Physics 334
Modern Physics
Credits: Material for this PowerPoint was adopted from Rick Trebino’s lectures from Georgia Tech which were
based on the textbook “Modern Physics” by Thornton and Rex. I have replaced some images from the adopted
text “Modern Physics” by Tipler and Llewellyn. Others images are from a variety of sources (PowerPoint clip art,
Wikipedia encyclopedia etc) and were part of original lectures. Contributions are noted wherever possible in the
PowerPoint file. The PDF handouts are intended for my Modern Physics class.
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CHAPTER 7
Atomic Physics
7.1 Schrödinger Equation in 3D
7.2 Quantization of Angular Momentum and
Energy in the Hydrogen Atom
7.3 Hydrogen Atom Wave Function
7.4 Electron Spin
7.5 Total Angular Momentum and the SpinOrbit Effect
7.6 The Schrödinger Equation for Two (or
more) Particles
7.7 Ground States of Atoms: The Periodic
Table
7.8 Excited States of Spectra and Atoms
Werner Heisenberg
(1901-1976)
The atom of modern physics can be symbolized only through a partial differential
equation in an abstract space of many dimensions. All its qualities are inferential; no
material properties can be directly attributed to it. An understanding of the atomic world
in that primary sensuous fashion…is impossible.
- Werner Heisenberg
http://dir.niehs.nih.gov/ethics/images/photo-heisenberg.jpg
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7.1: The Schrödinger Equation in Three
Dimension
The wave function must be a function of all three spatial coordinates.
Now consider momentum as an operator acting on the wave function.
In this case, the operator must act twice on each dimension. Given:
So the three-dimensional Schrödinger wave equation is
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The 3D infinite potential well
It’s easy to show that:
ψ ( x, y, z ) = A sin(k x x) sin(k y y) sin(k z z )
where:
k x = π nx / Lx
k y = π n y / Ly
k z = π nz / Lz
and:
π 2 2  nx2
nz2 
E=
 + + 
2m  L2x L2y L2z 
n y2
When the box is a cube:
E=
π 2 2
2mL2
(n
2
x
2
y
+n +n
2
z
)
Try 10, 4, 3
and 8, 6, 5
Note that more than one wave function can have the same energy.
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Degeneracy
The Schrödinger wave equation in three dimensions introduces
three quantum numbers that quantize the energy. And the same
energy can be obtained by different sets of quantum numbers.
A quantum state is called degenerate when there is more than
one wave function for a given energy.
Degeneracy results from particular properties of the potential energy
function that describes the system. A perturbation of the potential
energy can remove the degeneracy.
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Schrödinger Equation in Spherical
Coordinates
The potential energy of the electron-proton system is electrostatic:
Use the three-dimensional time-independent Schrödinger Equation.
For Hydrogen-like atoms (He+ or Li++)
Replace e2 with Ze2 (Z is the atomic number).
Replace m with the reduced mass, µ.
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Spherical Coordinates
The potential (central force)
V(r) depends on the distance r
between the proton and
electron.
Transform to spherical polar
coordinates because of the
radial symmetry.
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The Schrödinger Equation in Spherical
Coordinates
Transformed into
spherical coordinates,
the Schrödinger
equation becomes:
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7-2 Quantization of Angular Momentum
and Energy in the Hydrogen Atom
The wave function ψ is a function of r, θ, φ. This is a potentially
complicated function.
Assume instead that ψ is separable, that is, a product of three
functions, each of one variable only:
This would make life much simpler, and it turns out to work.
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Solution of the Schrödinger Equation
for Hydrogen
Separate the resulting equation into three equations: R(r), f(θ), and g(φ).
The derivatives:
Substitute:
Multiply both sides by r2 sin2 θ / R f g:
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Solution of the Schrödinger Equation for H
r and θ appear only on the left side and φ appears only on the right side
The left side of the equation cannot change as φ changes.
The right side cannot change with either r or θ.
Each side needs to be equal to a constant for the equation to be true.
Set the constant to be −mℓ2
azimuthal equation
It is convenient to choose the solution to be
.
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Solution of the Schrödinger Equation for H
satisfies the azimuthal equation for any value of mℓ.
