Chapter 9 Chraged particles in Magnetic Fields

1
Chapter 9 Chraged particles in Magnetic Fields
9.1 Coupling to the Electromagnetic Field 9.2
The Hydrogen Atom 9.3 The Spectrum of Hydrogen
Atoms 9.4 Current in the Hydrogen Atoms 9.5
The Magnetic Moment
2
9.1 Coupling to the Electromagnetic Field
In an electromagnetic field, the Lorentz force
which acts on a charged particle of charge e is

The Hamiltonian
Where
3
In the electromagnetic, the canonical momentum
(????) p is replaced by
In general, the gradient (??) and vector
potentials (??) do not commute,
4
The electromagnetic potential A and ? are not
unique,but are gauge independent (?????).
Due to
H0 is Hamiltonian without magnetic field. The
product Ap represent the coupling of the motion
of a particle to magnetic.
5
when the field is small, the third term can be
dropped. If A describe a plane electromagnetic,
the coupling term lead to radiative transition
(emission or absorption) (????(?????)).
Schrodinger equation
If we apply to the potential transformation
Where f(r,t) is an arbitrary function. The
solution of Schrodinger equation describe the
same physical state.
Schrodinger equation possesses gauge invariable
(?????)
6
9.2 The Hydrogen atom
Since the electron of hydrogen atom moves in a
central potential, we choose spherical
coordinate, and m represent the reduced mass
(???????).
The stationary Schrodinger equation is
where
7
In spherical coordinate (P78)
So the Schrodinger equation can be written
With the following separation variable, we can
separate the above equation into a radial and an
angular part.
8
Hence we can obtain
9
According to the property of spherical harmonic
function Ylm(?,?) (????)
Finally we get the differential equation about
the radial function
The radial function depends on the total angular
momentum quantum number l, and is independent on
magnetic quantum number m.
10
Energy E only depends on the radial part R(r) of
wave function. Due to the orthonormality of
spherical harmonic function,
We only consider the discrete (bound) state, so
energy eigenvalue is negative.
(1) When r?0,the angular momentum is dominant
11
Due to
Hence
(2) When r??, the asymptotic solution of R(r) is
given by
The abbreviation
So
12
Due to
Hence
(3) According to the asymptotic solution of R(r)
in the case of
We try the solution of Schrodinger equation
Insert this into
13
The abbreviation
(???????)
When the following condition is satisfied
The above equation has solution.
Hence
14
n is the principal quantum number (????)
nr is radial quantum number (?????)
l is angular momentum number (????)
a0 is Bohr radius (????)
n 1, the ground state energy (binding energy) of
hydrogen atom is
15
(No Transcript)
16
Wave function of hydrogen
For the energy given (i.e. principal quantum
number n is fixed), angular momentum quantum
number l and magnetic quantum number m are
The degeneracy of En is
17
The wave functions of low energy are
18
9.6 Hydrogen like Atom (P213)
hydrogen like atom having only one valence
electron in the outermost shell. Consider that
the inner electrons screen the nuclear potential,
the effective nuclear charge number Zeff is used
to described proton number Z in the nucleus.
Stationary Schrodinger equation in many-body
system
19
7.7 Spectrum of a Diatomic Molecule (??????)
Born-Oppenheimer approximation
Schrodinger equation
20
Introducing the centre-of mass coordinate (????)
R and the relative coordinate (????) r.
Take the x coordinate as an example,
21
µ is the reduced mass (????)
Hence the Schrodinger equation is
Separate variable
Energy split into
22
The centre-of-mass motion equation
The relative motion equation
Due to no potential being contained, the motion
of the centre of mass is free and described by a
plane wave
23
For the relative motion, use usual separation
variables for a central potential
The radius equation
where
Wl(r ) is the effective potential, and is sum of
true potential (????) V(r ) and the rotational
energy (???) L2/2µr2.
24
We constitute a reasonable potential Vr
r lt r0, repulsion between two nucleuses, r gt
r0, attraction
When l0,Wl(r ) depends on position potential Vr.
. When l? 0,Wl(r ) depends on Vr and angular
momentum L. the more L is, the higher position
of minimum potential is.
25
We expand Wl(r ) around rl.
Using the abbreviation
26
Set ? µrl2, is called the moment of inertia
(????)
substitutiing
Linear harmonic oscillator equation
27
The eigenvalue of harmonic oscillator
The whole energy is
Where E is composed of potential, vibration and
rotation. The energy value is valid for small
numbers n and l.
28
????, ???????????,???????????????????????
??,??????????????
????????????????????????,???????????1234.?
29
Example 1
One particle motions in an infinite spherical
potential,
Problem (1) the energy eigenvalue and
eigenfunction of ground state
(2) Consider the general case.
30
Solution (1)
In a central force field, the wavefunction of a
particle is described by
and Schrodinger equation is
(1)
When a particle stays in ground state, l 0
Set
So (1) becomes
(2)
31
With boundary conditions
The solution of equation (2) is
Where C is constant, k satisfied the following
condition
So the energy level of ground state E and the
radial wavefunction R(r) are
32
The wavefunction is
(2) In general case (l ? 0)
Set
Schrodinger equation is
(3)
33
With boundary condition
Introduce variable
Equation (3) becomes
This is spherical Bessel function (??????)
Set
34
u(?) satisfied the following equation
This half odd order Bessel function of (l1/2),
(?????????)
So
Because
Hence
is l order spherical Bessel function
35
According to boundary condition
We obtain
Set the solution of equation
are
The energy eigenvalue is
36
Exercise
1. Hydrogen atom stays in ground state, solve the
probability finding electron outside the range
which classical mechanics is not allowed, i.e,
2. Set potential is
a is Bohr radius.
Solve the energy level.