# How does QM solve the bohr atom problem?

• welkin
In summary, the original electron orbiting around the nucleus like planets around the sun model was rejected partly because the radiation (energy loss) was not taken into consideration. I thought the new QM model was supposed to fix it. However with the hydrogen atom we learned, there are still these stationary eigenstates of energy. Even though we don't have a point electron anymore (replaced with a wave of probabibility), doesn't the electron still have angular momentum/acceleration/etc which allows it to radiate? If this is true, then how can there be any stationary states? Why don't the solutions to the hydrogen atom we learn involve a decay in energy? Thanks a
welkin
I have only taken QM for a semester so I'm sure there are a lot of gaps in my understanding. Here is my question: The original electron orbiting around the nucleus like planets around the sun model was rejected partly because the radiation (energy loss) was not taken into consideration. I thought the new QM model was supposed to fix it. However with the hydrogen atom we learned, there are still these stationary eigenstates of energy. Even though we don't have a point electron anymore (replaced with a wave of probabibility), doesn't the electron still have angular momentum/acceleration/etc which allows it to radiate? If this is true, then how can there be any stationary states? Why don't the solutions to the hydrogen atom we learn involve a decay in energy? Thanks a lot in advance!

Hi Welkin,

The prediction of radiation from electron orbits comes from the classical dynamics of charged particles. If you start from quantum mechanics instead of classical electrodynamics you don't even encounter the problem. So I suppose the answer to your question is that QM "fixes" the problem by making it a non-issue.

Hmm... does that mean there is no radiation from the electron in the QM model?
And from that shall one draw the conclusion that the effective magnetic field created by the electron does not change?
Somewhere I must be confused... v_v

welkin said:
I have only taken QM for a semester so I'm sure there are a lot of gaps in my understanding. Here is my question: The original electron orbiting around the nucleus like planets around the sun model was rejected partly because the radiation (energy loss) was not taken into consideration. I thought the new QM model was supposed to fix it. However with the hydrogen atom we learned, there are still these stationary eigenstates of energy. Even though we don't have a point electron anymore (replaced with a wave of probabibility), doesn't the electron still have angular momentum/acceleration/etc which allows it to radiate? If this is true, then how can there be any stationary states? Why don't the solutions to the hydrogen atom we learn involve a decay in energy? Thanks a lot in advance!

1.Bohr-Sommerfeld's theory of the H-atom is called "old QM"."New" (1925-->) QM doesn't "solve the problems of the H atom".

2.We DO have a pointlike electron.

3.Measurable quantities for the H atom in a stationary state are time independent.

Daniel.

welkin said:
Hmm... does that mean there is no radiation from the electron in the QM model?

Unless the Hamiltonian is perturbed,no.

welkin said:
And from that shall one draw the conclusion that the effective magnetic field created by the electron does not change?
Somewhere I must be confused... v_v

Guess what ?It doesn't.But spin-orbit coupling is a relativistic effect.

Daniel.

dextercioby said:
Unless the Hamiltonian is perturbed,no.

Guess what ?It doesn't.But spin-orbit coupling is a relativistic effect.

Daniel.
isn't all magneticism relativistic effect anyway?
so how does the magnetic field from the Bohr model differ from the one used in QM?

Magnetism derives from angular momentum,not necessarily spin (orbital,too).So it's not a relativistic effect.Pauli's Hamiltonian can accommodate it pretty niftly.

What magnetic field in the Bohr model ...?

Daniel.

Hydrogen Atom Stability

Excellent question! QM avoids the issue by discounting classical physics and Maxwell's predictions, which is an extraordinary claim which requires convincing evidence. I suggest a new theory is needed to explain the stability of the hydrogen atom. Continue to ponder this question and you will eventually resolve this issue.

rlduncan said:
I suggest a new theory is needed to explain the stability of the hydrogen atom.

Did I misunderstand you or do you literally mean what you're saying here?

How does QM solve the photoelectric effect and the question of a cut-off in light intensity?

All possible electron-photon interactions are handled remarkably by QED...

Daniel.

welkin said:
However with the hydrogen atom we learned, there are still these stationary eigenstates of energy. Even though we don't have a point electron anymore (replaced with a wave of probabibility), doesn't the electron still have angular momentum/acceleration/etc which allows it to radiate? If this is true, then how can there be any stationary states? Why don't the solutions to the hydrogen atom we learn involve a decay in energy? Thanks a lot in advance!

