1.1
Conservation of Energy
Objective 1: Understand the transfer of energy from one form to another
Objective 2: Understand how energy of an object can be transformed into work done by that
object on its surroundings.
Objective 3: Compare the Kinetic Energy of particles of different mass.
Objective 4: Recall the Hamiltonian Equation for Total Energy
Objective 5: Understand and Calculate Momentum and Kinetic Energy for Relativistic
conditions.
1.1.1 Total Mechanical Energy
Objective 1: Understand the transfer of energy from one form to another
Objective 2: Compare the Kinetic Energy of particles of different mass.
The total mechanical energy of a particle can be defined as the sum of its Kinetic and Potential
energies.
. . . .
where the kinetic energy is equal to the expression,
1
. . 2
and the potential energy is expressed using the variable U and is a function of position (x),
(In quantum mechanics, the potential energy is represented by the symbol V, but we will use U
to avoid confusing the potential energy with potential.)
When a force is exerted on an object, this can result in a change in the kinetic energy and
potential energy (total mechanical energy) of the particle. Based on the law of conservation of
energy, the final energy is equal to the initial energy or,
In terms of the initial and final kinetic (K) and potential (U) energies,
or
and
∆ ∆
This expression implies that energy can be transformed from kinetic energy to potential energy
and vice versa. A ball thrown up in the air will have zero kinetic energy at its highest point,
where the potential energy is maximum. As the ball begins to descend, the kinetic energy of
the ball will increase as its potential energy goes to zero. Energy can be expressed in joules (J)
or eV, where 1.6 x 10-19 J is equal to 1 eV.
Example 1.1: Determine the velocity of an electron with kinetic energy equal to (a) 10 eV, (b)
100 eV, (c) 1,000 eV.
Solution:
The kinetic energy of the electron is equal to . . 2 . .
, and therefore v is equal to
Since the mass of an electron is 9.11 x 10-31 kg, the velocity of an electron with kinetic energy of
10 eV is equal to,
2 10 9.11 10
1.6 10# $ % ! '
!
1 $ % &
1.87 10+ ⁄&
,-. . . 100 , 5.93 10+ ⁄&
,-. . . 1000 , 1.87 102 ⁄&
⁄
Example 1.2: Compare the values of velocity for an electron with m = 9.11 x 10-31 kg and
kinetic energy equal to those specified in Example 1.1 with the velocities for a proton with m =
1.6 x 10-27 kg assuming the same kinetic energy values as those given in Example 1.1: (a) 10 eV,
(b) 100 eV, (c) 1,000 eV.
Solution:
From Example 1.1, the values of velocity for an electron are,
,-. . . 10 ,
,-. . . 100 ,
,-. . . 1000 ,
1.87 10+ ⁄&
5.93 10+ ⁄&
1.87 102 ⁄&
Similar calculations for a proton with heavier mass result in the following velocity values,
,-. . . 10 ,
,-. . . 100 ,
,-. . . 1000 ,
4.47 104 ⁄&
1.41 105 ⁄&
4.47 105 ⁄&
Therefore, a heavier particle with similar kinetic energy as the electron will result is a lower
velocity.
We will talk about how an electron gains this kind of energy in Section 1.2.
1.1.2 Work
Objective 1: Understand the transfer of energy from one form to another
Objective 2: Understand how energy of an object can be transformed into work done by that
object on its surroundings.
The work (W) done on an object or particle is equal to the force exerted on the particle times
the distance the particle moves.
6 , ·8
When a force moves an object from point a to point b, the work done is equal to the change in
the Total Mechanical Energy (assuming friction is equal to zero). Therefore,
69:; ∆ ∆
This implies that the energy of a particle can be converted to work done by the particle, or work
done on a particle can be converted to energy of the particle.
Energy can be expressed in units of Joules (J), where 1 J = 1 N m, or it can be expressed in units
of electron volts (eV), where 1 eV = 1.6 x 10-19 J.
