Is the Bohr model of the atom still valid in quantum mechanics?

[reilly]

I wouldn't know. I've never solved for R(r) inside the nucleus. Have you?

Zz.

Yes, in my nuclear physics course in grad school, with the assumption of a constant potential inside a spherical proton. It's basically a problem of matching inside -- free electrons -- with Coulomb wave functions outside. It's not that different than scattering from two radial square wells of different depths.

Regards,
Reilly Atkinson

 

[Sean Torrebadel]

I have a question related to this discussion. It seems that because quantum theory deals with the electrons position and momentum in a statistical and probability way, that the argument becomes that the electron is defined by the method.

What would happen to quantum's interpretation of the electron if someone came up with a way of reproducing the spectra of the elements using Bohrian terms, where each orbit of the electron is given a specific energy? So for instance, instead of a probability or smeared out version of the electron, what if someone came up with a method that treated the electron as an orbiting body- and it worked?

What would happen to the Copenhagen interpretation if Bohr's model, was modified, corrected for its inherent flaws, and adapted to explain the spectra of the elements? And don't just say that this has never been done, that the best of the best tried... I am asking in theory, what would happen to quantum's pre-emptory interpretation of the electron if a classical analog worked?

 

[Xeinstein]

The radius of the proton is estimated to be roughly 10^^ -13 cm, while a0 is roughly 10^^-8. If, for some reason, you wanted to compute the reverse beta decay matrix element, p + [e] --> n + v(neutrino), a reaction that cannot happen for a bound electron,[e].

As a first approximation, simply use the standard 1S wave function to compute the probability of finding the electron within the proton's radius, or whatever multiple thereof you wish. You find, ballpark, this probability to be roughly 10^^-15. So, even if it could happen, it won't. But, if you wanted to compute the reverse beta decay matrix element, you could legitimately consider the combined wave functions of the electron and proton as constant over a very small volume, and so forth.

ZapperZ:If there's a problem here, it's avoided discovery for almost a century of intense work in both nuclear and atomic physics.

What is the problem?
Regards, Reilly Atkinson

>> What is the problem?
The problem is this. When electron comes close to the radius of the proton (estimated to be roughly 10^^ -13 cm), it's speed will exceed the speed of light if non-relativistic quantum mechanics is used for calculation. That's nonsensical. So the Schroedinger equation won't work in this case

>> You find, ballpark, this probability to be roughly 10^^-15.

The ballpark number of probability (roughly 10^^-15) may be invalid

 

[Xeinstein]

Who told you it's not already "fallen"? What exactly is an electron in an atom? Did you know that, at least for the fundamental state of hydrogen atom, the electron has a non zero probability to be located in the nucleus? Teachers at school, as well as school books, don't always explain things correctly.

Can you tell me why the electron has a non zero probability to be located in the nucleus?
(at least for the fundamental state of hydrogen atom)

 

[Gokul43201]

Can you tell me why the electron has a non zero probability to be located in the nucleus?
(at least for the fundamental state of hydrogen atom)

Why should it not? When we solve the TISE, we only find zeros of the wavefunction at points (called nodes), or at regions beyond an infinite potential barrier (or at points at infinity). If you model the nuclear potential as finite everywhere, you will naturally get non-zero probabilities everywhere (except at nodes and points at infinity).

 

[Xeinstein]

Why should it not? When we solve the TISE, we only find zeros of the wavefunction at points (called nodes), or at regions beyond an infinite potential barrier (or at points at infinity). If you model the nuclear potential as finite everywhere, you will naturally get non-zero probabilities everywhere (except at nodes and points at infinity).

Can you tell me if the Schroedinger equation can still work in this case?
It won't faze you if the electron gets close to the proton, its speed may exceed the speed of light

 

[Gokul43201]

Its speed won't have to exceed c because of how the probabilities are distributed. There's only a very tiny probability that the electron exists inside the nucleus. The variance of the distribution is almost exactly the same as the variance you get from assuming a point-nucleus (which makes the RMS speed about 10^6 m/s). It's only when you force the electron probability to be almost entirely crammed within the nucleus that you start running into this problem.