This is a classic one. Conform classical physics, stable atoms would not exist because the electron would radiate; lose energy and spiral inward towards the nucleus. Eventually the electron will crash onto the nucleus. What does this tell you ? Well, the fact that classical physics is NOT able to explain atomic scaled phenomena. This is the biggest thing that you need to realize. Then, mr Bohr came with his electrons moving on shells. The angular momentum is quantized as well as the energy. When an electron is on a certain shell, it does not radiate, it has one constant energy. Radiation only occurs when the electron goes from one energy level to the other (ie absorption or emission of EM-radiation)...Then, the wavefunctions came and the particle-wave duality. Electrons do not move on definite circular orbits but we can only talk about a region in space where we have a certain probability of finding an electron. These are the orbitals...

regards
marlon

I am utterly confused. So I'll just ask one more time and if I can't get it, then so be it... v_v

So... even though we only have probability when we're dealing with QM, the probability cloud still moves, and the electron would still have some expectation value for velocity/momentum/angular momentum, etc... so why wouldn't this produce a current and changing magnetic field --> radiation?

welkin said:
So... even though we only have probability when we're dealing with QM, the probability cloud still moves,

No. The probability distribution for the electron for a hydrogen atom in the ground state, for example, is

$$P(r) = \frac {1}{\pi a_0^3}e^{-2r/a_0}$$

where $a_0$ is the radius of the ground-state orbit in the Bohr model. (Of course, there's no such thing as an "orbit" here, but it's a convenient constant for simplifying formulas.)

Note there's no time dependence, so the "probability cloud" doesn't move. or change its shape. Nor do any of the other energy eigenstates have any time dependence in their probablity distributions.

However, if you describe an atom that is in the process of making a transition between two energy eigenstates, by a wave function that is a linear combination (a "mixture," so to speak) of the two states, that wave function is not a stationary state. The probabilty distribution for such an "in-between" state does have a time dependence. In fact, it oscillates in shape, with a frequency equal to that of the photon that is emitted in the transition!

The basic QM premise regarding electron energies within the atom are that the electron may possesses energy $$E_{1}$$ or energy $$E_{2}$$ but no intermediate energies – all values of energy E satisfying the condition $$E_{1}< E < E< E_{2}$$ are forbidden. This caused problems : With regard to the jump of an electron from one orbit in the atom to another. Whatever the speed of the transition it has to last for some finite period of time (otherwise it would be a violation of the basic requirements of the theory of relativity ). But then it is hard to understand what the energy of the electron should be doing during this intermediate period – the electron no longer occupies the oribit corresponding to $$E_{1}$$ and has not yet arrived at the orbit corresponding to $$E_{2}$$ . This caused Schrodinger , the originator of the probability wave to make his famous remark about ”these damned quanum jumps”
The problem was eventually solved by resorting to the Heisenberg Uncertainty relation $$\Delta {E}\Delta {t}\leq {h}$$. This means that the possibility exists for virtual transitions to take place. According to the Uncertainty relation it is possible for the electron to go from$$\E_1 to \E_2$$, without getting any external energy , provided it immediately returns to$$\E_1$$. Such a journey is possible if its duration \Delta{t} is such that the inequality$$h/\Delta {t} > \E_{1} - \E_{2}$$, because in this case the uncertainty in the energy of the electron is greater than the difference in the energies of the levels under consideration.

inha said:
Did I misunderstand you or do you literally mean what you're saying here?

You did not misunderstand me. The hydrogen atom's stabililty must be explained based on sound physics. This new theory should be as simple as the Bohr Atom but takes into account the wave properties of the electron.

welkin said:
I am utterly confused. So I'll just ask one more time and if I can't get it, then so be it... v_v

So... even though we only have probability when we're dealing with QM, the probability cloud still moves, and the electron would still have some expectation value for velocity/momentum/angular momentum, etc... so why wouldn't this produce a current and changing magnetic field --> radiation?

Because Heisenberg's uncertainty principle trumps Maxwell's equations.

Electrons in atoms don't continuously emit radiation (even though Maxwell's equations say they should) because quantum theory (which is more accurate than classical EM theory) says they shouldn't.

One should simply stop trying to apply classical conceps to quantum systems, because
it doesn't work. The only definite thing that an electron has is a wavefunction, and from that one finds observables. The magnitude squared (|psi|^2)of the wavefunction gives it's probability of being found at a given position, and other operators give other things. In a stationary state, the expectation value of energy has no time dependence, so it's energy stays constant in time. Also, in a stationary state, although the wave function may have time depedence, the magnitude squared does not, and so the probability cloud doesn't move.

This isn't true of all states which solve schrodingers equation, but as long as the hamiltonian (operator which describes energy) is independent of time, so too will the probability cloud, and so the state doesn't change with time.

forgive me if I'm wrong (i'm still a junior in high school w/ only physics B under my belt..)

i thought the problem was resolved through wave duality?

because if we consider the electron to be a wave.. then it must go around the center of its orbit with a circumference = n * wavelength... where n = any integer > 0 (eventually became the principal quantum level)...

thus the orbits are quantisized and the wave cannot travel at any other circumference (and thus radius) otherwise it will destructively interfere with itself and kill itself off...

