Lecture 14' Hydrogen Atom

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Lecture 14. Hydrogen Atom
Atom is a 3D object, and the electron motion is
three-dimensional. Well start with the simplest
case - a hydrogen atom. An electron and a proton
(nucleus) are bound by the central-symmetric
Coulomb interaction. Because mpgtgtme, we neglect
the proton motion (the reduced mass is very close
to me, the center of mass of this system is 2000
times closer to the proton than to the electron).
Thus, we can treat this problem as the motion of
an electron in the 3D central-symmetric Coulomb
potential.
Three dimensions we expect that the motion will
be characterized with three quantum numbers (in
fact, there will be the fourth one, the electron
spin, which corresponds to the internal degree
of freedom for an electron the spin will be
added later by hand, it is not described by the
non-relativistic quantum mechanics).
Elt0 bound motion, discrete spectrum, Egt0
unbound motion, continuous spectrum
The task solve the t-independent S.Eq. for Elt0,
find the energy eigenvalues (the spectrum) and
eigenfunctions (stationary states).
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Orbital Motion in Classical Mechanics
We consider the case of a central force the
force is directed along the line that connects
the electron and proton or, in our Figure, the
electron and the coordinate origin.
Coulomb interaction is responsible for
acceleration
Kinetic energy
pr and pt the radial and tangential components
of the momentum
The angular momentum
For a circular motion
In general, a non-zero torque leads to the time
dependence of L
In the field of a central force, the torque is
zero, and the angular momentum about the center
is conserved
This makes L especially useful for analyzing the
central force motion.
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Quantum-Mechanical Approach
As usual, we start with the time-dependent
Schrödinger Equation (but now it is 3D case)
- potential energy of Coulomb interaction between
electron and proton
Laplacian ?2
K operator
U operator
The potential is time-independent, thus we can
separate time and space variables
- solution of the t-independent S. Eq. gives us
the eigenfunctions (the orthogonal basis ?i) and
eigenvalues (spectrum Ei) of the Hamiltonian.
Eigenfunctions correspond to the stationary
states with a well-defined energy.
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Spherical Polar Coordinates
The Coulomb potential has central symmetry (U(r)
depends on neither ? nor ? ). Its to our
advantage to use spherical polar coordinates
(this will allow us to separate variables).
- distance from origin to point P
- zenith angle
- azimuth angle
- a (small) volume element used for integration
(no relation to the torque ?)
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t-independent Schrödinger Equation in Polar
Coordinates
After multiplying by r2sin?
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Separation of Variables
For a spherically-symmetric potentials (U U(r)),
one can separate the variables by using a trial
function
To show that the variables are separable, lets
rewrite
Since the left and right sides depends on
different variables, both parts should be equal
to the same constant
Divide by sin2? and regroup
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Separation of Variables (contd)
Thus
Equation for R(r)
Equation for ?(?)
Equation for ?(?)
Or, in a more conventional form
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Azimuth Part of the Eigenfunctions
- the angular momentum along this axis is the
generator of rotation around this axis
The wavefunction must be single-valued, thus
ml must be integer
ml the magnetic quantum number
What is the range of ml variations?
- this Eq. has a solution provided
Thus,
Discrete eigenvalues of the angular momentum
along z axis
- to be discussed in more detail at the next
lecture solution provided
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Equation for the Radial Part R(r)
Note that only this equation includes the total
energy.
or, being expressed as the radial portion of
the Hamiltonian
What is the structure of this Hamiltonian? Recall
that in classical mechanics
- the eigenvalues of the operator
this term corresponds to the kinetic energy of
the radial motion (total kinetic energy minus
the kinetic energy of tangential motion)
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Equation for the Radial Part R(r) (contd)
substitute
formally, this equation coincides with the S.Eq.
for 1D motion in the potential
If U(r) is everywhere finite, R should be finite
as well, and, thus, ? should ?0 at r0 (also at
r??)
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Quantization and Eigenfunctions
- the spectrum of this equation is continuous for
Egt0 (unbound motion) and discrete for Elt0 (bound
motion).
For the bound motion, solutions exist if
and
The latter requirement can be expressed as a
condition on l
Thus, for the bound states in H atom
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Degeneracy Feature of 1/r Potential
- the energy of the electron in H atom does NOT
depend on its angular momentum!
Different eigenfunctions with the same n but
different l must have the same eigenvalue of
Hamiltonian (the electron total energy).
Degeneracy if two or more quantum states have
the same energy, they are said to be degenerate.
For the electron states in H atom, the degeneracy
is a consequence of the 1/r potential.
In Newtonian mechanics, bound states in a 1/r
potential are ellipses, and there is also a
degeneracy the energy of the bound states
depends only on the semi-major axis of the
ellipse and NOT on the angular momentum
- the motion along these two orbits with the same
2a and different values of L would correspond to
the same total energy
This degeneracy shows up in quantum mechanics in
that one finds the same eigenvalue of Hamiltonian
for several different values of the angular
momentum quantum number.
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Electron Spectrum
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Ground State and Other l 0 States (s-states)
n1
For these so-called s-states, the angular
momentum is zero, thus
ground state
The radial equation is reduces to
n2
n1, l 0
n1, l 0
n3
- recall that we got this estimate from the
uncertainty principle
- the Bohr radius
l 0 (s-states)
Note the difference between the probability
density I?I2 and the probability of finding the
electron within ?r (6.7)
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The Uncertainty Principle applied to an H Atom
- classical approach (orbits, separation of E
into K and U)
Quantum approach
?
the minimum of Equantum
non-relativistic motion
The speed of the electron motion in H atom
The dB wavelength of the electron is comparable
with R
Thus, we need Q.M. to describe the system!
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Comparison with Bohrs Model (Beiser 4.4)
Niels Bohr (1913) (a decade before de Broglies
work !) proposed the first successful
semi-classical model of an atom circular
electron motion as a standing wave.
Below we consider this model using de Broglie
terminology an electron can circle a nucleus
only if its orbit contains an integer number of
de Broglie wavelengths.
Combining these two equation, we get the correct
formula for the energy eigenvalues
(one of the most successful wild guesses ever
made)
This model has a huge historical importance, but
because of its semi-classic nature (orbits,
etc.), it is a poor and misleading picture of
what is really going on in H atom. The difference
between the probability to find an electron at
some distance r from the nucleus according to
Bohr model and that in the quantum model is
dramatic at lower n e.g., at n0 (the ground
state), according to quantum mechanics the
angular momentum is 0, and the probability peaks
at the nucleus position!
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Eigenfunctions for n ?2, l ?0
p
d
s
1
2
l 1 p state l 2 d state etc.
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