A. La Rosa
PSU-Physics
Lecture Notes
PH 411/511 ECE 598
INTRODUCTION TO
QUANTUM MECHANICS
________________________________________________________________________
From the Hamilton’s Variational Principle to
the Hamilton Jacobi Equation
Ref: Saletan and Cromer, “Theoretical Mechanics” Wiley, 1971.
This is one of the best book I have ever encountered. I highly recommend it.
H. Golstein, Classical Mechanics, Addison Wesley
4.1A Classical specification of the state of motion
The spatial configuration of a system composed by N point masses
is completely described by
3N Cartesian coordinates (x1, y1, z1), (x2, y2, z2), …, (xN , yN , zN ).
If the system is subjected to constrains then the 3N Cartesian coordinates are not
independent variables. If n is the least number of variables necessary to specify the
most general motion of the system, then the system is said to have n degrees of
freedom.
The configuration of a system with n degrees of freedom is fully
specified by n generalized position-coordinates q1, q2, … , qn
The objective in classical mechanics is to find the trajectories,
q = q (t)
 = 1 , 2, 3, …, n
or simply q = q (t) where q stands collectively for the set (q1, q2, … qn).
(1)
4.1B Time evolution of a classical state: Hamilton’s
variational principle
The classical action
 One first expresses the Lagrangian L of the system in terms of the generalized
position and the generalized velocities coordinates q and q ( =1,2,…, n).
L( q , q , t )
Typically (but no always) L=T-V,
(2)
L (x, x , t) = (1/2)m x 2 - V(x,t)
(3)
 Then, for a couple of fixed end points (a, t1) and (b, t2) the classical action S is
defined as,
( b, t 2 )
S (q ) ≡

L( q(t ) , q (t ) , t ) dt
Classical action
(4)
( a, t1 )
Number (1-dimension case)
or
set of numbers (n-dimension case)
Function
or
set of functions (n-dimension case)
Hamilton’s variational principle for conservative systems
 Out of all the possible paths that go from (a, t1) to (b, t2), the system takes only
one.
On what basis such a path is chosen?
Answer: The path followed by the system is the one that makes the functional S
an extreme (i. e. a maximum or a minimum).
S q  q = 0
The variational principle
(5)
x(t)
xC
b
xC(t)
a
t1
t2
t
Figure 1. For a fixed points (a, t1) and (b, t2), among all the
possible paths with the same end points, the path xC makes
the action S an extremum.
Example: The variational principle leads to the
Newton’s law
S ( x) ≡
( b,t 2 )
 a,t
(
1
)
L( x, x,t )dt =
( b,t 2 ) 1
(a,t ) 2

[ mx 2  V ( x )]dt
(6)
1
x(( t) = xC(t) +  h(t)
(7)’
h(t1) = h(t2) =0
(8)
( b, t 2 )
S(x ) = (a, t
(
1)
1
{ 2 m [ x C(t) +  h (t) ]2 – V( xC(t ) +  h(t ) ) }dt
(9)
The condition that xC is an extremum becomes,
dS
0
d  0
(10)
x(t)
b
xC(t)
 h(t)

a
t1
t2
t
Figure 2. For an arbitrary function h, a parametric family of trial paths x(t ) =
xC(t ) +  h(t ) is used to probe the action S given by expression (6).
( b, t 2 )
dS
 –
{ m x C(t) + V ’( xC(t) ) } h(t )dt
( a, t 1 )
d  0
dV
m xC (t) = – dx ( xC(t ) ) = F
(12)
More general: The variational principle leads to
the Lagrange equations of motion
In a more general case, the system may be composed by n particles. Using,
x (t)= (x1(t), x2(t), … , xn(t), x1 (t), x 2 (t), ,…, x n (t) )
the action is defined as

S( x ) ≡
( b, t 2 )
( a, t 1 )
L( x (t), x (t) , t) dt
(14)
where a and b are two fixed point in the configuration space.
Using a family of trial functions of the form
x ( (t) = x (t) +  h (t)
where
(15)
h (t1) = h (t2) = 0
Expression (15) is a compact notation of,
x (t)= ( x1(t), x2(t), … , xn(t), x1 (t), x2 (t), ,…, x n (t) )
h (t)= ( h1(t), h2(t), … , hn(t), h1 (t), h2 (t), ,…, hn (t) )
We look for an extreme value of the function
t2

