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QM 480
“On the Shoulders of Giants”
An Introduction to Classical Mechanics
QM 480
If I have seen further it is by standing on the shoulders of giants. Isaac Newton, Letter to Robert Hooke,
February 5, 1675English mathematician & physicist (1642 - 1727)
QM 480
Quantum Mechanics (QM) is based on classical mechanics. It combines classical mechanics with statistics and statistical mechanics.
For native English-speakers, it is somewhat unfortunate that it uses the word “quantum”. A better English word which describes the thrust of this approach would be “pixel”.
QM 480 Lights! Camera! Action!
2nd Century BC Hero of Alexandria found that light, traveling from one point to
another by a reflection from a plane mirror, always takes the shortest possible path.
1657 Pierre de Fermat reformulates the principle by postulating that
the light travels in a path that takes the least time! In hindsight, if c is constant then Hero and Fermat are in
complete agreement. Based on his reasoning, he is able to deduce both the law of
reflection and Snell’s law (nsin = n’ sin’)
QM 480 An Aside
Fermat is most famous for his last theorem:
Xn +Yn = Zn where n=2 and …
On his deathbed, he wrote:
And n= arrgh! I’m having a heartattack!
His last theorem was only solved by computer in the last 10 years…
QM 480 Now we wait for the Math
1686 The calculus of variations is begun by Isaac
Newton
1696 Johann and Jakob Bernoulli extend
Newton’s ideas
QM 480 Now we can get back
1747 Pierre-Louise-Moreau de Maupertuis asserts a
“Principle of Least Action” More Theological than Scientific “Action is minimized through the Wisdom of God” His idea of action is also kind of vague
Action (today’s definition)— Has dimensions of length x momentum or energy x
time Hmm… p * x or E*t … seems familiar…
QM 480 To the Physics
1760Joseph Lagrange reformulates the principle
of least action
The Lagrangian, L, is defined as L=T-V where T= kinetic energy of a system and V=potential energy of a system
QM 480 Hamilton’s Principle
1834-1835 William Rowan Hamilton’s publishes two
papers on which it is possible to base all of mechanics and most of classical physics.
Hamilton’s Principle is that a particle follows a path that minimizes L over a specific time interval (and consistent with any constraints).
A constraint, for example, may be that the particle is moving along a surface.
QM 480 Lagrange’s Equations
02dt
d
2dt
d
dt
d
00
gRearrangin
Recall
2
2
xm
xdx
d(-U)
so
xm
xxmxm
xmdx
d(-U)and xm
dx
dU-
dx
dU(x)-xmF
QM 480 Lagrange’s Equations
0)(02
2
xUx
andxm
x
Now
And I can add zero to anything and not change the result
0dt
d
2
0)(2dt
d)(
2
2
22
x
L
dx
dL
Thus
VTLandTxm
but
xUxm
xx-U
xm
dx
d
QM 480 Expanding to 3 Dimensions
Since x, y, and z are orthogonal and linearly independent, I can write a Lagrange’s EOM for each. In order to conserve space, I call x, y, and z to be dimensions 1, 2, and 3.
So
Amusingly enough, 1, 2, 3, could represent r, , (spherical coordinates) or r, , z (cylindrical) or any other 3-dimensional coordinate system.
3,2,10dt
d
iq
L
dq
dL
ii
QM 480Example: Simple Harmonic
Oscillator Recall for
SHO: V(x)= ½ kx2 and let T=1/2 mv2
Hooke’s Law: F=-kx
xmkxorxmkx
so
xmxmandxmxmxx
L
kxkxdx
d
dx
dL
kxxmL
q
L
dq
dL
0
dt
d
2
1
2
12
1
2
1
0dt
d
2
2
22
QM 480 Tip
The trick in the Lagrangian Formalism of mechanics is not the math but the proper choice of coordinate system.
The strength of this approach is that
1. Energy is a scalar and so is the Lagrangian
2. The Lagrangian is invariant with respect to coordinate transformations
QM 480Two Conditions Required for
Lagrange’s Equations
1. The forces acting on the system (apart from the forces of constraint) must be derivable from a potential i.e. F=-dU/dx or some similar type of function
2. The equations of constraint must be relations that connect the coordinates of the particles and may be functions of time.
QM 480 Your Turn
Projectile: Go to the board and work a simple projectile
problem in cartesian coordinates. Don’t worry about initial conditions yet.
Now do the same in polar coordinates.
Hint:
sin2
1
2
1 22
mgrU
rmrmT
QM 480 Introducing the Hamiltonian
First, any Lagrangian which describes a uniform force field is independent of time i.e. dL/dt=0.
L
q
L
dt
dq
dt
dL
q
L
dt
d
q
LSince
Lq
q
L
dt
dL
t
L
t
q
q
L
t
q
q
L
dt
dL
qqLL
),(
QM 480 Introducing the Hamiltonian
Hmmm… H for Hornblower or Hamilton?
