Transcript 1.1 Conservation of Energy

1.1
Conservation of Energy
1.1.1
Total Mechanical Energy
1.1.2
Work
1.1.3
Momentum and Hamiltonian Equation
1.1.4
Rest Mass
1.1.5
Summary
1.1.5
Homework
1.1.1 Total Mechanical Energy
(Most calculations in quantum mechanics are
assumed to be non-relativistic)
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,
and the potential energy is expressed using the variable U and is a
function of position (x),
(In quantum mechanics, the potential energy is expressed in terms of
V, but we will use U to avoid confusing the potential energy with
potential.)
1.1.1 Total Mechanical Energy
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
Kinetic Energy can be converted to Potential Energy and vice
versa!!!
1.1.1 Total Mechanical Energy - Example
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
therefore v is equal to
, and
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,
1.1.1 Total Mechanical Energy – Class Exercise
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.
1.1.2 Work
The work (W) done on an object or particle is equal to the force on
the particle times the distance the particle moves.
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),
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.
1.1.3 Momentum and Hamiltonian Equation
1.1.3 Momentum and Hamiltonian Equation
Since
The equation for total energy can be written as
When the total energy is expressed in terms of momentum and
position, it is called the Hamiltonian Total Energy or
This equation is important because it will be used to derive
Schrödinger’s equation in section 3.
1.4.1 Rest Mass
E = mc2
Rest mass – the mass of an object at rest; the rest mass is the same
in all frames of reference.
The total energy of a particle is equal to
E  mc  mc KE
2
2
Rest mass
where

1
2
v
1
c2
Relativistic effects need to be considered for particles traveling
at speeds comparable to the speed of light (c = 3 x 108 m/s)
1.4.1 Rest Mass
E = mc2
Example 1.3: Determine the rest energy associated with one
electron.
Therefore, the rest energy of a single electron is approximately
equal to 0.5 MeV
1.1.4 Rest Mass
Classical vs Relativistic Mechanics Calculations
Momentum and Kinetic Energy
Classical
Momentum, kgm/s
mv
E  KE  PE
Energy, J or eV
Kinetic energy,
J or eV
where

1
mv 2
2
1
2
v
1
c2
Note: As v increases, γ increases.
Relativistic
mv
E  mc 2
PE = 0
KE  mc2  mc2
1.1.4 Rest Mass
Limits - Classical vs Relativistic Mechanics
Calculations
v
c
p
Classical model: v increases indefinitely as p increases
Relativistic model: v is never greater than c no matter how much p
increases since γm increases with increase in v.
1.1.4 Rest Mass
Limits - When do we need to use relativistic
calculations?
• Momentum
If v << 0.1c → non-relativistic calculations/classical calculations
If v ≥ 0.1 c → relativistic calculations (> 0.5% difference)
or in terms of Energy…
• Kinetic Energy
If KE << 0.01 mc2 → non-relativistic calculations/classical
If KE ≥ 0.01 mc2 → relativistic calculations
1.1.4 Rest Mass
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  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.
Solution:
For an electron traveling at v = 5 x 107 m/s (where v/c = 0.17),
1.1.4 Rest Mass
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  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.
Solution con’t:
For and electron traveling at v = 2.94 x 108 m/s (where v/c = 0.98),
It is clear from these calculations, that using the classical equation for
momentum for v = 5 x 107 m/s would result in an accurate value, and
that the relativistic equation for momentum is required for v = 2.94 x 108
m/s.
1.1.4 Rest Mass
Classroom Exercise
What is the maximum bias value we can use to accelerate electrons and
still be able to use classical calculations to determine the kinetic energy
and velocity of the electrons. Identify the critical bias value, and the
corresponding kinetic energy (J and eV) and the velocity of the
electrons.
1.1.5 Summary
1. Conservation of energy means that energy can be transferred from
potential energy to kinetic energy and vice versa or
2. Energy of an object can be transformed into work done by that
object on its surroundings.
3. For a given kinetic energy, particles of higher mass have lower
velocities.
4. When the kinetic energy is expressed in terms of momentum, the
energy equation becomes the Hamiltonian Equation for Total
Energy.
5. Relativistic effects need to be considered for momentum
calculations when v ≥ 0.1 c.
1.1.5 Summary
6. Relativistic effects need to be considered for kinetic energy
calculations when KE≥ 0.01 mc2.
7. Momentum increases indefinitely using classical model.
8. Momentum reaches a limit using relativistic model because γm
with increase in the particle velocity.
9. There is a limit to the bias applied to accelerate electrons between
2 parallel plates above which relativistic effects need to be
considered when calculating the kinetic energy and velocity of
electrons.
1.1.6 Homework
1. If the kinetic energy of a particle is equal to its rest mass, what is the
velocity of the particle. Do you need to consider classical or
relativistic calculations?
2. If an electron has kinetic energy equal to 0.1 MeV, what is it’s
velocity? Calculate the velocity using classical and relativistic
calculations.
3. Calculate the velocity of a proton (m = 1.67 x 10-27 kg) with kinetic
energies equal to (a) 10 eV, (b) 100eV, (c) 1,000 eV. (d) Compare
the velocities to those of an electron with the same energies.
Construct a table like the one shown below. (e) Do any of these
velocities require relativistic calculations? Explain why or why not.
Type of Particle
Kinetic Energy, eV
10 eV
Proton velocity, m/s
(mp=1.67 x 10-27 kg)
Electron velocity, m/s
(mp=9.11 x 10-31 kg)
100 eV
1000eV
1.1.6 Homework
4. Create the following table for electrons for the velocities specified
in the table. me = 9.11 x 10-31 kg.
v
m/s
v/c
γ
p
classical
p
rel
%
diff
KE
classical
KE
rel
3 x105
3 x106
3 x 107
1.1 x
108
(rel  classical )
% Diff 
*100%
rel
%
diff
References
1 D.A.B. Miller, Quantum Mechanics for Scientists and Engineers,
Cambridge University Press, New York, 2008.
2. A. Beiser, Concepts of Modern Physics, McGraw Hill, New York,
2003.
3. F.W. Sears, Zemansky, Young, Addison Wesley Education Publishers,
1991.
4. J.R. Taylor, C.D. Zafiratos, M.A. Dubson, Modern Physics for Scientists
and Engineers (2nd Ed.), Prentice Hall, New Jersey, 2004.