文档介绍：Coupling to the ic Field
The Hydrogen Atom
The Spectrum of Hydrogen Atoms
Current in the Hydrogen Atoms
The ic Moment
Chapter 9 Chraged particles in ic Fields
Coupling to the ic Field
In an ic field, the Lorentz force which acts on a charged particle of charge e is
Where
The Hamiltonian
In the ic, the canonical momentum (正则动量) p is replaced by
In general, the gradient (梯度) and vector potentials (矢势) do mute,
The ic potential A and are not unique,but are gauge independent (规范相关的).
H0 is Hamiltonian without ic field. The product A·p represent the coupling of the motion of a particle to ic.
Due to
when the field is small, the third term can be dropped. If A describe a plane ic, the coupling term lead to radiative transition (emission or absorption) (辐射跃迁(发射或吸收)).
If we apply to the potential transformation
Schrodinger equation
Where f(r,t) is an arbitrary function. The solution of Schrodinger equation describe the same physical state.
Schrodinger equation possesses gauge invariable (规范不变性)
Since the electron of hydrogen atom moves in a central potential, we choose spherical coordinate, and m represent the reduced mass (折合或约化质量).
The stationary Schrodinger equation is
The Hydrogen atom
where
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.
Hence we can obtain
According to the property of spherical harmonic function Ylm(,) (球谐函数)
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 ic quantum number m.
(1) When r0,the angular momentum is dominant
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.