It is a delightful fact that one can get both the fundamental equation of classical mechanics (Newton’s Second Law) and the fundamental equation of quantum mechanics (Schrödinger’s equation) by solving very simple variational problems based on the familiar conservation of mechanical energy equation
In the present note I want to briefly set these out emphasising the common underlying structure provided by the conservation of mechanical energy and the calculus of variations. The kinetic energy will be taken to be
where is the particle’s velocity, is its momentum, and is its mass. The potential energy will be regarded as some function of only.
To obtain Newton’s Second Law we find the stationary path followed by the particle with respect to the functional
The function is usually termed the `Lagrangian’ in classical mechanics. The functional is usually called the `action’. The Euler-Lagrange equation for this calculus of variations problem is
and this is Newton’s Second Law in disguise! We have
so substituting these into the Euler-Lagrange equation we get Newton’s Second Law, .
To obtain Schrödinger’s equation we introduce a function
where is again the momentum of the particle and is the reduced Planck’s constant from quantum mechanics. (Note that has units of length mass time so we need to remove these by dividing by which has the same units. The function in quantum mechanics is dimensionless). We then have
and differentiating both sides gives
Therefore we can write the kinetic energy as
and putting this into the conservation of mechanical energy equation gives
We now find the stationary path followed by the particle with respect to the functional
The Euler-Lagrange equation for this calculus of variations problem is
and this is Schrödinger’s equation in disguise! We have
so substituting these into the Euler-Lagrange equation we get
and this is the (time-independent) Schrödinger equation for a particle of mass with fixed total energy in a potential on the line .