Download Homework No. 01 (Spring 2016) PHYS 530A: Quantum Mechanics II

Homework No. 01 (Spring 2016)
PHYS 530A: Quantum Mechanics II
Due date: Wednesday, 2016 Jan 27, 4.30pm
1. (30 points.) The motion of a particle of mass m undergoing simple harmonic motion is
described by
d
(mv) = −kx,
(1)
dt
where v = dx/dt is the velocity in the x direction.
(a) Find the Lagrangian for this system that implies the equation of motion of Eq. (1)
using Hamilton’s principle of stationary action.
(b) Determine the canonical momentum for this system
(c) Determine the Hamilton H(p, x) for this system.
2. (10 points.) The Hamiltonian is defined by the relation
X
pi q̇i − L(qi , q̇i , t).
H(pi , qi , t) =
(2)
i
Show that
∂H
∂L
dH
=
=− .
dt
∂t
∂t
Under what circumstances is H interpreted as the energy of the system?
(3)
3. (30 points.) A relativistic charged particle of charge q and mass m in the presence of a
known electric and magnetic field is described by


q
d  mv 
q
= qE + v × B.
(4)
2
dt
c
1− v
c2
(a) Find the Lagrangian for this system that implies the equation of motion of Eq. (4)
using Hamilton’s principle of stationary action.
(b) Determine the canonical momentum for this system
(c) Determine the Hamilton H(p, r) for this system.
4. (30 points.) Consider the Lagrangian
1
L= m
2
dr
dt
1
2
− V (r, t).
(5)
(a) Show that principle of stationary action with respect to δr implies Newton’s second
law
d2 r
(6)
m 2 = −∇V.
dt
(b) Show that principle of stationary action with respect to δt implies
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d 1
∂V
dr
m
+V =
,
(7)
dt 2
dt
∂t
which for a static potential, ∂V /∂t = 0, is the statement of conservation of energy.
(c) Show that the invariance of the total time derivative term, that gets contributions
only from the end points, under an infinitesimal rigid rotation
r′ = r − δr,
δr = δω × r,
(8)
implies the conservation of total angular momentum, L = r × p.
5. (30 points.) The Hamiltonian for a hydrogenic atom is
H=
p21
p2
Ze2
+ 2 −
,
2m1 2m2 |r1 − r2 |
(9)
where r1 and r2 are the positions of the two constituent particles of masses m1 and m2
and charges e and Ze.
(a) Introduce the coordinates representing the center of mass, relative position, total
momentum, and relative momentum:
R=
m1 r1 + m2 r2
,
m1 + m2
r = r1 − r2 ,
P = p1 + p2 ,
p=
m2 p1 − m1 p2
,
m1 + m2
(10)
respectively, to rewrite the Hamiltonian as
H=
P2
p2
Ze2
+
−
,
2M
2µ
r
where
M = m1 + m2 ,
1
1
1
+
.
=
µ
m1 m2
(11)
(12)
(b) Show that Hamilton’s equations of motion are given by
dR
P
=
,
dt
M
dP
= 0,
dt
dr
p
= ,
dt
µ
dp
r
= −Ze2 3 .
dt
r
(13)
(c) Verify that the Hamiltonian H, the angular momentum L = r × p, and the LaplaceRunge-Lenz vector
1
r
p × L,
(14)
A= −
r µZe2
are the three constants of motion for a hydrogenic atom.
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