[QMSE] W01 - Classical Mechanics Review

• 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

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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 $r$
  • $V\left(r\right)$

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 $V\left(r\right)$
  • $H=\frac{p^2}{2m}+V\left(r\right)$
• 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 $F_{pushx}$ driving a ball
  • up a positive slope
  • by distance $\Delta x$ up the slope
• the change in potential energy $V$
  • $\Delta V=F_{pushx}\Delta x$
• this can also be twisted to be understood as
  • $F_{pushx}=\frac{\Delta V}{\Delta x}$
  • in the infinitesimal limit: $F_{pushx}=\frac{dV}{dx}$
• 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
• $F=-\nabla V=-[\frac{\partial V}{\partial x}i+\frac{\partial V}{\partial y}j+\frac{\partial V}{\partial z}k]$