Question

5. [25 pts total Consider the H-atom system of an electron freely moving about a nu When solve, looking at this system in spherical coordinates we obtain three differential equations to representing the radial distance (r), axial angle (θ), and azimuthal angle (d) coo rdinates. a) 12 pts] What is the approximation typically invoked to treat this system as just the behavior of the electron motion? Why is it acceptable to do so? b) [3 pts] Sketch a representation of the system under study when considering H-like systems. Label all relevant parts and show how this system originally in (x y, z)-coordinates can be converted into spherical coordinates (r, θ ). c [5 pts] The Phi-equation to be solved is: I dub Show that Φ", (d) = Acos(m4) + isin(m,φ) is a solution for this differential equation. d) [5 pts] Determine A that normalizes Φ from part c). e) [5 pts] How is the magnetic angular quantum number, m, related to the orbital angular momentum quantum number, I? What values can mi take on in relation to ? f) [5 pts] Provide a simple explanation relating the angular quantum numbers, / and mi, in relation to the structure of the periodic table.

Answer 1

a) The schrodinger equation can be solved completely for hydrogen atom and for related ion which have only one eletron such as He^+. For other atom only approximate solution can be obtained.

b)

-なーーーープ(xyz)

now $z=rcos\theta$

$y= rsin\theta.sin\phi$

$x= r.sin\theta .cos\phi$

$\Phi = A\left \lfloor cos\left ( mi\phi \right )+\imath sin\left \left ( mi\phi \right ) \right \rfloor$

ACsin mi φ , mi "ricas mpt.rn 2 mi ce s 2 2

e) Value of m= -l, -l+1......., 0, ... l, number= 2l+1

f) structure of the periodic table is depend upon the the electron filling e.g. on basis of energy which can think in term of n+l. when n is same the l tell energy level which is exactly aufbau principle says.

Aufbau principle

You may consider an atom as being "built up" from a naked nucleus by gradually adding to it one electron after another, until all the electrons it will hold have been added. Much as one fills up a container with liquid from the bottom up, the orbitals of an atom are filled from the lowest energy orbitals to the highest energy orbitals.

Orbitals with the lowest principal quantum number ({\displaystyle n}) have the lowest energy and will fill up first, in smaller atoms. Larger atoms with more subshells will seem to fill "out of order", as the other factors influencing orbital energy become important. Within a shell, there may be several orbitals with the same principal quantum number. In that case, more specific rules must be applied. For example, the three p orbitals of a given shell all occur at the same energy level. So, how are they filled up? ans: all the three p orbitals have same energy so while filling the p orbitals we can fill any one of the Px, Py or Pz first. it is a convention that we chose to fill Px first, then Py and then Pz for our simplicity. Hence you can opt for filling these three orbitals from right to left also. Aufbau principle state that "atomic orbitals are filled with electrons in order of increasing energy level "