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Published : Tuesday, March 27, 2012
Reading/s : 16
Origin : umich.edu
Number of pages: 12
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Chapter 1 – The Principles of Classical Physics

Quantum mechanics was developed over the first four decades of the 20-th century,
following the surprising failure of classical physics to provide a consistent description of
the behavior of light and matter on the atomic scale. It is therefore important to
understand the basic principles underlying classical physics, before proceeding to
quantum mechanics.
Roughly speaking, classical physics is split into two major, seemingly unrelated,
branches: (1) Classical mechanics, pioneered in the 17-th century by Newton, which
describes the motion of particles, and (2) The classical theory of electromagnetism,
brought to completion in the 19-th century by Maxwell, which, among other things,
describes electromagnetic radiation in terms of waves. This chapter provides a brief
outline of each of those branches, with emphasis on concepts that will later help us
understand quantum mechanics.

The postulates of classical mechanics

Classical mechanics was meant to provide the general rules that govern the
dynamics of all material bodies, such as cannon balls, planets, and pendulums. It is
important to understand that classical mechanics is a theory. The rules that it laid down
for describing dynamics were justified by the fact that they were consistent with
practically all known observations (with a few exceptions that were attributed to
misunderstanding of the observations that could be later resolved within the framework
of classical physics).
Theories are based on postulates. Postulates are rules of nature, which cannot be proven,
and derive their justificaction form the fact that they are consistent with experiment.
Classical mechanics is based on two postulates:

1. The state postulate, which defines what constitutes a complete description of the
state of a system.

2. The time evolution postulate, which tells us how to predict the future state of the
system, or reconstruct its past, based on the knowledge of its present state.

A more detailed discussion of those postulates follows.







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The classical state postulate

A material body consists of tiny point particles. The state of each point particle at time
rt( ) p(t)t is completely defined by its position, , and momentum, .

Important points:

1. Actually, a complete description must also include the masses, charges, etc. of each
of the particles. However, unlike rt( ) and p()t , those parameters do not change over
time.
2. All other dynamical variables of interest, such as kinetic and potential energy,
forces, and angular momenta, can be expressed in terms of rt( ) and p(t) .
3. rt( ) and p(t) can be represented by their coordinates relative to an inertial Cartesian
1coordinate system :
GG
rt()=[x(t),y()t ,z(t)], p(t)=[p ()t ,p (t),p ()t ]. (1) xy z
4. For the systems that we will be interested in, p()tm=v()t =m(dr/dt) , where m is the
mass of the point particle and vt( ) = [v (t),v (t),v (t)] is its velocity.
xy z
5. The state of a classical system is often described by a point in a 6D space whose
coordinates correspond to x,,yz,p,p,p . The latter is known as the system’s phase xy z
space.

The classical time evolution postulate

Given that the state of a point particle is given by r(0) and p(0) at time t = 0, one can
predict its state at any other time t (t may follow or proceed t = 0, i.e. t > 0 or t < 0 ,
respectively), by using the classical equation of motion (Newton’s second law):
2dd
mr()t==p()t F[r(t),t]. (2)
2dt dt
Here, Fr[,t] is the force that the particle experiences when in position r at time t .

Important points:

1. The classical equation of motion, Eq. (2), is a rule of nature, or a postulate, and as
such cannot be derived.

1 The above discussion applies to non-relativistic classical mechanics. The latter is assumed to be valid in at
least one geometrical frame, which is known as the Galilean or inertial frame. According to the Galilean
relativity principle, if a Galilean frame exists, all frames that move in constant velocity relative to it are also
Galilean frames (i.e. classical mechanics is valid in them). Thus, there is no one unique and absolute
Galilean frame. Those assumptions are no longer valid at velocities that approach the speed of light,
where special relativity theory is required. We will not deal with relativistic corrections in this course.
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2. The time evolution postulate assumes that there is a force, but doesn’t give it to us
explicitly. In other words, in order to make predictions (or reconstruct the
particle’s past), we have to know what force the system experiences at any given
point throughout its trajectory.
3. It is a common practice to express the force in terms of the potential energy,
Vr(,t) , such that
GG ⎡∂∂∂V V V ⎤
Frt(, ) =−∇V(r,t) = − , , . (3) r ⎢ ⎥∂∂x yz∂⎣ ⎦
Here, ∇ is the gradient, which operates on a function of x,y and z, and outputs a r
vector whose components are the partial derivatives of the function with respect
to x,y and z. As for the force, the potential at the position of the particle is needed
at every point in time in order to solve the classical equation of motion.
4. The dynamics of the system is often described in terms of the trajectory that the
point that represents the state of the system traces in phase space.
5. From a mathematical point of view, the classical equation of motion for a point
particle corresponds to a set of three coupled second order differential
equations, which require six initial conditions in order to be solved,
namelyxy(0), (0),z(0),p (0),p (0),p (0). Note that the solution will be xy z
explicitly dependent on the particular choice of initial conditions (if you start at
different places and/or with different velocities, you will most probably end up at
different places).