HttanconsaLq
Lq
So
Lq
Lq
dt
d
dt
dL
q
Lq
dt
d
q
Lq
dt
dq
q
L
q
L
dt
dq
dt
dL
0
QM 480 Introducing the Hamiltonian
)!(
)(2
22
1
)()(
22
energymechanicalEVT
but
HVTVTT
So
Tqmq
Tqqm
q
TqmTIf
q
T
q
qVqT
q
L
HLq
Lq
QM 480 H is only E when
It is important to note that H is equal to E only if the following conditions are met: The kinetic energy must be a homogeneous
quadratic function of velocity
The potential energy must be velocity independent
While it is important to note that there is an association of H with E, it is equally important to note that these two are not necessarily the same value or even the same type of quantity!
QM 480Making Simple Problems
Difficult with the Hamiltonian
Most students find that the Lagrangian formalism is much easier than the Hamiltonian formalism
So why bother?
QM 480Making Simple Problems
Difficult with the Hamiltonian
First, we need to define one more quantity: generalized momenta, pj
LqpHorLqpH
becomesLq
LqHSo
jwhereq
Lp
jjj
jj
3
1
3,2,1
QM 480 SHO with the Hamiltonian
22
222
22
2
22
2
1
2
2
1
2
2
1
2
1
kxm
pH
kxm
p
m
pL
m
ppH
becomesLqpHSo
xm
px
m
pxm
q
Lp
kxxmL
Big deal, right? But look what we did
L=f(q,dq/dt,t) H=f(q,p,t) So our mechanics all
depend on momentum but not velocity
Recall light has constant velocity, c, but a momentum which is p=hc/ !
QM 480 The Big Deal
So if we are going to define mechanics for light, it does not make any sense to use the Lagrangian formulation, only the Hamiltonian!
QM 480 That Feynman Guy!
Richard Feynman thought that Lagrangian mechanics was too powerful a tool to ignore.
Feynman developed the path integral formalism of quantum mechanics which is equivalent to the picture of Schroedinger and Dirac.
So which is better? Both and Neither There seems to be no undergraduate treatment of
path integral formalism.
QM 480Hamilton’s Equations of
Motion Just like Lagrangian formalism, the Hamiltonian
formalism has equations of motion. There are two equations for every degree of freedom
They are
q
Hp
p
Hq
QM 480 Finishing the SHO
kxF
pdt
dpFSince
kxporkxq
Hp
m
p
p
Hx
kxm
pH
22
2
1
2
Hooke’s Law again!
QM 480 Symmetry
Note that Hamilton’s EOM are symmetric in appearance i.e. that q and p can almost be interchanged!
Because of this symmetry, q and p are said to be conjugate
q
Hp
p
Hq
QM 480 Definition of Cyclic
Consider a Hamiltonian of a free particle i.e. H=f(p)… then – dp/dt=0 i.e. momentum is a “constant of the motion”
Now in the projectile problem, U=-mgy and for x-component, H=f(px) only!
Thus, px= constant and the horizontal variable, x is
said to “cyclic”!
A more practical definition of cyclic is “ignorable” and modern texts sometimes use this term.
QM 480 Definition of canonical
Canonical is used to describe a simple, general set of something … such as equations or variables.
It was first introduced by Jacobi and rapidly gained common usuage but the reason for its introduction remained obscured even to contemporaries
Lord Kelvin was quoted as saying “Why it has been so called would be hard to say”
QM 480 Poisson Brackets
},{},{
0},{
0},{
0},{
p? and q of functions were vandu ifWhat
},{
as defined is
p and q variablescanonical therespect to with vandu ofBracket Poisson
, jijiji
ji
yx
ji
qppq
pp
p
x
y
y
p
y
x
xyxExample
p
u
q
v
p
v
q
uvu
QM 480 Kronecker Delta
i,k=1 if i=k
i,k=0 if i≠k
QM 480 Back to Fish
Consider two continuous functions g(q,p) and h(q,p) If {g,h}=0 then h and g are said to commute In other
words, the order of operations does not matter
If {g,h}=1 then quantities are canonically conjugate
• A look ahead: we will find that canonically conjugate quantities obey the Uncertainty principle
QM 480 Properties of Fish
},{ c)
},{ b)
p)H(q,Hn HamiltoniaH where
},{ a)
BracketPoisson theof properties are following The
Hpp
Hqq
t
gHg
dt
dg
jj
jj
QM 480 Levi-Civita Notation
1
1
0
order) of(out n permutatio oddan isk j, i, if 1-
2,3 1, of permuationeven an isk j, i, if 1
otherany equalsindex any if 0
C
notation
compact ain expressed becan components individual The
B AC
B and A ofproduct vector heConsider t
132321213
231312123
133112122
,i
ijk
kjkjijk
where
BA
QM 480 Levi-Civita Notation
23132321231
132
123
23321
321
321
1
1
Consider
3̂2̂1̂
C
B and A ofproduct vector heConsider t
BABAC
BABAC
BBB
AAA
