Motion of a Particle in a Central Potential - General Formalism of Wave Mechanics, Quantum Mechanics | Physics for IIT JAM, UGC - NET, CSIR NET PDF Download

 Page 1


A. STATIONARY STATES OF A PARTICLE IN A CENTRAL POTENTIAL
We will be dealing with the following three topics:
A) Stationary states of a particle in a central potential
V(~ r) is invariant under any rotation about the origin, that is

H;L
k

= 0, and thus
the eigenfunctions of
ˆ
L
2
and
ˆ
L
z
are also eigenfunctions of H.
Page 2


A. STATIONARY STATES OF A PARTICLE IN A CENTRAL POTENTIAL
We will be dealing with the following three topics:
A) Stationary states of a particle in a central potential
V(~ r) is invariant under any rotation about the origin, that is

H;L
k

= 0, and thus
the eigenfunctions of
ˆ
L
2
and
ˆ
L
z
are also eigenfunctions of H.
B) Motion of the center of mass and relative motion for a system of two inter-
acting particles
(i) a two particle system in which interaction energy depends only on the particles’
relative position can be replaced by a simpler problem of one ?ctitious particle;
(ii) in addition, when the interaction depends only on the distance between parti-
cles, then the ?ctitious particle’s motion is governed by a central potential.
Page 3


A. STATIONARY STATES OF A PARTICLE IN A CENTRAL POTENTIAL
We will be dealing with the following three topics:
A) Stationary states of a particle in a central potential
V(~ r) is invariant under any rotation about the origin, that is

H;L
k

= 0, and thus
the eigenfunctions of
ˆ
L
2
and
ˆ
L
z
are also eigenfunctions of H.
B) Motion of the center of mass and relative motion for a system of two inter-
acting particles
(i) a two particle system in which interaction energy depends only on the particles’
relative position can be replaced by a simpler problem of one ?ctitious particle;
(ii) in addition, when the interaction depends only on the distance between parti-
cles, then the ?ctitious particle’s motion is governed by a central potential.
C) Exactly solvable problems
(i) V(~ r) is a Coulomb potential: hydrogen, deuterium, tritium, He
+
,Li
+
;
(ii) V(~ r) is a quadratic potential: isotropic three-dimensional harmonic oscillator.
Page 4


A. STATIONARY STATES OF A PARTICLE IN A CENTRAL POTENTIAL
We will be dealing with the following three topics:
A) Stationary states of a particle in a central potential
V(~ r) is invariant under any rotation about the origin, that is

H;L
k

= 0, and thus
the eigenfunctions of
ˆ
L
2
and
ˆ
L
z
are also eigenfunctions of H.
B) Motion of the center of mass and relative motion for a system of two inter-
acting particles
(i) a two particle system in which interaction energy depends only on the particles’
relative position can be replaced by a simpler problem of one ?ctitious particle;
(ii) in addition, when the interaction depends only on the distance between parti-
cles, then the ?ctitious particle’s motion is governed by a central potential.
C) Exactly solvable problems
(i) V(~ r) is a Coulomb potential: hydrogen, deuterium, tritium, He
+
,Li
+
;
(ii) V(~ r) is a quadratic potential: isotropic three-dimensional harmonic oscillator.
1. Outline of the problem
a. REVIEW OF SOME CLASSICAL RESULTS
Page 5


A. STATIONARY STATES OF A PARTICLE IN A CENTRAL POTENTIAL
We will be dealing with the following three topics:
A) Stationary states of a particle in a central potential
V(~ r) is invariant under any rotation about the origin, that is

H;L
k

= 0, and thus
the eigenfunctions of
ˆ
L
2
and
ˆ
L
z
are also eigenfunctions of H.
B) Motion of the center of mass and relative motion for a system of two inter-
acting particles
(i) a two particle system in which interaction energy depends only on the particles’
relative position can be replaced by a simpler problem of one ?ctitious particle;
(ii) in addition, when the interaction depends only on the distance between parti-
cles, then the ?ctitious particle’s motion is governed by a central potential.
C) Exactly solvable problems
(i) V(~ r) is a Coulomb potential: hydrogen, deuterium, tritium, He
+
,Li
+
;
(ii) V(~ r) is a quadratic potential: isotropic three-dimensional harmonic oscillator.
1. Outline of the problem
a. REVIEW OF SOME CLASSICAL RESULTS
Force on the particle located at the point M
~
F = 
~
rV(r) =
dV
dr
~ r
r
(2.1)
is always directed to the origin O. In this case the angular momentum theorem
implies that the angular momentum
~
L =~ r~ p is a constant of motion:
d
~
L
dt
=
~
0 (2.2)
and the particle trajectory is on the plane through the origin and perpendicular to
~
L.