1.1.3 Momentum
Objective 1: Recall the Hamiltonian Equation for Total Energy
The Kinetic Energy can also be expressed in terms of momentum as shown below.
. . <
. . where p2 is the vector dot product = · =.
1
<
2
=
1
? > @
2
. . ?
2
The equation for total energy can be written as
?
2
When the total energy is expressed in terms of momentum and position, it is called the
Hamiltonian Total Energy or
?
A
2
This equation is important because it will be used to obtain Schrödinger’s equation in Section
3.3.
1.1.4 Rest Mass and Relativistic Calculations
Objective 1: Understand the transfer of energy from one form to another
Objective 2: Understand and Calculate Momentum and Kinetic Energy for Relativistic
conditions.
The terms for rest mass and rest energy are introduced here in order to be able to discuss
relativistic calculations of momentum and energy, i.e. calculations where the speed of an object
(v) is comparable to the speed of light (c = 3 x 108 m/s). These calculations obey the special
theory of relativity and one of the major conclusions drawn from this theory is that an object
being accelerated through space by a force can never reach the speed of light (in contrast, the
classical laws of motion assume that the speed of a particle can increase indefinitely).
The rest mass is usually discussed when talking about the special theory of relativity and is
defined as the mass at rest of a single body (elementary particles such as electrons, atoms,
balls, spaceships or the sun) in a given inertial frame. The mass of an object in any inertial
frame is measured at rest (v = 0) and is therefore equal for all inertial frames. For classical
theories regarding the law of conservation of mass, the mass is conserved in all processes. In
reality, there is a very small change in the rest mass associated with species involved in
chemical reactions, and very large changes in the rest mass for particles in nuclear reactions,
where the mass can be eliminated completely.
Similarly, the rest energy is defined as the energy of an object when v = 0 and is equal to
B This implies that an object at rest has energy and that this energy can be converted to other
forms of energy such as kinetic energy or heat. For instance, the rest energy for 1 kg of
uranium is 9 x 106 joules. This is a significant amount of energy and is equal to the amount of
energy generated by a power plant in 1 year. Is it possible for an object at rest to have that
much energy and to convert that energy into other forms of energy? Actually yes, this value is
comparable to the amount of energy converted to heat by the nuclear fission of uranium 235,
where 1 kg can produce 9 x 1013 joules of heat.
Example 1.3:: Determine the rest energy associated with one electron.
Solution:
Therefore, the rest energy of a single electron is approximately equal to 0.5 MeV.
In classical mechanics, the Galilean transformation is used to rel
relate
ate the laws of motion between
two inertial frames or two points of origin corresponding to where the observer is located. The
measurement of time (t) and distance (x) differ when the measurement is made by an observer
moving with the object being measure
measured
d vs a measurements made by an observer at rest
relative to the moving object.
In Fig. 1.1, frame S is at rest and frame S’ is moving at velocity equal to v1. An object in frame S’
that appears to be at rest by an observer in frame S’, will appear to be moving by an observer at
rest in frame S.
S
y
v=0
S’
v = v1
y
v1
x
x
Figure 1.1 – Two inertial frames S (at rest) and S’ (moving at velocity equal to v1) with an
object in S’. An observer in frame S’ would not observe that the object is moving, while an
observer at rest in frame S, would observe the movement of the object in S’.
The Galilean transformation equations (shown below) do not correctly model the laws of
motion at velocities approximating the speed of light (v → c).
The correct transformation equations were developed by Hendrik Lorentz and explained by
Einstein using the factor γ, where
γ 1
C1 B
This factor appears in many relativistic formulas, including those for relativistic momentum and
kinetic energy, which are the focus of this section.
Based on Einstein’s special theory of relativity, the measurement of time (t) and distance (x)
differ from the Galilean transformation by the factor γ, which includes the speed of light (c).
