• many CM (Classical Mechanics) concepts are still useful in understanding QM (Quantum Mechanics)
    • discarding them will apparently increase the effort to understand QM
    • QM also has to explain what CM can already explain
  • specifically, understanding of waves and oscillations is very important
    • energy and
    • momentum hold as well
  • a part of learning QM is parts of CM that are needed to build concepts of QM
    • they are used differently than used in CM
  • compared to the math, we do not need much physics
    • partly because it is defining new physical concepts
  • the core of quantum mechanics can be understood without the additional complexities of the physics of general relativity
    • relativistic views of the world are provided by general world

Elementary Classical Mechanics (ECM)

  • conceptually this is equivalent to
    • newtonian classical mechanics (NCM)
    • hamiltonian classical mechanics (HCM)
    • lagrangian classical mechanics (LCM)
  • HCM and LCM are mathematically more sophisticated in the way they express ECM
  • NCM also distinguishes from RCM (Relativistic Classical Mechanics)
    • only non-relativistic CM i.e. mass moving much slower than speed of light
    • also distinguishes form QM
  • photons travel at light speed
    • but have no mass
    • most of their physics can be handled with QM without relativistic complexities

momentum and kinetic energy

  • for particle of mass \(m\)
    • momentum is \( p = mv \)
    • \(v\) is the velocity
    • both \( v \) and \( p \) are both velocities
  • energy due to motion is kinetic energy
    • \( KE = \frac{p^2}{2m} \)
    • \( p^2 = p \times p \):
      • vector product of momentum with itself
  • in CM, kinetic energy can never be negative

potential energy

  • energy due to position
    • \(V \)
    • units = Joules
    • not to be confused with Voltage \(V\) whose units are Joules/Coulomb
  • can be written as a function of position vector \( r \)
    • \( V(r) \)

conservative-fields

  • in CM, fields like which potential energy exists in
    • are called conservative or irrotational fields
    • the change in a potential energy around a closed path is \(0\)
  • gravitational
  • electrostatic

non-conservative fields

  • going round a vortex i.e. water going around a bath tub drain
    • a vortex field has a rotational aspect to it
    • so the potential energy change along a closed path is not zero
  • work has to be done to move against the field
  • work will be done on by the field when going along with the field

hamiltonian

  • the sum total of potential and kinetic energies
    • written as a function of position and momentum
    • is called the classical “Hamiltonian”
    • denoted by “\(H\)”
  • for a classical particle of mass \( m \) in a conservative potential \( V(r) \)
    • \( H = \frac{p^2}{2m} + V(r) \)
  • the zero chosen for potential energy is arbitrary
    • there is not absolute origin for potential energy
    • in both CM and QM
    • only delta potential energy is worked with
    • the zero has to be kept consistent
    • there is however a best practice of choosing the origin to simplify the algebra

Force

  • in CM, newton’s II law related force and acceleration
    • \( F = ma \)
    • \( F \): force
    • \( m \): mass
    • \( a \): acceleration
  • equivalently force is the rate of change of momentum

  • \( F = \frac{dp}{dt} \)

  • in QM, the idea of force is seldom used directly
    • even if there is behavior that corresponds to force like in CM
    • forced are setup as a gradient in the potential

force and potential energy

  • assume a force \( F_{push x} \) driving a ball
    • up a positive slope
    • by distance \(\Delta x \) up the slope
  • the change in potential energy \(V\)
    • \( \Delta V = F_{push x}\Delta x \)
  • this can also be twisted to be understood as
    • \( F_{push x} = \frac{ \Delta V }{ \Delta x } \)
    • in the infinitesimal limit: \( F_{push x} = \frac{ d V }{ d x } \)
  • the direction of the force of the potential on the ball is downhill
    • due to gravity
    • so force due to gravity field \( F_{x} = - \frac{dV}{dx} \)

force as a vector

  • use gradient operator to generalize force to three dimensions
    • to construct the components along the three directions X, Y and Z
  • \( F = -\nabla V = - \left[ \frac{\partial V}{\partial x} i + \frac{\partial V}{\partial y} j + \frac{\partial V}{\partial z} k \right] \)