S( x () =
L( x ((t),
t1
x ((t), t) dt
(16)
From (15) and (16) one obtains,
dS

d 
dS

d
t2
{
t1


i
t2
t1
{
i
L
L 
hi  
hi } dt
xi
xi
i
L
hi } dt +
xi

i
The variational principle requires that,
t2
L
hi - 
xi
t1
t2
t1
{ (
i
d L
)hi } dt
dt xi
dS

d  0
L
  t { x
t2
i
1
i
-(
d L
) } hi (t ) dt = 0
dt xi
(17)
This could be satisfied only if,
d L L

0
dt xi xi
for i=1,2, …, n. The Lagrange Equations
(18)
These are second order equations. The motion is completely specified if the initial
values of the n coordinates xi and the n velocities xi are specified. That is, the xi and xi
form a complete set of 2n independent variables for describing the motion.
Remark:
Hamilton’s variational principle involves physical quantities T
and V, which can be defined without reference to a particular
set of generalized coordinates. The set of Lagrange equations
is therefore invariant with respect to the choice of
coordinates. !
4.2 The Hamiltonian formulation of classical mechanics
An alternative to the Lagrangian description of mechanics, outlined above, is the
Hamiltonian formulation. Instead of dealing with a set of n differential equations of 2ndorder given in (18), one is resorted to solve 2n differential equations of 1st order, as we
will see below. However one may end up with a similar intensity of difficulty when
solving the corresponding equations. The advantages of the
Hamiltonian
formulation lie not necessarily in its use as a calculation tool, but rather in the
deeper insight it affords into the formal structure of
mechanics.1 Its more abstract formulation is of interest because of their essential
role in constructing the more modern theories in physics. In this course it is used as a
point of departure for elaborating a quantum theory.
In this section we show that the new formulation is implemented through:
 a change of variables (to be specified later),
(x1, x2, …, xn, x1 , x2 , …, xn , t)
specified later
tobe
 (x1, x2, … xn, p1, p2, … pn,, t)
 instead of
to be specified later
L( x, x , t ) , 
H=H(x1, x2, …, xn, p1, p2, …,pn, t) = H(x, p, t)
How to obtain such a transformation? How to choose H?
To get an idea of what transformation is convenient (i. e. what particular combination
between the ( x, x , t ) and ( x, p, t ) variables is suitable for our purpose), let’s familiarize
with the following, much simpler, Legendre transformation
4.2B The Hamilton Equations of Motion
The equation of motion of a system is described by a Lagrange function
 
L  L( x, x , t ) , leading to a set of differential equations of second order,
d L L

0,
dt xi xi
i = 1, 2, …, n
Consider the following transformation of coordinates,
 
( x, x , t ) 
 
( x, p, t )
where pi 
L 
(39)
L  
( x, x, t )
 xi
H
 
 
H ( x, p, t )   x i pi - L( x, x , t )
i
   
 
  x i ( x, p, t ) pi - L( x, x ( x, p, t ), t )
i
The Hamiltonian function
(43)
On one hand

dH =
i
H
H
H
dxi  
dpi 
dt
xi
pi
t
i
(41)
 
 
On the other hand, the expression for H ( x, p, t )   xi pi - L( x, x , t ) given in (40)
i
implies,
dH =
 p dx   x dp  
i
i
i
i
i
i
i
L
dxi
xi
L
(d
)
dt xi


i
L
L
dxi 
dt
xi
t
pi
( d pi )
dt
p i
Thus, the first and fourth sum-terms cancel out
dH =
 x dp
i
i


i
p i dxi
i

L
dt
t
From (41) and (42) one obtains,
 
 
H ( x, p, t )   x i pi - L( x, x , t )
i
   
 
  x i ( x, p, t ) pi - L( x, x ( x, p, t ), t )
i
(42)
xi 
H
pi
Hamilton canonical
equations
H
p i  
xi
(43)
(2n first-order equations)
i = 1, 2, …, n
and
H
L

t
t
(44)
Notice if L does not depend on t explicitly, neither does H.
Recipe for solving problems in mechanics
 
i) Set up the Lagrangian L( x, x , t ) .
ii) Obtain the canonical momenta pi 
iii) Obtain
L  
( x, x, t )
 xi
 
 

H ( x, p, t )   xi pi - L( x, x , t ) in terms of x and p .
i
4.2C Definition of the Poisson bracket
Finding constants of motion before calculating the
motion itself
 
