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Lectures 3-4: One-electron atoms. Schrödinger equation for one-electron atom. Solving the Schrödinger equation. Wavefunctions and eigenvalues. Atomic orbitals. See Chapter 7 of Eisberg & Resnick. The Schrödinger equation. One-electron atom is simplest bound system in nature.

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lectures 3 4 one electron atoms
Lectures 3-4: One-electron atoms
  • Schrödinger equation for one-electron atom.
  • Solving the Schrödinger equation.
  • Wavefunctions and eigenvalues.
  • Atomic orbitals.
  • See Chapter 7 of Eisberg & Resnick.


the schr dinger equation
The Schrödinger equation
  • One-electron atom is simplest bound system in nature.
  • Consists of positive and negative particles moving in 3D Coulomb potential:
  • Z =1 for atomic hydrogen, Z =2 for ionized helium, etc.
  • Electron in orbit about proton treated using reduced mass:
  • Total energy of system is therefore,


the schr dinger equation1
The Schrödinger equation
  • Using the Equivalence Principle, the classical dynamical quantities can be replaced with their associated differential operators:
  • Substituting, we obtain the operator equation:
  • Assuming electron can be described by a wavefunction of form,

can write


where, is the Laplacian operator.


the schr dinger equation2
The Schrödinger equation
  • Since V(x,y,z) does not depend on time, is a solution to the Schrödinger equation and the eigenfunction is a solution of the time-independent Schrödinger equation:
  • As V = V(r), convenient to use spherical polar coordinates.


  • Can now use separation of variables to split the partial

differential equation into a set of ordinary differential equations.



separation of the schr dinger equation
Separation of the Schrödinger equation
  • Assuming the eigenfunction is separable:
  • Using the Laplacian, and substituting (2) and (1):
  • Carrying out the differentiations,
  • Note total derivatives now used, as R is a function of r alone, etc.
  • Now multiply through by and taking transpose,




separation of the schr dinger equation1
Separation of the Schrödinger equation
  • As the LHS of Eqn 3does nor depend on r or  and RHS does not depend on  their
  • common value cannot depend on any of these variables.
  • Setting the LHS of Eqn 3 to a constant:
  • and RHS becomes
  • Both sides must equal a constant, which we choose as l(l+1):
  • We have now separated the time-independent Schrödinger equation into three
  • ordinary differential equations, which each only depend on one of  (4),  (5) and R(6). .





summary of separation of schr dinger equation
Summary of separation of Schrödinger equation
  • Express electron wavefunction as product of three functions:
  • As V ≠ V(t), attempt to solve time-independent Schrodinger equation.
  • Separate into three ordinary differential equations for and .
  • Eqn. 4 for () only has acceptable solutions for certain value of ml.
  • Using these values for mlin Eqn. 5, () only has acceptable values for certain values of l.
  • With these values for l in Eqn. 6, R(r) only has acceptable solutions for certain values of En.
  • Schrödinger equation produces three quantum numbers!


azimuthal solutions
Azimuthal solutions (())
  • A particular solution of (4) is
  • As the einegfunctions must be single valued, i.e.,  =>

and using Euler’s formula,

  • This is only satisfied if ml = 0, ±1, ±2, ...
  • Therefore, acceptable solutions to (4) only exist when ml can only have certain integer

values, i.e. it is a quantum number.

  • ml is called the magnetic quantum number in spectroscopy.
  • Called magnetic quantum number because plays role when atom interacts with magnetic fields.


polar solutions
Polar solutions (())
  • Making change of variables (z = rcos, Eqn.5 transformed into an associated Legendre equation:
  • Solutions to Eqn. 7 are of form

where are associated Legendre polynomial functions.

  •  remains finite when = 0, 1, 2, 3, ...

ml = -l, -l+1, .., 0, .., l-1, l

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

00 = 1

10 = cos1±1 = (1-cos2)1/2

    • 20 = 1-3cos22±1 = (1-cos2)1/2cos
    • 2±2 = 1-cos2



spherical harmonic solutions
Spherical harmonic solutions
  • Customary to multiply () and () to form so called spherical harmonic functions
  • which can be written as:

i.e., product of trigonometric and polynomial


  • First few spherical harmonics are:

Y00= 1

Y10= cos Y1±1= (1-cos2)1/2 e±i

Y20= 1-3cos2 Y2±1= (1-cos2)1/2cos e±i


radial solutions r r
Radial solutions (R( r ))
  • What is the ground state of hydrogen (Z=1)? Assuming that the ground state has n = 1, l = 0 Eqn. 6can be written
  • Taking the derivative


  • Try solution , where A and a0are constants. Sub into Eqn. 7:
  • To satisfy this Eqn. for any r, both expressions in brackets must equal zero. Setting the second expression to zero =>
  • Setting first term to zero =>

Same as Bohr’s results



radial solutions r r1
Radial solutions (R( r ))
  • Radial wave equation

has many solutions, one for each positive integer of n.

