Interpetation of wavefunction - atom

[liquidFuzz]

It's now years since I studied this, if anyone could help me remember.

If I look at a hydrogen atom and it's shells. In the ground state there's 1s, the wave function is then:
[itex]\Psi_{nlm}(r,\theta,\phi) = Y_{lm}(\theta,\phi)R_{nl}(r) = Y_{00}R_{10}[/itex]

[itex]Y_{00} = \frac{1}{\sqrt{4\pi}}[/itex]

[itex]R_{10} = 2 \left(\frac{Z}{a_0}\right)^{3/2} e^{-\frac{\rho}{2}}[/itex]

With hydrogen's single proton and n = 1 etc.


[itex]\Psi_{100} = \frac{1}{\sqrt{\pi}} \left(\frac{1}{a_0}\right)^{3/2} e^{-\frac{r}{a_0}}[/itex] - I just looked up this in Physics Handbook. Now I can't remember how to interpret this.

My stumbling interpretations
I the Bohr model there's fixed energy levels, but this calculation shows a high likeliness of the atom being in the centre. Is this right? Further, is the model predicting a likely radial position for a electron sitting in the 1s energy state?

 

[jtbell]

Now I can't remember how to interpret this.

The square of this expression gives the probability distribution for the electron being at location [itex](r, \theta, \phi)[/itex] with respect to the nucleus. Obviously in this case (n = 1, l = 0, m = 0) it doesn't actually depend on either of the angles; but for other combinations of n, l, m it does.

For this case (the ground state) the most likely location is in fact at the center of the atom, i.e. inside the nucleus, because this is the location that gives you the maximum value for the probability distribution.

 

[vanhees71]

The Bohr model is wrong in many respects. Particularly it's wrong to still learn about it in the 21st century. Better forget everything about the Bohr model first and then remember the modern version of quantum mechanics which has been discovered in 1925/26 by Heisenberg, Born, Jordan, Dirac, and Schrödinger.

Quantum mechanics can be described in the position representation as representing the states of the electron in the hydrogen atom by a square-integrable wave function. In the context of atomic physics most important states are the energy-eigen states, which (according to the dynamics of quantum theory) represent stationary states of the electron. To determine the energy states completely, in addition to the energy eigenvalue you need also to specify the angular-momentum state. This is given by some quantum numbers, [itex]l[/itex] and [itex]m[/itex]. The eigenvalues of the modulus of the orbital angular momentum are given by [itex]l(l+1)\hbar^2[/itex], where [itex]l \in \{0,1,2,\ldots \}[/itex], and then for each [itex]l[/itex] the "magnet-quantum number" can run in [itex]m \in \{0,\pm 1,\pm 2,\ldots,\pm l \}[/itex], and the [itex]z[/itex] component of orbital angular momentum is given by [itex]m \hbar[/itex]. The lowest energy state has necessarily [itex]l=m=0[/itex], and thus is non-degenerate (I leave out the discussion of spin here!).

Now, according to the funcamental rules of quantum theory, the squared modulus of the wave function is the position-probability distribution for the electron (Born's probability interpretation), if it is in the state discribed by this wave function. In your case, you are looking at the ground state, i.e., the energy eigenstate with the lowest possible energy value, and this is the most stable stationary state of the atom.

Now, if you plot the position-probability distribution according to your ground-state wave function, you'll find it to be spherically symmetric (that's what [itex]l=0[/itex] is telling you) and having its maximum in the center (i.e., at the nuclues). The Bohr radius of the first orbit is only the average distance of the electron from the nucleus, and the atom is spherical and not a little disk as suggested by the Bohr model.

The probablity-density distributions, you can fine nicely depicted in many books on introductory quantum mechanics. Nice colorful pictures can be found in Wikipedia, where you also find a nice description of the quantum theory of the hydrogen atom:

http://upload.wikimedia.org/wikipedia/commons/7/70/HAtomOrbitals2.png
http://en.wikipedia.org/wiki/Hydrogen_atom

 

[liquidFuzz]

jtbell - Thanks for shedding a light on this.

vanhees71 - Thanks for the I'll redeem this fella approach in your vigorous post. You're right about the Bohr model and the quantum model. But regarding the tutorial aspect of learning obsolete models are completely wrong. The atom as we see it underwent big changes, one step lead to an other. Te Bohr model was a big leap, for man kind. Ops, I got carried away in the last sentence. ;-)

 

[vanhees71]

Well, the question, whether one should learn/teach historically important but outdated views on nature in the physics curriculum. On the one hand, I'm strongly objecting the historical approach to teach physics. It's misleading to teach a beginner in quantum theory Bohr's and Sommerfeld's model of atoms. As much misleading is it to teach students the "relativistic-mass" concept or that temperature depends on the motion of the observer wrt. the body, whose temperature one likes to describe etc. although these were well founded ideas in the early times of quantum mechanics and relativity. The modern theories are better to comprehend and have at the same time a broader range of applicability since they represent the progress in our understanding of the fundamental laws of nature, and it does not make much sense to teach first some outdated points of view that lead to more confusion than they help to comprehend the subject.

On the other hand, I love history of science since it is a very interesting subject. To figure out, how physics came to the knowledge about the fundamental laws of nature is not only very fascinating but can also help to understand better the more modern points of view, but as I said above, I think, one should present physics not in a historical but in a logical order. Of course, a good physics course also includes some historical context, but this should be presented after the students have learned about the most logical and comprehensible models first. Then it is the more fascinating to learn how much effort of the greatest minds in history went into the world view we have today.

I think the best example of this approach to teach (theoretical) physics are the textbooks by Steven Weinberg (e.g., his 3 volumes on quantum field theory).

 

[liquidFuzz]

vanhees71 - yes, something along those lines.

On topic
Lets say my precious hydrogen atom finds a friend and becomes: [itex]H_2^+[/itex]

To find out how they behave, or bond to each other I use the wave model.
[itex]\Psi = \frac{1}{\sqrt{2}}(1s_1 + 1s_2)[/itex] and [itex]\Psi = \frac{1}{\sqrt{2}}(1s_1 - 1s_2)[/itex]

I actually feel somewhat unsure of what the minus tells me. Up wave down wave..? Further, how do I determine wish states are possible?