Hessam said:
forgive me if I'm wrong (i'm still a junior in high school w/ only physics B under my belt..)

i thought the problem was resolved through wave duality?

because if we consider the electron to be a wave.. then it must go around the center of its orbit with a circumference = n * wavelength... where n = any integer > 0 (eventually became the principal quantum level)...

thus the orbits are quantisized and the wave cannot travel at any other circumference (and thus radius) otherwise it will destructively interfere with itself and kill itself off...

This was DeBroglie's reasoning, and you have expressed it well. It is essentially correct, and he won a Nobel Prize for it, but note that it assumes one fixed wave length at a time. What if the wave length can change? what if it is dynamic , meaning it depends on forces acting in time, what happens then? This is the problem that Schroedinger set himself. To represent dynamic quantities, physicists use differential equations, equations between rates of change. And Schroedinger derived his famous Schroedinger equation for a particle with a variable, dynamic, wave length.

Thus the reasoning you have seen here doesn't really contradict DeBroglie, it more accurately extends it to the general case.

So... even though we only have probability when we're dealing with QM, the probability cloud still moves, and the electron would still have some expectation value for velocity/momentum/angular momentum, etc... so why wouldn't this produce a current and changing magnetic field --> radiation?

(This all must be qualified with "If I understand correctly")

The cloud does not move -- the probability of finding an electron at your favorite point does not vary with time, and the probability of finding the electron with your favorite momentum does not vary with time. The net effect is that the charge and current distributions are constant.

The expected value for the momentum, incidentally will be zero. (The expectation for the magnitude of momentum might not be zero, but that's a different question all together!)

But that doesn't matter -- radiation only occurs when the charge and/or current distributions are nonconstant. (For example, a uniform current running through a closed loop does not radiate!)

Electrons in atoms don't continuously emit radiation (even though Maxwell's equations say they should) because quantum theory (which is more accurate than classical EM theory) says they shouldn't.

I thought Maxwell's equations remained unchanged in QM?

Yes,it's true.Radiation is treated classically,both in nonrelativistic QM and in Dirac's relativistic theory.

Daniel.

rlduncan said:
You did not misunderstand me. The hydrogen atom's stabililty must be explained based on sound physics. This new theory should be as simple as the Bohr Atom but takes into account the wave properties of the electron.

Perhaps Bohmian Mechanics (i.e. "Bohm's theory" or the "de Broglie - Bohm interpretation") would be of interest to you. In that interpretation of QM, wave-particle duality is explained in the most obvious way possible: there's both a wave and a particle (with the wave, obeying Schroedinger's equation, guiding the trajectory of the particle according to a simple first-order DE). And it turns out that when the wave configuration is given by an energy eigenstate, the Bohmian particle velocity is zero -- so if the electron in a hydrogen atom is in one of its energy eigenstates, the electron particle is just sitting there, not moving.

This can be understood in terms of the quantum wave exerting a kind of repulsive force on the electron that just cancels the attractive Coulomb force and allows it to be in equilibrium (without orbiting and hence without centripetal acceleration). And *this* means that the stability of the atom (in regard to the radiation that would be expected classically from a centripetally accelerating electron) is no mystery. There is no radiation because the electron doesn't move.

See

http://plato.stanford.edu/entries/qm-bohm

for more details on Bohm's alternative to the standard (Copenhagen) way of understanding quantum theory.

## 1. What is the Bohr atom problem?

The Bohr atom problem refers to the inability of classical physics to fully explain the behavior of atoms. Classical physics predicted that electrons should orbit the nucleus in a stable, circular path, but this was not observed in experiments.

## 2. How does quantum mechanics solve the Bohr atom problem?

Quantum mechanics introduces the concept of discrete energy levels for electrons, rather than continuous orbits. This explains why electrons do not spiral into the nucleus, as classical physics predicted. It also allows for the observation of phenomena such as electron tunneling and wave-particle duality.

## 3. What is the role of the Schrödinger equation in solving the Bohr atom problem?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes the behavior of particles at the atomic level. It takes into account the wave-like nature of particles and allows for the prediction of energy levels and probabilities of finding particles in certain states.

## 4. How does the uncertainty principle relate to the Bohr atom problem?

The uncertainty principle, a key principle in quantum mechanics, states that it is impossible to know both the position and momentum of a particle with absolute certainty. This helps to explain why electrons do not have specific, defined orbits around the nucleus, but rather exist in a cloud of probability.

## 5. What are some limitations of quantum mechanics in solving the Bohr atom problem?

While quantum mechanics has been successful in explaining many phenomena at the atomic level, it still has limitations. For example, it cannot fully explain the behavior of large or complex systems, and there are still unanswered questions about the nature of quantum mechanics itself, such as the measurement problem.

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