  • Solutions are of the form (see Appendix N of Eisberg & Resnick):

where a0is the Bohr radius. Bound-state solutions are only acceptable if

where n is the principal quantum number, defined by n = l +1, l +2, l +3, …

  • Enonly depends on n: all l states for a given n are degenerate (i.e. have the same energy).



radial solutions r r2
Radial solutions (R( r ))
  • Gnl(Zr/a0) are called associated Laguerre polynomials, which depend on n and l.
  • Several resultant radial wavefunctions (Rnl( r )) for the hydrogen atom are given below


radial solutions r r3
Radial solutions (R( r ))
  • The radial probability function Pnl(r ), is the probability that the electron is found between r and r + dr:
  • Some representative radial probability functions are given at right:
  • Some points to note:
    • The r2factor makes the radial probability density

vanish at the origin, even for l = 0 states.

    • For each state (given n and l), there are n - l - 1

nodes in the distribution.

    • The distribution for states with l = 0, have n maxima,

which increase in amplitude with distance from origin.


radial solutions r r4
Radial solutions (R( r ))
  • Radial probability distributions for an electron in several of the low energy orbitals of hydrogen.
  • The abscissa is the radius

in units of a0.

s orbitals

p orbitals

d orbitals


hydrogen eigenfunctions
Hydrogen eigenfunctions
  • Eigenfunctions for the state described by the quantum numbers (n, l, ml) are therefore of form:

and depend on quantum numbers:

n = 1, 2, 3, …

l = 0, 1, 2, …, n-1

ml = -l, -l+1, …, 0, …, l-1, l

  • Energy of state on dependent on n:
  • Usually more than one state has same

energy, i.e., are degenerate.


born interpretation of the wavefunction
Born interpretation of the wavefunction
  • Principle of QM: the wavefunction contains all the dynamical information about the system it describes.
  • Born interpretationof the wavefunction: The probability (P(x,t)) of finding a particle at a position between x and x+dx is proportional to |(x,t)|2dx:

P(x,t) = *(x,t) (x,t) = |(x,t)|2

  • P(x,t) is the probability density.
  • Immediately implies that sign of wavefunction has no

direct physical significance.




born interpretation of the wavefunction1
Born interpretation of the wavefunction
  • In H-atom, ground state orbital has the same sign everywhere => sign of orbital must be all positive or all negative.
  • Other orbitals vary in sign. Where orbital changes sign,  = 0 (called a node) => probability of finding electron is zero.
  • Consider first excited state of hydrogen: sign of

wavefunction is insignificant (P = 2 = (-)2).


born interpretation of the wavefunction2
Born interpretation of the wavefunction
  • Next excited state of H-atom is asymmetric about origin. Wavefunction has opposite sign on opposite sides of nucleus.
  • The square of the wavefunction is identical on

opposite sides, representing equal distribution

of electron density on both side of nucleus.


atomic orbitals
Atomic orbitals
  • Quantum mechanical equivalent of orbits in Bohr model.


s orbitals
s orbitals
  • Named from “sharp” spectroscopic lines.
  • l = 0, ml = 0
  • n,0,m = Rn,0 (r ) Y0,m (, )
  • Angular solution:
  • Value of Y0,0is constant over sphere.
  • For n = 0, l = 0, ml = 0 => 1s orbital
  • The probability density is


p orbitals
p orbitals
  • Named from “principal” spectroscopic lines.
  • l = 1, ml = -1, 0, +1 (n must therefore be >1)
  • n,1,m = Rn1 (r ) Y1,m (, )
  • Angular solution:
  • A node passes through the nucleus and separates the two lobes of each orbital.
  • Dark/light areas denote opposite sign of the wavefunction.
  • Three p-orbitals denoted px, py , pz


d orbitals
d orbitals
  • Named from “diffuse” spectroscopic lines.
  • l = 2, ml = -2, -1, 0, +1, +2 (n must therefore be >2)
  • n,2,m = Rn1 (r ) Y2,m (, )
  • Angular solution:
  • There are five d-orbitals, denoted
  • 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
Quantum numbers and spectroscopic notation
  • Angular momentum quantum number:
    • l = 0 (s subshell)
    • l = 1 (p subshell)
    • l = 2 (d subshell)
    • l = 3 (f subshell)
  • Principal quantum number:
    • n = 1 (K shell)
    • n = 2 (L shell)
    • n = 3 (M shell)
  • If n = 1 and l = 0 = > the state is designated 1s. n = 3, l = 2 => 3d state.
  • Three quantum numbers arise because time-independent Schrödinger equation contains three independent variables, one for each space coordinate.
  • The eigenvalues of the one-electron atom depend only on n, by the eigenfunctions depend on n, l and ml, since they are the product of Rnl(r ), lml () and ml().
  • For given n, there are generally several values of l and ml => degenerate eigenfunctions.


orbital transitions for hydrogen
Orbital transitions for hydrogen
  • Transition between different energy levels of the hydrogenic atom must follow the following selection rules:

l = ±1

m = 0, ±1

  • A Grotrian diagram or a term diagram shows the allowed transitions.
  • The thicker the line at right, the more probable and hence more intense the transitions.
    • The intensity of emission/absorption lines could not be explained via Bohr model.


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