Electron in Hydrogen Atom - De broglie wavelength

Petrushka

I had a physics exam today in which we were presented with a model of a hydrogen atom with a single electron orbiting a single proton. We were told the radius of the "orbit" of the electron, and subsequently had to calculate the electrostatic force between the proton and the electron and the speed of the electron.

Then for the last part of the question, we had to calculate the de broglie wavelength of the electron, and then the ratio of this wavelength to the circumference of the electron's orbit - i got the answer to be 1.0 to 2sf.

Assuming this is the correct answer (which it may well not be), it seems to be a like a very interesting result. Can somebody perhaps explain to me if there is an underlying reason (which, if it exists, I assume to be quantum mechanical), as to why this is the case.

Thanks.

ZapperZ

What you found was the fundamental harmonic of the "standing wave" corresponding to the ground state. Only multiples of the wavelengths can exist over a fixed circumference.

Zz.

jtbell

It is correct, and it has great historical significance. Your textbook surely has some discussion of it. Briefly, de Broglie showed that if you assume that the electron is associated with a wave that has wavelength [itex]\lambda = h / p[/itex], and that the allowed "orbits" in a hydrogen atom correspond to standing waves wrapped around in a circle so that the circumference is an integral multiple of the wavelength, you can derive the observed energy levels! This turned out to be the starting point for modern quantum mechanics. De Broglie won the Nobel Prize for this.

dextercioby

Can you prove that ? I have reasons to believe that the first ever treatment of the H atom in the '20-s was due to Heisenberg and Pauli within the formalism of matrix mechanics,in the second half of 1925.

Daniel.

George Jones

I don't think that Jon is saying that de Broglie's work completed quantum theory. Bohr originated his model of the hydrogen atom in 1913, and de Broglie postulated his matter waves some time around 1923/24. By the end of 1924, physicists had probably combined these developments along the lines given by Jon. However, this is a collection of somewhat ad hoc principles that gives the correct energy levels for hydrogen, that, if I remember correctly, has some inconsistencies, and that does work for more general systems.

By the end of 1925, Heisenberg had a fairly complete treatment of matrix mechanics that, in early 1926, Pauli used to solve for the energy spectrum of hydrogen. Schrodinger used his wave to arrive at the energy spectrum a few days after Pauli. Dirac also used matrix mechanics to obtain the energy levels of hydrogen a few days after Pauli. Wave mechanics, matrix mechanics, and the relationship between the two constitute a general framework that is much more than a collection of ad hoc principles.

My version of the history of quantum theory may have some mistakes. A definitive treatment is given by the multi-volume set, The Historical Development of Quantum Theory, by Mehra and Rechenberg.

The story behind Schrodinger's solution has some interesting personal details. For Christmas vacation of 1925, Schrodinger went to a chalet in the Alps, and, instead of taking his wife, he took his mistress. He returned home with his equation in hand. He separated variables, solved the angular equation, but could not solve the radial equation, so he sought help from Hermann Weyl, one of the top mathematicians of the day. Weyl also happened to be Schrodinger's wife's lover.

Regards,
George

jtbell

Right. The context to de Broglie's idea is the Bohr model of the hydrogen atom, with its discrete orbits. Bohr postulated that the the electron has specific allowed orbits because the orbital angular momentum is quantized: [itex]L = mvr = n \hbar[/itex]. Bohr simply postulated this. De Broglie's hypotheses were an attempt to explain this more "deeply" in terms of wave behavior. This got people thinking about applying wave mechanics to the hydrogen problem, and Schrödinger came up with what we now know as non-relativistic quantum mechanics.

So I think a reasonable statement would be that de Broglie's idea was an inspiration for Schrödinger's QM.

George Jones

A very direct inspiration - Debye suggested to Schrodinger (After Schrodinger gave a seminar on de Broglie's work???) that he look for the wave equation for de Boglie's matter waves. Only a few weeks passed between this suggestion and Schrodinger's discovery of his infamous equation.

Regards,
George

drv

Hello Petrushka,

Please define your units. I am not familiar with sf units.

jtbell

I doubt that Petrushka will respond to your question because this thread dates from June 2005, and Petrushka's last posting on Physics Forums was in April 2006 according to her(?) profile.

Nevertheless, I think that "1.0 to 2sf" means "1.0 to 2 significant figures", that is, she calculated the result, rounded it off to two significant figures, and got 1.0.

drv

My question is about the speed of the electron in various stable states and the size(s) of the hydrogen atom. Bohr's atom assumes ever increasing sizes, which does not correlate with measurements. Orbitals do not fully define the movements of the electron, and there can be many possible orbits for an orbital. This leaves many questions unanswered.