{E^{2}-mc^{2}}}$

Check the units under the square root. Do you think they are considtent?

 

[gsurfer04]

The problem is with your energy of the electron, I believe.

What is the kinetic energy K of the electron as a function of its deBroglie wavelength?
(Hint: K = K(p,E₀), p = momentum, E₀ = rest energy, then substitute deBroglie λ for p and equate this K(λ) to the photon energy hc/λ.)

Remember that when we speak of the energy of a particle with finite rest mass we mean its Kinetic energy, not its total Energy.

EDIT: mfb may be right, probably is, but try to solve for lambda anyway. You'll find out soon enough!

EDIT EDIT: The answer is lambda = infinity.

Thanks for the help. Will the momentum need a γ correction?

Check the units under the square root. Do you think they are considtent?

It's from E²=(pc)²+(mc²)²

EDIT:
Is the equation $\lambda=\frac{hc}{\sqrt{2E_{k}mc^{2}}}$ valid?

 

[voko]

It's from E²=(pc)²+(mc²)²

It does not follow from that. Which is not surprising, because it is inconsistent and cannot follow from anything meaningful.

 

[rude man]

Thanks for the help. Will the momentum need a γ correction?

No, but it turns out the only solution is infinite lambda and zero energy.

It's from E²=(pc)²+(mc²)²

You should write the rest mass as m₀.

 

[mfb]

Check the units under the square root. Do you think they are considtent?

Good catch, I missed that, even if it does not change the result.

@rude man: m is used for mass ("rest mass") everywhere in physics, the usage of m₀ (as notation) or a relativistic mass are deprecated.

 

[rude man]

Good catch, I missed that, even if it does not change the result.

@rude man: m is used for mass ("rest mass") everywhere in physics, the usage of m₀ (as notation) or a relativistic mass are deprecated.

No.

When dealing in relativity, m is relativistic mass. As in E = mc².
Rest mass is then distinguished by giving it the symbol m_0.. Check any textbook, e.g. Weidner and Sells.

Obviously, in dealing with Newtonian mechanics, m is rest mass.

 

[gsurfer04]

No.

When dealing in relativity, m is relativistic mass. As in E = mc².
Rest mass is then distinguished by giving it the symbol m_0.. Check any textbook, e.g. Weidner and Sells.

Obviously, in dealing with Newtonian mechanics, m is rest mass.

mfb is right, the relativistic correction is on the momentum, not the mass.

 

[rude man]

mfb is right, the relativistic correction is on the momentum, not the mass.

That's what I said, and why I insisted (and insist) that the mass here be denoted m₀ instead of m, to clarify the fact that it is not the relativistic but the rest mass.

The correct expression thus becomes, as I said, E² = (pc)² + (m₀c²)².

 

[gsurfer04]

There is no such thing as relativistic mass.

 

[rude man]

There is no such thing as relativistic mass.

Gee, I hope you're kidding ...

 

[Dick]

Gee, I hope you're kidding ...

Of course there is a notion of relativistic mass. It just isn't used much anymore. So more modern books just use m as rest mass and don't talk about 'relativistic mass'. It causes more confusion than it is useful.

 

[rude man]

Of course there is a notion of relativistic mass. It just isn't used much anymore. So more modern books just use m as rest mass and don't talk about 'relativistic mass'. It causes more confusion than it is useful.

Thanks for the clue-in, Dick. Amazing!

So you can't say m = γm₀ any more? So how do they e.g. write kinetic energy? Used to be mc² - m₀c². What is it now?

Oh my.

So when the OP wrote E2 = (pc)2 + (mc2)2, that was right because m is now rest mass? And what symbol do you use for (may God forgive me for saying it) relativistic mass?

And E = mc² is now "energy = rest mass times c²"?

I am impressed. I hope Einstein knows about this ...

 

[Dick]

Thanks for the clue-in, Dick. Amazing!

So you can't say m = γm₀ any more? So how do they e.g. write kinetic energy? Used to be mc² - m₀c². What is it now?

Oh my.

So when the OP wrote E2 = (pc)2 + (mc2)2, that was right because m is now rest mass? And what symbol do you use for (may God forgive me for saying it) relativistic mass?

And E = mc² is now "energy = rest mass times c²"?

I am impressed. I hope Einstein knows about this ...