None of this says that the probability for v>c is zero. But then, we're only doing non-relativistic calculations, so that's expected.

 

[lightarrow]

Can you tell me if the Schroedinger equation can still work in this case?
It won't faze you if the electron gets close to the proton, its speed may exceed the speed of light

The electron is not pointlike when is bound in the atom.

 

[Xeinstein]

The electron is not pointlike when is bound in the atom.

Who told you that electron is point-like? The fact that it's not point-like is well-known
The electron (particle or wave) must move inside atom, and its speed can be calculated using momentum operator

 

[lightarrow]

Who told you that electron is point-like? The fact that it's not point-like is well-known
The electron (particle or wave) must move inside atom, and its speed can be calculated using momentum operator

And what value do you use as distance from the nucleus in that case? You use r or <psi|r|psi>?

 

[erastotenes]

You shouldn't take the "wave-nature" of the electron too litteraly. When we talk about particles and waves in QM we are really referring to classical analogies that are often convenient since they help us understand what is going on, it doesn't mean that an electron is a "wave" in the classical sense (waves in water etc); it simply means that electrons (and everything else) has wave-like (and at the same time particle-like) properties.
In the case of the bubble chamber it is probably more conventient to think of the electron as a particle since its particle-like properties "dominates" (i.e. it behaves more or less like a classical particle).

This can be quite confusing. However, it is important to remember that this confusion only arises because we are trying to describe QM phenomena- and the math that is needed to describe these phenomena- using analogies from our "classical" world.

I think the QM related conceptual problem is not merely rooted in the classicallly strange statement that "it has both wavelelike and particlelike properties but it is actually none of them", but the conceptual problem is that it gives us no criteria "under which objective physical conditions" it behaves like wave and "under which other physical conditions" it behaves like a particle.

By the way the math itself (namely the Schrödinger eq. for example) uses only the wave function not the particle like aspect. And the wave function in QM has no classical analogy (complex value,3n dimensional configuration space for n particles) at all.

 

[erastotenes]

I have a question related to this discussion. It seems that because quantum theory deals with the electrons position and momentum in a statistical and probability way, that the argument becomes that the electron is defined by the method.

What would happen to quantum's interpretation of the electron if someone came up with a way of reproducing the spectra of the elements using Bohrian terms, where each orbit of the electron is given a specific energy? So for instance, instead of a probability or smeared out version of the electron, what if someone came up with a method that treated the electron as an orbiting body- and it worked?

What would happen to the Copenhagen interpretation if Bohr's model, was modified, corrected for its inherent flaws, and adapted to explain the spectra of the elements? And don't just say that this has never been done, that the best of the best tried... I am asking in theory, what would happen to quantum's pre-emptory interpretation of the electron if a classical analog worked?

Bohm's theory where wave function guides the pointlike particle so that the particle velocity is quantum mechanical current density calculated from the wave function is actually such a theory. But it has its own shortcomings in my opinion.

 

[Archimedes546]

Who told you it's not already "fallen"? What exactly is an electron in an atom? Did you know that, at least for the fundamental state of hydrogen atom, the electron has a non zero probability to be located in the nucleus? Teachers at school, as well as school books, don't always explain things correctly.

Have you never heard of murphy's law?, Probability is no guarentee. Therefore, you absolutely CAN'T say the electron isn't in the nucleus based on probability.

 

[lightarrow]

Who told you it's not already "fallen"? What exactly is an electron in an atom? Did you know that, at least for the fundamental state of hydrogen atom, the electron has a non zero probability to be located in the nucleus? Teachers at school, as well as school books, don't always explain things correctly.

Have you never heard of murphy's law?, Probability is no guarentee. Therefore, you absolutely CAN'T say the electron isn't in the nucleus based on probability.

No, I didn't mean that, I meant we cannot think of the electron as an orbiting particle around the nucleus; however I have already explained my statements in a previous post.
(P.S. Did you really mean "you absolutely CAN'T say the electron isn't in the nucleus based on probability"?)