Download general properties of the solution: quantum numbers:

The Hydrogen Atom:
structure:
- one proton with charge q = +e in nucleus
- one electron with charge q = -e in shell
assumptions:
- proton is stationary
- the electron is bound to the proton (Ekin < Eion)
potential energy:
- electron in field of nucleus
- known from classical mechanics: energy and angular momentum of a particle moving in
a spherically symmetric potential U(r) are conserved
Schrödinger equation:
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general properties of the solution:
quantum numbers:
- in 1D problems (particle in a box, harmonic oscillator) one
quantum number is sufficient to describe the solution
- a particle in a 3D box requires three quantum numbers for
the specification of its wave function because of the three
independent boundary conditions
- the electron in the hydrogen atom is confined by a 3D
potential generated by the nucleus
- thus three quantum numbers are required to describe the
spatial solutions of the Schrödinger equation for the
hydrogen atom
- a fourth quantum number is required to take into account
the electron spin (to be discussed later)
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simplification:
polar coordinates:
- use polar coordinates since the potential is spherically symmetric,
i.e. it depends only on the separation r between the nucleus and the
electron
- spherical polar coordinates for the point P = (x,y,z) are r, θ, φ
- radius r: distance between nucleus and point P with
coordinates (x,y,z)
- polar angle θ: angle between the vector along positive z-axis
and vector pointing along r
- azimuth angle φ: angle between the projection of the r vector
into the xy-plane and the positive x-axis
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- Laplace operator in Cartesian coordinates
- in polar coordinates
- kinetic energy:
- with the angular momentum operator L in polar coordinates
- eigenfunctions and eigenvalues of the angular momentum operator
- with spherical harmonic functions Yl,m and total angular momentum quantum number l
and magnetic quantum number ml
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angular momentum:
in Cartesian coordinates
angular momentum operator:
x, y, z components of angular
momentum:
square of angular momentum operator:
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Commutation Relations and Simultaneous Eigenfunctions:
consider two operators A and B with simultaneous eigenfunctions ψ and eigenvalues a and b
apply A to the second equation and B to the first equation and subtract
commutation relation:
If such a commutation relation is fulfilled simultaneous eigenfunctions of A and B can be
found and the eigenvalues of A and B can be measured simultaneously. If
there are no simultaneous eigenfunctions ψ of the two operators A and B.
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Angular Momentum Commutation Relations:
- The components lx, ly, lz of the angular
momentum do not commute. There are no
simultaneous eigenfunctions and eigenvalues.
Therefore the physical observables cannot be
determined with arbitrary precision simultaneously.
This observation is related to the Heisenberg
uncertainty principle.
The components lx, ly, lz of the angular momentum do commute individually with the
square of the total angular momentum L2. There are simultaneous eigenfunctions and
eigenvalues. Therefore the associated physical observables can be determined
simultaneously.
similarly:
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Schrödinger equation in polar coordinates:
general:
for the hydrogen
atom:
- use electrostatic potential
- multiply by
- the solutions ψ to the Schrödinger equation of the hydrogen atom
specifies the behavior of the single electron in the potential of the
hydrogen nucleus
- ψ has to fulfill the usual conditions
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Separation of variables
Ansatz:
- separate the wave function into a product of three components that only
depend on r, θ and φ independently
- with derivatives
- using the product wave function in Schrödinger equation results in
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note:
- all terms in the Schrödinger equation depend on one of the functions R, Θ
and Φ only
- rearrange the equation such that the term involving Φ is on the r.h.s.
- the equation can only be correct if the l.h.s. and the r.h.s. are
simultaneously identical to the same constant that we choose as ml2
- equation for Φ
- use result and separate variables again
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continue:
- both equations can only be true if they are equal to the same
constant l(l+1) (which is a clever choice the reason for which will
become apparent later)
- differential equation for Θ
- differential equation for R
- find solutions for differential equations for R, Θ and Φ
- only R depends on the potential
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solutions to Φ:
- Φ has to be single valued
- this is only valid for ml = 0, ±1, ±2, ±3, ...