The extension of the above discussion to systems that consist of N point particles is as
follows:

1. The state of the system is completely given by the positions and momenta of all
theNpoint particles, at timet :{,r r }and {,p p}. As a result, the 1 N 1 N
corresponding phase space is 6N-dimensional.
2. The classical equation of motion is given by:
2dd GG GG
m r ()t==p ()t F[rt( ),…,r (t),t]=−∇V[rt(),…,r ()t ,t] , (4)
jj j j11N r N2 jdt dt
where rx=(,y,z) is the position of particle j (j=1,…,N) and
jjjj
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F V ⎡Vx/,Vy/ ,Vz/ ⎤=−∇ = − ∂ ∂ ∂ ∂ ∂ ∂ (5) jr j j jj ⎣ ⎦
is the force that particle j experiences (either due to an external potential, or due to
interactions with the other particles).

The classical Hamiltonian and the law of energy conservation

The classical Hamiltonian function of a given system gives its energy as a function of
positions and momenta. For a point particle of mass m moving under the influence of the
potential energy Vr ( ) (which we assume, for simplicity, to be time-independent), the
Hamiltonian is given by:
() HT= +Vr , (6)
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where
22 2 2 2 T==mv /2 p /2m=ppp++ /2m (7) ( )xy z
is the kinetic energy. The classical Hamiltonian of a system that consists of N point
particles is given by
HT= +V(,r …,r ) , (8) 1 N
where
NN
2 TT== p /2m . (9) ∑∑j j j
jj11
The importance of the Hamiltonian, and the reason for why it is defined in the way it is,
has to do with the fact that energy remains constant over time (as long as the potential
energy is not explicitly dependent upon time):
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NNNpdp drdd d2 jj j , (10) Hp=+ /2m V(r,…i,r )= +∇Vi= 0∑ ∑∑jj 1 N jdt dt dt m dt dtjj==11j=1j
The second equality in Eq. (10) is based on the chain rule, according to which
df x()t /dt =(df /dx)(dx/dt) . (11) [ ]
The last equality in Eq. (10) is based on the fact that p =mdr /dt and jjj
dp /dt =−∇ V [Cf. Eq. (4)]. The above result is known as the energy conservation law j j
in classical mechanics, and can be summarized as follows:

The law of energy conservation: dH /dt = 0 (as long as V is time-independnet). (12)

Example – classical mechanics of the harmonic oscillator.

Consider a point particle of mass m , which is attached to one end of a spring. The other
end is assumed to be attached to a wall, and other forces, such as gravity, are ignored.
The particle moves along a line, which we can choose to coincide with the x axis. Thus,
the state of the particle is described by x and p , and the corresponding phase space is x
2D.
Based on empirical observation, we know that the following potential energy is
appropriate in this case (as long as the displacement from the equilibrium distance is not
too large):
1 2 Vx() = kx , (13)
2
where k > 0 is the spring constant (whose actual value depends on the spring’s
stiffness). It should be noted that the potential energy has a minimum at the origin
(x = 0 ). This minimum is known as the equilibrium point.
The force in this case is linear in the displacement (note that ∇→dd/ x in this case, r
since the motion is in 1D):
d 1⎛⎞ 2 F =− kx=−kx (14) ⎜⎟dx 2⎝⎠
4The negative sign on the R.H.S. of Eq. (14) implies that this is a returning force, namely
– the force opposes displacements relative to the equilibrium position (the force is zero at
the equilibrium point).



