1.1Conservation of Energy 1.1.1Total Mechanical Energy 1.1.2Work 1.1.3Momentum and Hamiltonian Equation 1.1.4Rest Mass 1.1.5Summary 1.1.5Homework

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, and therefore v is equal to Since the mass of an electron is 9.11 x 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 kg and kinetic energy equal to those specified in Example 1.1 with the velocities for a proton with m = 1.6 x 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

This equation is important because it will be used to derive Schrödinger’s equation in section 3. When the total energy is expressed in terms of momentum and position, it is called the Hamiltonian Total Energy or The equation for total energy can be written as Since

1.4.1Rest Mass E = mc 2 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 Rest mass Relativistic effects need to be considered for particles traveling at speeds comparable to the speed of light (c = 3 x 10 8 m/s) where

1.4.1Rest Mass E = mc 2 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

1.1.4Rest Mass Classical vs Relativistic Mechanics Calculations Momentum and Kinetic Energy ClassicalRelativistic Momentum, kgm/s Energy, J or eV PE = 0 Kinetic energy, J or eV where Note: As v increases, γ increases.

1.1.4Rest Mass Limits - Classical vs Relativistic Mechanics Calculations Classical model: v increases indefinitely as p increases Relativistic model: v is never greater than c no matter how much p increases since γ m increases with increase in v. c v p

1.1.4Rest 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 mc 2 → non-relativistic calculations/classical If KE ≥ 0.01 mc 2 → relativistic calculations

1.1.4Rest Mass Example 1.4: The typical velocity of an electron being accelerated in a cathode ray tube is 5 x 10 7 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 10 8 m/s. Compare the values of  for the electron moving at a velocity equal to 5 x 10 7 m/s to that of an electron moving at 2.94 x 10 8 m/s. Solution: For an electron traveling at v = 5 x 10 7 m/s (where v/c = 0.17),

1.1.4Rest Mass Example 1.4: The typical velocity of an electron being accelerated in a cathode ray tube is 5 x 10 7 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 10 8 m/s. Compare the values of  for the electron moving at a velocity equal to 5 x 10 7 m/s to that of an electron moving at 2.94 x 10 8 m/s. Solution con’t: For and electron traveling at v = 2.94 x 10 8 m/s (where v/c = 0.98), It is clear from these calculations, that using the classical equation for momentum for v = 5 x 10 7 m/s would result in an accurate value, and that the relativistic equation for momentum is required for v = 2.94 x 10 8 m/s.

1.1.4Rest 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.

1.1.5Summary 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.

1.1.5Summary 6.Relativistic effects need to be considered for kinetic energy calculations when KE≥ 0.01 mc 2. 7.Momentum increases indefinitely using classical model. 8.Momentum reaches a limit using relativistic model because γ m 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.

1.1.6Homework 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 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 ParticleKinetic Energy, eV 10 eV100 eV1000eV Proton velocity, m/s (m p =1.67 x kg) Electron velocity, m/s (m p =9.11 x kg)

1.1.6Homework 4.Create the following table for electrons for the velocities specified in the table. m e = 9.11 x kg. v m/s v/c γ p classical p rel % diff KE classical KE rel % diff 3 x x x x 10 8

References 1 D.A.B. Miller, Quantum Mechanics for Scientists and Engineers, Cambridge University Press, New York, A. Beiser, Concepts of Modern Physics, McGraw Hill, New York, F.W. Sears, Zemansky, Young, Addison Wesley Education Publishers, J.R. Taylor, C.D. Zafiratos, M.A. Dubson, Modern Physics for Scientists and Engineers (2 nd Ed.), Prentice Hall, New Jersey, 2004.