L let’s consider a dynamical function F= F ( x, p ,t ) and calculate its total differential
change with time,
dF

dt

i
F
F
F
xi  
p i 
xi
pi
t
i
H
pi
-
H
xi

i
dF

dt
F H
F H F


xi pi i pi xi t
 F H F H  F
 


p

p

x
t
i
i 
 i i
  x
i
(48)
It makes sense to define
F, H     F H  F H 
i
 xi pi
(49)
pi xi 
Poisson bracket between F and the
Hamiltonian H.
In terms of the Poisson bracket, the time dependence of a physical quantity is expressed
as,
dF
F
 F, H  
dt
t
(48)’
Looking for physical quantities (F) whose Poisson bracket
with the Hamiltonian vanishes
A quantity F that does not depend explicitly on time, will be a constant of motion if the
Poisson bracket between F and the Hamiltonian H vanishes. Certainly, one would have
to have a lot of intuition to figure out such a function F. We will see later, however, that
there exist systematic methods to find just that. Here we just want to show the
possibility of finding constant of motion without solving the equations.
Example: Two dimensional, symmetric simple harmonic
oscillator. 2 For simplicity take k=1 and m=1. Then,
1
1
L( x1 , x2 , x1 , x2 , t )  ( x1  x2 )  ( x1  x2 )
2
2
2
2
2
2
(49)
pi 
L
( x1 , x2 , x1 , x2 , t )  xi for i= 1,2
 xi
 
(50)
 
H ( x, p, t )   xi pi - L( x, x , t )
i
H ( x1, x2 , p1, p2 , t ) 
1
2
1
( p1  p2 )  ( x1  x2 )
2
2
2
2
(51)
2
Without finding the explicit solution, let’s show that
F ( q1 , q2 , p1, p2 , t )  x1 p2  x2 p1
angular momentum
(52)
Is a constant of the motion.
Proof. First, let’s calculate,
F H F H

 p2 p1  x2 x1
x1 p1 p1 x1
F H F H

  p1 p2  x1 x2
x2 p2 p2 x2
F
0
t
which gives
 F H F H  F

=0

pi xi  t
i 1 
i pi
2
  x
Accordingly, expression (48) gives,
dF
0
dt
(53)
That is, without explicitly calculating the solution, we know that, for this problem, the
angular momentum is a constant of motion.
4.2D The modified Hamilton’s principle: Derivation of the
Hamilton’s equations from a variational principle.
Hamilton’s canonical equation can be obtained from a variational principle, similar
to the way the Lagrange equations were obtained in Section 4.1B above. However, the
variations will be over paths in the ( x, p) phase-space, which has 2n dimensions, twice

the n dimensions of the x configuration-space. Interestingly enough, the function inside
the integral, upon which the variational principle will be applied, is again the Lagrangian
L, but now considered as a function of ( x, x , p, p ,t ) .
L( x, x , p, p ,t )   x i pi - H ( x, p, t )
(55)
i
As a first step, let’s apply the variational principle to,
( x2 , p2 ,,t2 )

S ( x, p) ) ≡
L( x, x , p, p ,t ) dt
(56)
( x1 , p1 , t1 )
( x2 , p2 ,,t2 )
S ( x, p) ) ≡



 x i pi - H ( x, p, t )  dt
( x1 , p1 , t1 )  i

Inside the integral, we have just written ( x, x , p, p ,t ) for
simplicity, but
respectively.
( x(t ), x (t ), p(t ), p (t ) ,t ) should be used instead,
In applying the variational principle, we realized it is very similar to the case when we
applied it in the configuration space. This time we just have more independent
variables.
p i  
H
xi
xi 
H
pi
That is, we have obtain in (58) the canonical Hamiltonian equations (43).
4.3 The Poisson bracket between two arbitrary variables
 
In general, the Poisson bracket [S,R] of the dynamical variable S= S ( x, p) with the
 
dynamical variable R= R ( x, p) is defined as,
S, R    S R  S R 
pi xi 
i  xi pi
Poisson bracket of the
dynamic quantities R
and S
(55)
4.3A The Hamiltonian equations in terms of the Poisson brackets
dxi
 [ xi , H ]
dt
Hamilton canonical
equations
(2n first-order equations)
dpi
 [ pi , H ]
dt
T
(59)
i = 1, 2, …, n
constitutes an alternative way to express the Hamilton canonical equations.
4.3B Fundamental brackets
x , x     xx



i

i
x x x 
= 0

pi pi xi 
 p p p p 
,
p

i  x p  p x  = 0


i
i
i 
 i
p

for any 1, 2, ..., n
for any 1, 2, ..., n
x , p     xx
for ≠
i
p x p 
= 0

pi pi xi 
x , p     xx
p x p 
= 1

pi pi xi 
for =



i




i

i
1
2
H. Goldstein, Classical Mechanics, Addison-Wesley Publishing (1959).
Saletan and Cromer, “Theoretical Mechanics” Wiley, 1971.), page 180.