Born interpretation of the wavefunction

Born interpretation of the wavefunction
o In H-atom, ground state orbital has the same sign everywhere => sign of orbital must be all positive or all negative.
o Other orbitals vary in sign. Where orbital changes sign, T = 0 (called a node) => probability of finding electron is zero.



Consider first excited state of hydrogen: sign of wavefunction is insignificant (P = T² = (-T)²).

v²

electron density profile

wavefimction. y (for (a))

(c)

Summary of separation of Schrodinger equation

o Express electron wavefunction as product of three functions: _'•,〜)= o As V(t),attempt to solve time-independent Schrodinger equation.
o Separate into three ordinary differential equations for /?(r),0(0) and ^(0). o Eqn. 4 for only has acceptable solutions for certain value of m_t.
o Using these values for m! in Eqn. 5, &(0) only has acceptable values for certain values of I.
o With these values for I in Eqn. 6, R(r) only has acceptable solutions for certain values of E_n.
o Schrodinger equation produces three quantum numbers!

o Transition between different energy levels of the hydrogenic atom must follow the following selection rules:
Al= ±1
Am = 0, ±1
o A Grotrian diagram or a term diagram shows the allowed transitions.
o The thicker the line at right, the more probable and hence more intense the transitions.
o The intensity of emission/absorption lines could not be explained via Bohr model.
d orbitals
o Named from ‘‘diffuse” spectroscopic lines.
o / = 2,/n, = -2,-1,0,+1,+2 (n must therefore be >2)
o K.2._m = R_nl (r)Y_2jn(0,^)
o Angular solution: -(3cos²0-V)
o There are five ^/-orbitals, denoted
o m = 0 is z². Two orbitals of m = -1 and +1 are xz and yz. Two orbitals with m = -2 and +2 are designated xy and x²-y².
Quantum numbers and spectroscopic notation



o Principal quantum number: o
o n = 1 (K shell)
o n = 2 (L shell) o n = 3 (M shell) o ...

Angular momentum quantum number: o I = 0 (s subshell)
o I =] (p sub shell)
o I = 2 (d subshell)
o 1 = 3 (f subshell)
o ...


o If = 1 and / = 0 = > the state is designated Is. /? = 3, / = 2 => 3d state.
o Three quantum numbers arise because time-independent Schrodinger equation contains three independent variables, one for each space coordinate.
o The eigenvalues of the one-electron atom depend only on n, by the eigenfunctions depend on n,I and m_Jt since they are the product of R_nl(r ),(I)"," ((p) and 0_w/(0).
o For given n, there are generally several values of I and m₍ => degenerate eigenfunctions.
Radial solutions (R( r ))
o Radial probability distributions for an electron in several of the low energy orbitals of hydrogen.


s orbitals
o The abscissa is the radius in units of a₀.
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p orbitals d orbitals
Hydrogen eigenfunctions
o Eigenfunctions for the state described by the quantum numbers (n, l,m,) are therefore of form:


and depend on quantum numbers:
n = 1, 2, 3,...
1 = 0, 1, 2, .... n-1 in/ = -I. -l+l 0, .... l-l, I
o Energy of state on dependent on n:
o Usually more than one state has same energy, i.e.,are degenerate.
Lectures 3-4: One-electron atoms

Schrodinger equation for one-electron atom.

Atomic orbitals

o Quantum mechanical equivalent of orbits in Bohr model.
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are needed to see this picture.
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o Customary to multiply CD(^) and to form so called spherical harmonic functions which can be written as:
Y,W) = .W
i.e., product of trigonometric and polynomial
functions.

o First few spherical harmonics are:

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Y_o°= |

cos0 Y^^Cl-cos^'^e^
Y₂°= 1-3cos²0 Y₂^±,= (1-cos²0)^,/2cos0
o One-electron atom is simplest bound system in nature.
Consists of positive and negative particles moving in 3D Coulomb potential: o Z = 1 for atomic hydrogen, Z =2 for ionized helium, etc.



V = V(x,y,z) =

-Ze²
4 x² + y² + z²


o Electron in orbit about proton treated using reduced mass:
mM
m + M
o Total energy of system is therefore,
KE + PE = E
+ P.²) + V(x,y,z) = E

o Schrodinger’s QM treatment had a number of advantages over semi-classical Bohr model:
1. Probability density orbitals do not violate the Heisenberg Uncertainty Principle.
1. Orbital angular momentum correctly accounted for.
1. Electron spin can be properly treaded.
1. Electron transition rates can be explained.

s orbitals
o Named from “sharp” spectroscopic lines.
o I = 0, = 0
O ⁼ )





Angular solution:




Pnbn, (^r,) = (r, 0, )' (r, 0, )


o Value of Y_oo is constant over sphere, o For n = Q,1 = 0,rn! = 0 => Is orbital o The probability density is
Born interpretation of the wavefunction
o Next excited state of H-atom is asymmetric about origin. Wavefunction has opposite sign on opposite sides of nu

o The square of the wavefunction is identical on opposite sides, representing equal distribution of electron density on both side of nucleus.


nodal plane


angular node

p orbitals

o Named from “principal” spectroscopic lines.
o I = = -1, 0, +1 (n must therefore be >1)
° Ki._in = (r)Y_lm(e, )
o Angular solution: y；₀ = — cos^
v 4^
o A node passes through the nucleus and separates the two lobes of each orbital.
o Dark/light areas denote opposite sign of the wavefunction.
o Three p-orbitals denoted p_x, p、,p:

Making change of variables (z = rcos0),Eqn. 5 transformed into an associated Legendre equation'.


₂//0


"7/²


(7)

Solutions to Eqn. 7 are of form
where 6i"',i(^cos^) are associated Legendre polynomial functions.



o 0 remains finite when

o Can write the associated Legendre functions using quantum number subscripts:


0_1±1 = (l-cos²9)^l/2
0_2±1 = (1-cos²6)^,/2cosG
©00= ¹
0₁₍₎= cosO O₂₀= 1 -3cos²0 ©_2±2= l-cos²0
o A particular solution of (4) is
o As the einegfunctions must be single valued, i.e., O(0)=

=e

ini/ 27T

1 = cosm, 2 f si n w, 2
and using Euler’s formula,
This is only satisfied if m, = 0, ±1, ±2,...
o Therefore, acceptable solutions to (4) only exist when m, can only have certain integer values, i.e. it is a quantum number.
o is called the magnetic quantum number in spectroscopy.
o Called magnetic quantum number because plays role when atom interacts with magnetic
fields.

o G_nl(Zr/a₀) are called associated Laguerre polynomials, which depend on n and I.
o Several resultant radial wavefunctions (R_n/( r)) for the hydrogen atom are given below
n i 4(r)