The Lorentz transformation or Lorentz-Einstein transformation is used to transform coordinates
for a given event observed by an observer in S (x, y, z and t) to those observed by an observer in
S’(x’, y’, z’ and t’) for velocities approaching the speed of light as follows:
D γ E
FD F
GD G
E D γ HE I
B
In this section we will be concerned mainly with the application of γ to calculate momentum (p)
and kinetic energy (K.E.) under relativistic conditions, and we will identify the velocity limit
where relativistic calculations are required to accurately model the motion, and thus the energy
of an object.
Based on the concept of special relativity and the conservation of momentum, relativistic
momentum, where v → c is defined as
? γ C1 B
When v << c, then γ = 1 and momentum is equal to the value calculated by the classical formula
? However, as the velocity of an object approaches the speed of light (v → c), the value of
C1 JK
LK
approaches zero and the momentum approaches infinity. This means that it would
take an infinite amount of energy to accelerate an object to the speed of light, which is not
possible.
Figure 1.2 illustrates the increase in momentum based on classical (p = mv) and relativistic (p =
γmv)
mv) calculations. Based on classical mechanics, the velocity can increase indefinitely. In the
relativistic model, the velocity approaches the speed of light, but an infinite amount of energy
would be required to actually achieve this value (c), which is not possible.
Figure 1.2 – Comparison of classical momentum and relativistic momentum models.
models
Example 1.4: The typical velocity of an electron being accelerated in a cathode ray tube is 5 x
107 m/s. An electron that is accelerated for the purpose of creating high energy radiation for
cancer treatment can reach velocities as high as 2.94 x 108 m/s. Compare the values of γ for the
electron moving at a velocity equal to 5 x 107 m/s to that of an electron moving at 2.94 x 108
m/s.
γ Solution:
1
C1 B
For an electron traveling at v = 5 x 107 m/s (where v/c = 0.17),
γ 1
m H5 x 102 s I
M1 H3 10Q & I
1.01
For and electron traveling at v = 2.94 x 108 m/s (where v/c = 0.98),
γ M1 1
m sI
H2.94 x 10Q
H3 10Q & I
5
Relativistic calculations are required for both velocity values because v>0.1c.
A similar equation was derived for the energy of a moving object (v ≠ 0),
γ B This equation models the total energy of a moving object (v ≠ 0) with zero potential energy. A
comparison of γ B , the energy of a moving body, to B , the energy of a body at rest, yields
the relativistic equation for kinetic energy (K.E.),
. . γ B B B γ 1
Based on this equation for kinetic energy, as the velocity increases, the kinetic energy increases,
but the velocity of any object can never reach the speed of light (c).
The relativistic equations for momentum, ? γ , and the corresponding equation for total
energy, γ B B . ., follow several principles consistent with the theory of
relativity. The equations or models must
1. be valid in all inertial frames.
2. reduce to the classical values at low velocities, when γ 1.
3. must agree with experimental results.
One other important conclusion from both models, is that the speed of an object can never
reach the speed of light.
If mechanics equations exist for classical and relativistic regimes, how do we know when to use
the classical models and when to use the relativistic models? Some simple guidelines are
provided in Table 1.1 in terms of speed and energy.
Table 1.1 – Guidelines for using classical vs relativistic models for velocity and energy.
Velocity Limit
Energy Limit
Classical Models
v << 0.1 c
E << 0.01 B Relativistic Models
E ≥ 0.01 B v ≥ 0.1 c
Table 1.2 - Values of γ for various velocity (v) values.
v
γ
m/s
0
1.000
0.01c
1.000
0.1c
1.005
0.5c
1.15
0.90c
2.3
0.99c
7.1
We will look at massless particles that possess energy in Part II. These happen to be photons
which are electromagnetic waves with particle-like properties that travel at the speed of light (v
= c).
References
1. D.A.B. Miller, Quantum Mechanics for Scientists and Engineers, Cambridge University Press,
New York, 2008.
2. F.W. Sears, Zemansky, Young, Addison Wesley Education Publishers, 1991.
3. J.R. Taylor, C.D. Zafiratos, M.A. Dubson, Modern Physics for Scientists and Engineers (2nd
Ed.), Prentice Hall, New Jersey, 2004.