Figure 1: A particle on a 1D harmonic potential.
The classical equation of motion is given by
2d
mx()t =−kx(t) . (15)
2dt
It is easy to verify (by substitution) that the following is a solution of this differential
2equation (in fact, one can also show that this is the most general form of the solution):
x(tA) =cos( ωt) +Bsin( ωt) , (16)
where ω = k /m is the angular frequency, and A,B are constants, which can be
directly related to x(0),p(0) :
xA(0) =
dp(0) . (17)
pm(0)==x(t) m ωB⇒B=
dt m ωt =0
Thus, the state of the harmonic oscillator at time t is given by:
p(0)
xt( )=+x(0)cos(ωωt) sin( t)
. (18) m ω
p(tm) =− ωx(0)sin()t +p(0)cos()t
The classical Hamiltonian of the harmonic oscillator is given by
2p 1 2 2 Hm=+ ωx . (19)
22m
It should be noted that because x and p are continuous variables (can get any value on
the real axis), so is the energy. It should also be noted that the energy is non-negative
(H ≥ 0 ), and that the lowest possible energy is zero.
It is also easy to verify that energy is conserved, i.e. thatdH/0dt = :

2 The fact that the classical equation of motion of the harmonic oscillator can be solved analytically, makes
it an exception. The classical equation of motion of most relevant systems can not be solved analytically.
Nowadays, the solution is often found numerically, on a computer.
5

2⎛⎞d d p 1 p dp dx p p22 2 2 2 (20) Hm=+ωωx= +mx=−mωx+mωx= 0()⎜⎟
dt dt22m m dt dt m m⎝⎠
The harmonic oscillator is an example of a system subject to a bounded potential. This
means that the potential energy confines the particle to a finite region of space. More
specifically, the particle moves between two turning points, located at x =±x , where 0
the momentum vanishes:
1 22 2 Hm=⇒ωx x=± 2/Hm ω . (21) 002

Figure 2: The turning points on the harmonic potential
It should be noted that x increases with the energy, H (the amplitude of the motion 0
increases with the energy). However, for a given energy, a classical particle moving in a
harmonic potential cannot penetrate the region where −x >>x x . 00
x()tp, (t) in Eq. (18) trace an ellipsoidal trajectory in phase space. The trajectory can in
fact be made circular if plotted in terms of the renormalized position and momentum,
x = m ωx and p =p /m ω. The radius of the circle is
22 2 2xp+=m ωx+p/mω=2H/ω . Thus, energy conservation is manifested by the
fact that the radius of the circular trajectory is fixed.


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Figure 3: Phase space trajectory of the harmonic oscillator.


Classical wave theory

Mechanical waves

Long before the development of quantum mechanics, it was well known that some
phenomena are more naturally described in terms of waves. Common examples are:
1. Waves along a string (e.g. in a violin).
2. Waves on a liquid surface, which result from a local perturbation, such as that
caused by throwing a stone into a pond.
3. Sound waves, which correspond to propagating density disturbances in a material
medium.
All of these phenomena can be described in terms of the displacement of some quantity,
relative to a reference value, at different points in space and time. Thus, the waves along
a string can be described in terms of the displacement in height, relative to the height of
the unperturbed string, as a function of position along the spring and time: Hx ( ,t) (note
that this wave moves in 1D). The wave on a liquid surface can be described in terms of
the displacement in height, relative to the height of the unperturbed liquid, as a function
of position and time: Hx ( ,y,t) (note that this wave moves in 2D). Finally, a sound wave
can be described by the density variation relative to that of the unperturbed medium, at
every point throughout the medium, and at any time, Dx( ,y,z,t) =D(r,t) (note that this
wave moves in 3D).
Thus, generally speaking, a wave can be mathematically described by a function that
describes the displacement, or wave amplitude, at any point in space and time:Ar( ,t) . It
is important to note that waves represent a delocalized phenomenon, in the sense the
perturbation is “smeared” throughout the whole space, which should be contrasted with
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the localized nature of the trajectory in classical mechanics, which is given by rt( ) (only
one position at a time).
The examples described above correspond to mechanical waves, where the medium
consists of material particles. Thus, at least in principle, one could describe the
underlying motion that gives rise to the wave via classical mechanics (as long as the
forces between the particles are known). In such cases, the description in terms of waves
simply provides a more convenient way of describing the motion of a complex system in
cases where we are not really interested in what each of the medium particles is doing.