Kinetic energy is just $mc^2 (\gamma-1)$. I, personally, have always taken E=mc^2 to mean equivalence between rest mass and energy. Energy in a boosted frame is $\gamma mc^2$. I didn't make the choice to avoid the use of 'relativistic mass'. But that seems to be the way the market is choosing. I don't think Einstein would be upset. It's just a change in notation.

 

[rude man]

Kinetic energy is just $mc^2 (\gamma-1)$. I, personally, have always taken E=mc^2 to mean equivalence between rest mass and energy. Energy in a boosted frame is $\gamma mc^2$. I didn't make the choice to avoid the use of 'relativistic mass'. But that seems to be the way the market is choosing. I don't think Einstein would be upset. It's just a change in notation.

I am rendered speechless. And I do think Einstein would seriously not be pleased. But OK, we bend with the wind ...

 

[Dick]

I am rendered speechless. And I do think Einstein would seriously not be pleased. But OK, we bend with the wind ...

I don't see why. You sprinkle a few more gamma's around. Drop some m0's. It's still just classical special relativity. There's no conceptual change, just notation. If Einstein were rendered speechless by this, I'd say he had a pretty thin skin. :)

 

[gsurfer04]

If you think about it, an objects mass cannot increase as it's an extensive property based on its internal structure, which doesn't change with velocity.

 

[mfb]

@rude man: There is no need to get sarcastic. Check any recent (after 1980-1990) publication, you will not find the concept of a relativistic mass there.

 

[gsurfer04]

I'm not being sarcastic, it's just a reason why relativistic mass doesn't make sense.

 

[rude man]

No more F = d/dt (mv) (which was correct even relativistically)..
No more centripetal force = mv^2/R.
And, best of all ... no more p = mv!

All those traditional formulas rendered erroneous ... reminds me of the 'improvements' in Windows!

OK, I promise, this is my last comment on the subject. Apparently some folks are not happy with them.

 

[mfb]

$F=\frac{dp}{dt}$
Centripetal force $\frac{\gamma m v^2}{R}$
$p=\gamma m v$
Where is the issue?

@gsurfer04: I did not mean you in my post.

 

[voko]

There are a few papers by Okun on the whole "relativistic mass" issue. Quite an interesting read.

 

[rude man]

$F=\frac{dp}{dt}$
Centripetal force $\frac{\gamma m v^2}{R}$
$p=\gamma m v$
Where is the issue?

@gsurfer04: I did not mean you in my post.

You just stated 'where is the issue'. All the traditional formulas now have to be changed by multiplying by gamma.

And I wasn't sarcastic. Just flabbergasted at the way the physics community has dumbed-down.

 

[Dick]

You just stated 'where is the issue'. All the traditional formulas now have to be changed by multiplying by gamma.

And I wasn't sarcastic. Just flabbergasted at the way the physics community has dumbed-down.

I thought you were done. You promised. It's a frigging pedogical point. Many teachers of relativity feel that it causes more confusion than artificially preserving Newtonian expressions buys you. That's all. It makes the distinction between Newtonian mechanics and relativistic mechanics more explicit. It's the same damned theory. If you think this is 'dumbing down' you should probably take your flabbergastion elsewhere. It's just a change in the emphasis of notation.

 

[mfb]

To see how misleading it can be to try to keep Newtonian formulas:

No more F = d/dt (mv) (which was correct even relativistically)

m there now m(v), and in general F and dv/dt point in different directions - something really counterintuitive if you look at that equation. m is a scalar, but still the two vectors in that equation are not parallel to each other.

 

[rude man]

To see how misleading it can be to try to keep Newtonian formulas:

... in general F and dv/dt point in different directions.

They do? News to me and I would think to Mr. Newton also.

- something really counterintuitive if you look at that equation. m is a scalar, but still the two vectors in that equation are not parallel to each other

It's counterintuitive that v and dv/dt can point in opposite directions? Like a car moving at v and letting up on the gas?

And what about centripetal force? Is it counterintuitive that v and dv/dt are non-parallel?

Guess I'm just not appreciating the New Order here - again.

Oh how things are changing in the world of physics since my day ...

@Dick - sorry, yes, I broke my word, but as long as I get ripostes I feel obliged to answer them. I promise not to initiate posts on this subject ever again.

 

[mfb]

They do? News to me and I would think to Mr. Newton also.

Yes, I think special relativity would be new for Isaac Newton.

And what about centripetal force? Is it counterintuitive that v and dv/dt are non-parallel?

No, that formula does not uses vectors anyway (or you have to rewrite it - but v is squared anyway, so its direction does not matter).