Wavefunction of relativistic free particle

[I_wonder]

Could anyone please help with the following, rather unusual, query?

I know that for spin 0 bosons, the Klein Gordon equation gives solutions that are similar to the solutions of the Schrödinger equation for a non-relativistic free particle, the only difference being that the energy used when calculating the wave frequency (E/h-bar) is the relativistic energy m*c-square*gamma.

I don't know how the wavefunctions look for spin 1 or spin 1/2 particles. What I would like to know is if there is any kind of particle for which one gets a wavefunction where the frequency is different then m*c-square*gamma/h-bar? Also, is it possible to verify these things experimentaly? I mean, can an experiment get an eigenstate and measure both the energy and the wave frequency? Has that been done for relativistic particles? Have any discrepancies been found? Where could I find such data?

Thanks!

 

[Zacku]

Could anyone please help with the following, rather unusual, query?

I know that for spin 0 bosons, the Klein Gordon equation gives solutions that are similar to the solutions of the Schrödinger equation for a non-relativistic free particle, the only difference being that the energy used when calculating the wave frequency (E/h-bar) is the relativistic energy m*c-square*gamma.

I don't know how the wavefunctions look for spin 1 or spin 1/2 particles. What I would like to know is if there is any kind of particle for which one gets a wavefunction where the frequency is different then m*c-square*gamma/h-bar? Also, is it possible to verify these things experimentaly? I mean, can an experiment get an eigenstate and measure both the energy and the wave frequency? Has that been done for relativistic particles? Have any discrepancies been found? Where could I find such data?

Thanks!

As far as I know, for any relativistic free particle, the frequency $$\omega$$ that appears in $$e^{-i \omega t}$$ when regarding the evolution of a steady state (i.e. an eigenstate of the Hamiltonian operator) must be equal to $$\sqrt{m^2c^4+p^2c^4}/\hbar$$ because the hamiltonian is the generator of time translations.
However, the relation between $$\omega$$ and the wave vector $$\vec{k}$$ will be spin dependent.

P.S1 : Unfortunately the word "wave function" is no more appropriate to qualify a state in relativistic quantum theory. Results can be obtained with this point of vue only for the Klein-Gordon equation and for the Dirac equation (i.e. for a spin zero particle and a spin 1/2 particle) but they lead to fondamental problems that necessite a new approach of quantum mechanics ; that is quantum field theory.

P.S2 : Excuse me for the english, I live in France and as everybody knows french people are pretty bad in english :) .

 

[I_wonder]

As far as I know, for any relativistic free particle, the frequency $$\omega$$ that appears in $$e^{-i \omega t}$$ when regarding the evolution of a steady state (i.e. an eigenstate of the Hamiltonian operator) must be equal to $$\sqrt{m^2c^4+p^2c^4}/\hbar$$


Thankyou Zacku! Your english is just fine. It's not my language, either.

I realize of course that I will not be able to say anything inteligent about this subject before I study QFT, which I hope to do next year. What I'm trying to find now, for my own peculiar reasons, is if there are any experimental results that might suggest that some correction to the frequency may be required in the case of a free relativistic particle, that is, that the dependence of the frequency on the energy may be more complicated.

Thanks.

 

[Zacku]

I realize of course that I will not be able to say anything inteligent about this subject before I study QFT, which I hope to do next year. What I'm trying to find now, for my own peculiar reasons, is if there are any experimental results that might suggest that some correction to the frequency may be required in the case of a free relativistic particle, that is, that the dependence of the frequency on the energy may be more complicated.
Thanks.

Actually wathever the formalism (i mean the "classical" one or the "field" one) any states in QM has to satisfy the equation of evolution :
$$i \hbar \frac{d}{dt} |\psi(t) \rangle = \hat{H} |\psi(t)\rangle$$
You know that, if $$|\psi(t)\rangle$$ is an eigenket of $$\hat{H}$$ so that :
$$\hat{H} |\psi(t)\rangle = \hbar \omega |\psi(t) \rangle$$
you get obviously :
$$|\psi(t) \rangle = e^{-i \omega t} |\phi \rangle$$
Thus $$\omega$$ is directly linked to the eigenvalue of the hamiltonian in QM, in non relativistic QM but also in realtivistic QM.

So, if there exists an experiment which result is that $$\omega$$ is different from the energy, then QM is false and we have to find something else...

 

[akhmeteli]

Could anyone please help with the following, rather unusual, query?

I know that for spin 0 bosons, the Klein Gordon equation gives solutions that are similar to the solutions of the Schrödinger equation for a non-relativistic free particle, the only difference being that the energy used when calculating the wave frequency (E/h-bar) is the relativistic energy m*c-square*gamma.

I don't know how the wavefunctions look for spin 1 or spin 1/2 particles. What I would like to know is if there is any kind of particle for which one gets a wavefunction where the frequency is different then m*c-square*gamma/h-bar? Also, is it possible to verify these things experimentaly? I mean, can an experiment get an eigenstate and measure both the energy and the wave frequency? Has that been done for relativistic particles? Have any discrepancies been found? Where could I find such data?

Thanks!

For what it's worth, http://www.ensmp.fr/aflb/AFLB-301/aflb301m416.pdf . The experiment was performed with weakly relativistic electrons (80 MeV). The article is unusual, as it was published in 2005 in a rather obscure journal, whereas the experiment seems to have been performed in 1988, and at that time the authors published some of its results in Phys. Rev. B (I have not read that article). So use your judgement.

 

[lbrits]

For somewhat unrelated reasons, I invite you to write the Klein-Gordon equation in lightcone coordinates. The result is interesting

 

[I_wonder]

Thanks to everyone!

I_wonder...

 

[schieghoven]

I don't know how the wavefunctions look for spin 1 or spin 1/2 particles.

Peskin and Schroeder, An Introduction to Quantum Field Theory, Chap. 3, give a very accessible description of the free-particle solutions of the Dirac equation (the governing equation for relativistic spin-1/2 particles). It's one of the classic texts for relativistic quantum mechanics and field theory, very well written, and I recommend it wholeheartedly!

One of the subtleties of the Dirac equation (and other relativistic equations) is that it admits both positive and negative-frequency solutions, which correspond to electron and positron wavefunctions respectively.