The solution must be single valued to be a valid solution for any φ:
mℓ must be an integer (positive or negative) for this to be true.
Now set the left side equal to −mℓ2 and rearrange it [divide by sin2(θ)].
Now, the left side depends only on r, and the right side depends only
on θ. We can use the same trick again!
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Solution of the Schrödinger Equation for H
Set each side equal to the constant ℓ(ℓ + 1).
Radial equation
Angular equation
We’ve separated the Schrödinger equation into three ordinary secondorder differential equations, each containing only one variable.
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Solution of the Radial Equation for H
The radial equation is called the associated Laguerre equation and the
solutions R are called associated Laguerre functions. There are
infinitely many of them, for values of n = 1, 2, 3, …
Assume that the ground state has n = 1 and ℓ = 0. Let’s find this solution.
The radial equation becomes:
The derivative of
yields two terms.
This yields:
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Solution of the Radial
H
Try a solution
Equation for
A is a normalization constant.
a0 is a constant with the dimension of length.
Take derivatives of R and insert them into the radial equation.
⇒
To satisfy this equation for any r, both expressions in parentheses must
be zero.
Set the second expression
equal to zero and solve for a0:
Set the first expression equal
to zero and solve for E:
Both are equal to the Bohr results!
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Principal
Quantum
Number n
There are many solutions to the radial wave equation, one for
each positive integer, n.
The result for the quantized energy is:
A negative energy means that the electron and proton are bound
together.
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Quantum Numbers
The three quantum numbers:
n: Principal quantum number
ℓ: Orbital angular momentum quantum number
mℓ: Magnetic (azimuthal) quantum number
The restrictions for the quantum numbers:
n = 1, 2, 3, 4, . . .
ℓ = 0, 1, 2, 3, . . . , n − 1
mℓ = −ℓ, −ℓ + 1, . . . , 0, 1, . . . , ℓ − 1, ℓ
Equivalently:
n>0
ℓ<n
|mℓ| ≤ ℓ
The energy levels are:
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Hydrogen Atom Radial Wave Functions
First few
radial
wave
functions
Rnℓ
Subscripts
on R
specify
the
values of
n and ℓ.
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Solution of the Angular and Azimuthal
Equations
The solutions to the azimuthal equation are:
Solutions to the angular and azimuthal equations are linked
because both have mℓ.
Physicists usually group these solutions together into
functions called Spherical Harmonics:
spherical harmonics
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Normalized Spherical Harmonics
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Solution of the Angular and Azimuthal
Equations
The radial wave function R and the spherical harmonics Y determine
the probability density for the various quantum states. The total wave
function
depends on n, ℓ, and mℓ. The wave function
becomes
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Orbital Angular Momentum Quantum
Number ℓ
Energy levels are degenerate with respect to ℓ (the energy is
independent of ℓ).
Physicists use letter names for the various ℓ values:
ℓ=
0
1
2
3
4
Letter =
s
p
d
f
g
5...
h...
Atomic states are usualy referred to by their values of n and ℓ.
A state with n = 2 and ℓ = 1 is called a 2p state.
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Orbital Angular Momentum Quantum
Number ℓ
It’s associated with the R(r) and f(θ) parts of the wave function.
Classically, the orbital angular momentum
with L = mvorbitalr.
L is related to ℓ by
In an ℓ = 0 state,
This disagrees with Bohr’s
semi-classical “planetary”
model of electrons orbiting
a nucleus L = nħ.
Classical orbits—which do not
exist in quantum mechanics
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Magnetic Quantum Number mℓ
The angle φ is the angle from the z axis.
The solution for g(φ) specifies that mℓ is an integer and is related to the
z component of L:
Example: ℓ = 2:
Only certain orientations
of are possible. This is
called space
quantization.
And (except when ℓ = 0)
we just don’t know Lx and
Ly!
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Rough derivation of ‹L2› = ℓ(ℓ+1)ħ2
We expect the average of the angular momentum components
squared to be the same due to spherical symmetry:
But
Averaging over all mℓ values (assuming each is equally likely):
because:
∑m
2
= ( + 1)(2 + 1) / 3
n = −
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