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I fail to understand what the electron clouds actually signify. Such as the $p$ orbitals, which have a dumbbell like shape. Now I am aware that they aren't actual trajectories of electrons, but what is their need? Does it indicate the highest probability of finding an electron?

And also, concerning the Zeeman effect, which I believe is the splitting of spectral lines in a static magnetic field (correct me if I am wrong), we have been told that this occurs due to separation of orbitals into their components. Assuming the $p$ orbital is affected by a magnetic field, it would split into $p_x, p_y$ and $p_z$ components, what does this mean, and how do these probabilities (if that assumption is right) actually exist in its essence?

I would really appreciate some insight into this relatively new topic.

And one last thing concerning, the Radial probability density and Radial probability distribution curves, what does the graph of $4\pi r^2R^2$ vs. $r$ plots attempt to impart? The local maximas represent the radius of the nodes in an atom, but how is that specific graph derived?

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    $\begingroup$ What makes you think that probability distributions are not of physical relevance. The central anti-node of s-orbitals is essentially responsible for setting the rate of electron-capture in those isotopes where it is energetically admitted. $\endgroup$ – dmckee May 8 '14 at 16:48
  • $\begingroup$ Apologies, what i meant by physical relevance was whether it was pertaining to a purely mathematical abstract existence or not. Because if they are actually just probabilities, then i do not understand how Splitting of spectral lines equate to division of orbital components, and i do not completely understand what the division signifies... $\endgroup$ – Mathbreaker May 8 '14 at 16:54
  • $\begingroup$ If you are just starting out at quantum mechanics I would suggest that you not insist on understanding the full implication of the math yet. The usual route to quantum intuition is by working through a lot of computations and seeing how the results reflect/are-reflected-in observations. These distributions are of real significance. $\endgroup$ – dmckee May 8 '14 at 17:01
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    $\begingroup$ splittin of lines in the magnetic field means that a spectral line of energy E, becomes two lines, of energy E1 and E2. That happens because without a magnetic field in the QM equation, the magnetic quantum number m is degenerate, i.e. whether the electron spin is up or down does not affect the energy level. The magnetic field changes the potential and since the spin sees the field there is a difference with or against its direction. There are two orbitals, instead of one, to give to probability distribution of the electron. No px py pz splits. $\endgroup$ – anna v May 8 '14 at 17:32
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On the electron 'cloud' pictures: what is usually shown is a surface where the probability (per unit volume) of finding an electron (the magnitude of the wave function squared) is constant. 'Inside' the volume, the probability (per unit volume) is higher in this case, so indeed these shapes show the volume where the probability of finding an electron is greater than a certain value ('high').


Concerning the Zeeman Effect, instead of saying that orbitals are 'split' into sub-orbitals (or subshells), one can also start from the fact that there are $n^2$ solutions ('steady states') for the $n$'th energy level (without an external field). Each of these solutions is characterized by the following quantum numbers:

(I'm ignoring the intrinsic spin of the electron here)

Without any external field, the energy levels of the wave functions only depend on $n$, they are said to be degenerate (multiple solutions / wave functions with the same energy).

Applying an external magnetic field, the energy of the wave functions changes. The additional energy depends on magnitude of the electrons angular momentum (orbital momentum along the $z$ axis which is proportional to $m$ plus intrinsic spin) and the strength of the magnetic field (taken to be parallel to the $z$ axis). In this case, the solutions belonging to the same $n$ have different energies, now also depending on their value $m$ in addition to $n$.

In practice, if you were observing spectral lines (differences of energies between solutions), you would see that when applying a magnetic field, the spectral lines suddenly split (and that when increasing the strength of the magnetic field, the split also increases etc.).


On your third question, if you want to get the probability as function of the radial distance to the center, you have to integrate over the other two variables, i.e. the polar angle $\theta$ and the azimuthal angle $\phi$ (in spherical coordinates). Multivariate calculus says that the coordinate transformation from Cartesian $(x,y,z)$ coordinates to spherical $(r,\theta,\phi)$ is as follows:

$$ \int_{x=0}^\infty \int_y \int_z |\psi|^2 dx dy dz = \int_{r=0}^\infty \int_{\theta=0}^{\pi} \int_{\phi = 0}^{2\pi} |\psi|^2 r^2 \sin\theta d\phi d\theta dr $$

At this link you can see that all the solutions to the Schrödinger Equation of the Hydrogen atom can be written as the product of a radial part $R(r)$ (depending on the quantum numbers $n$ and $l$) and a spherical harmonic $Y(\theta, \phi)$ (depending on the quantum numbers $n$ and $l$), i.e. $\psi = R(r) \cdot Y(\theta, \phi)$.

The fact that the wave functions can be written as such a product means that we can also write the integral as a product of an integral of $|Y|^2$ over ($\theta$,$\phi$) and a integral of $R^2(r)$ over $r$:

$$ \left(\int_{\theta=0}^{\pi} \int_{\phi = 0}^{2\pi} |Y(\theta,\phi)|^2 \sin\theta d\phi d\theta \right) \cdot \left(\int_{r=0}^\infty R^2(r) r^2 dr \right) $$

The left integral is one (this actually depends on the exact definition of $Y$) while in the right part you can recognize the radial probability distribution function to be integrated over (up to a constant factor, depending on how $Y$ and $R$ are normalized).

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Orbitals (s, p, ...) are special wave functions $\psi$ that are used to approximately describe one electron in spherically symmetric potential due to nucleus (and for many-electron atoms, also due to other electrons). $\psi$ is a mathematical function that is useful in the process of finding some interesting numbers, like expected average electric moment of the atom or intensity of its emission lines. It does not necessarily have the physical significance in the sense flies or stones have it if that is what you mean. It only gives probabilities, which relate to possible state of the system, not to the actual state, and many things that are most easily derived with use of $\psi$ can be derived even without them just by using different formalism.

And also, concerning the Zeeman effect, which I believe is the splitting of spectral lines in a static magnetic field (correct me if I am wrong), we have been told that this occurs due to separation of orbitals into their components. Assuming the p orbital is affected by a magnetic field, it would split into px,py and pz components, what does this mean, and how do these probabilities (if that assumption is right) actually exist in its essence?

Splitting of spectral lines and separation of orbitals are just two consequences of changing the Hamiltonian to include additional (magnetic) term (or if it was there before, consequence of turning the magnetic field on). I do not think there is any subordination between these two effects, one has to calculate both of them from the new Hamiltonian.

And one last thing concerning, the Radial probability density and Radial probability distribution curves, what does the graph of 4πr2R2 vs. r plots attempt to impart?

Density and distribution usually mean the same thing in this context. The graph has this meaning: the probability that the electron is somewhere between sphere with radius $r_1$ and sphere of radius $r_2$ is given by the area under the plotted curve, delimited on the left and right by the abscissae $r_1,r_2$.

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