- ml is the magnetic quantum number (will be discussed)
Θ are solutions to the differential equation:
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associated Legendre functions P are solutions to the differential equation for Θ
- the Legendre functions are defined for quantum numbers -l ≤ ml ≤ l
- therefore ml = 0, ±1, ±2, … , ±l
- l is the orbital quantum number
- the combination of the solutions for Θ and Φ are the spherical harmonic functions Y and
solve the orbital part of the Schrödinger equation
normalized
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The radial Schrödinger equation for the Hydrogen atom
has solutions
whit the Laguerre polynomials
and their derivatives
normalization coefficients
and the Bohr radius a0
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- the differential equation for R has solutions for E > 0 when the electron
is not bound to the nucleus and for a discrete set of En < 0, when the
electron is bound to the nucleus
- this result is identical to the Bohr expression for the energy of the electron
in the Hydrogen atom with n being the principal quantum number
Spatial quantum numbers of the Hydrogen atom:
- principal quantum number:
- orbital quantum number:
- magnetic quantum number:
- dependence of wave function on quantum numbers:
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Energy Quantization and Principal Quantum Number:
for bound electron:
no quantization for free electron:
For the motion of a particle in an attractive central potential with a 1/r2 force not only the total
energy but also the total angular momentum is a conserved quantity.
Orbital Quantum Number:
rearrange radial Schrödinger equation:
total electron energy:
orbital kinetic energy:
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orbital kinetic energy:
electron angular momentum:
unit of angular momentum:
The electron orbital momentum is quantized according to the above equation. The total
angular momentum is conserved (as in classical mechanics).
The orbital quantum number l determines the absolute value of the angular momentum L
of the electron. The direction of the vector quantity L is determined by the magnetic
quantum number ml as we will see soon.
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naming convention for angular momentum states:
sharp principal diffuse fundamental …
historical names of different spectral series
electron states with different principal (n) and angular (l) quantum numbers:
this scheme is widely used in chemistry and physics to designate electronic states in atoms
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magnetic quantum number:
space quantization:
- Lz is the component of the angular moment L along the
quantization axis z
- the quantization axis can be determined by an external
magnetic field B pointing in the z-direction for example
- ml can take 2 l + 1 different values from -l, …, 0, …, l
- note that Lz is never equal to L and therefore can never be
pointing in the direction of the quantization axis
- physical interpretation considering the orbital magnetic
moment μ of an electron in an external field B (also see
discussion of Zeeman effect)
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Angular Momentum and the Uncertainty Principle:
- the angular momentum is only quantized in one spatial direction
- If L was aligned with the quantization axis the electron would be
certain to move in the x-y plane. The uncertainty principle would
require the momentum uncertainty Δpz to be infinite. Thus the
electron could not be bound to the nucleus.
- the uncertainty principle enforces that not all components of an
observable (a physically measurable quantity) are determined with
certainty
- only the projection Lz of L along the quantization axis z is
known, the other components Lx and Ly remain uncertain such
that L is only determined to lie on the cone in space determined by
the magnitude of |L| and its z component
- as a consequence the average values of Lx and Ly are 0 while Lz is
determined by the quantum number ml
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Electron probability density:
- in the Bohr model the electron is expected to be on a definite orbit
around the nucleus
- in the quantum mechanics approach the probability density is
determined by the wave function
- no dependence of |ψ|2 on azimuth angle φ
- electron probability density is symmetric around the quantization axis
- the radial component of the wave function depends on both n and l
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Radial Hydrogen wave functions:
- l = 0 (s) wave functions have a maximum
probability density at the nucleus (r = 0)
- l > 0 (p, d, f, g, ...) wave functions have a node in
the probability density at (r = 0)
- as a result the average <r> is non-zero giving the
electron angular momentum
- electron probability density:
- probability to find electron in Volume element dV
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- probability of finding an electron in a spherical shell
between r and r + dr
- P(r) dr plotted below is noticeably different
from Ψ(r)
- the most probable value of r for a s = 0
electron is the Bohr radius a0
- the average value of r for a s = 0 electron is
<r> = 1.5 a0
- the average value of 1/r for a s = 0 electron
is <1/r> = 1/a0
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n=2
Hydrogen Wave Functions
n=1
n=4
l=0
l=0
ml = 0
n=3
l=1
ml = 0
z
l=1
l=0
l=2
ml = 0
±1
±2
±1
±3
l=1
ml = 0
electron densities as function of r and θ
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±1
±2
l=2
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angular probability density:
- azimuth angle dependence
- polar angle dependence
- the polar angle dependence is somewhat complicated as can be seen on the last slide
- the spatial structure of the electron probability density can be used to explain the geometry of
chemical bonds
- For each n the electron in an orbital with maximum l and maximum |ml| is resembling the
Bohr description most. The most probable radius is then given by n2 a0 as in the Bohr model
and the electron angular momentum is most parallel to the quantization axis. Also, the
probability density is maximum in a circular region in the x-y plane
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