Electromagnetic waves

Classical physics offers at least one important example of a wave phenomenon, which
does not have a mechanical analogue. The latter corresponds to electromagnetic
radiation (Radio waves, microwave radiation, IR, visible and UV radiation, X-rays, etc.).
According to Maxwell’s celebrated theory of electromagnetism, electromagnetic
radiation can be described in terms of oscillating electric and magnetic fields (which can
be described as perpendicular to each other and to the direction of propagation).
However, such waves can be propagated in vacuum, in the absence of any material
medium! Thus, at least for electromagnetic radiation, a description in terms of classical
mechanics is not possible.

Figure 4: Electromagnetic waves.
Traveling waves

We will focus for simplicity on a 1D wave along a string and consider a 1D wave of the
following form:
Ax(,t) =A cos(kx−+ωt φ) . (22) 0
A wave that has this form is called a traveling wave. In order to visualize the behavior of
this wave, it is convenient to consider its time dependence at a fixed position, and its
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position dependence at a fixed time (the latter corresponds to taking a snap shot of the
wave at a given time):
1. At a fixed position, say x = 0 , the wave amplitude oscillates in time between +A 0
and −A , according to At(0, ) =A cos( ωt − φ). Thus, the displacement at a given 0 0
point has the same time-dependence as a harmonic oscillator of angular frequency
ω (which is measured in Radians per unit time). The circular frequency is defined
by ν =ω /2π , and is measured in units of cycles per unit time. The period of the
−1−1oscillation, that is the time per cycle, is given by τ==νω /2π . ()
2. At a fixed time, say t = 0, the wave amplitude oscillates in space between +A and 0
−A , according to Ax(,0)=A cos(kx+ φ) . k is the wave number, and λ =2/π |k| 0 0
is the wavelength (note that k can be positive or negative, depending on the
direction of the wave propagation). This means that if we take a snapshot of the wave
at a given time, then λ would be the distance between subsequent peaks.

The above discussion can be easily generalized to a 3D wave. The main difference is the
wave number becomes a vector:
Ar(,t) =A cos(k ir−+ωt φ) . (23) 0


Figure 5: A traveling wave in 1D.

The superposition principle

Not all waves have the form of a traveling wave. However, it can be shown that all
waves can be given in terms of a linear combination of traveling waves (stated without
proof):
Ax( ,t)=−g cos(kx ωt+ϕ)+g cos(kx−ωt+ϕ)+ ... (24) 11 1 2 2 2
In more mathematical terms, the complete set of traveling waves (corresponding to
different values of k and ω ) constitutes a basis, which can be used to represent any other
wave function.
For example, if two traveling waves, Ax(,t) =A cos(kx−+ωt φ) and 10,1 1 11
Ax (,t)=A cos(kx− ωt+ φ ), overlap in space and time, the overall wave amplitude 20,2 2 22
9superposition:Ax(,t) =A(x,t) +A(,xt) . The important point is is given by their sum, or 1 2
that although Ax( ,t) is a sum of two traveling wave, it is not a traveling wave (unless
kk= , ω = ω and φ = φ ). 12 1 2 12
The superposition principle is responsible for several unique phenomena that
distinguish waves from particles. One particularly simple example corresponds to the
superposition of two traveling waves that have the same amplitude, wave vector and
frequency, but are phase-shifted with respect to one another (i.e. AA= =A , 0,1 0,2 0
kk= =k , ω=ω=ω but φ ≠ φ ): 12 12 12
A(,x t)=−A cos(kx ωt+φω)+ A cos(kx− t+φ )
010 2
, (25) ∆ φ⎛⎞2cAkos cosx ωφt+( )0 ⎜⎟2⎝⎠
where φφ=+( φ)/2 and ∆ φ=−φ φ. The amplitude of A(,xt) is now 12 1 2
2cA os ∆ φ/2 , and depends explicitly on the phase difference between the two waves, ( )0
∆ φ . More specifically, the amplitude vanishes when cos ∆ φ / 2 = 0 , which corresponds ( )
to ∆=φ/2(n+1/2) π , or ∆=φ (2n+1) π (n =0,1±± ,2, …). This case corresponds to
destructive interference, where the two waves completely cancel each other out. The
opposite occurs when cos∆ φ / 2=±1, which corresponds to ∆ φ / 2 =n π , or ∆=φ 2n π . ( )
This case corresponds to constructive interference, where the two waves build each
other up.


Figure 6: Destructive (top) and constructive (bottom) interference.

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