Direct Energy, 2018a

11 CALCULUS OF VARIATIONS 263
The above two equations can be combinedalgebraically.
( )
−
2
2m ∇2 + E potential energy ψ = E total ψ (11.70)
With some more algebra, Eq. 11.70 can be rewritten.
∇ 2 ψ + 2m
2 (E total − E potential energy ) ψ =0 (11.71)
Equation 11.71 is the time independent Schrödinger equation, andit is one
of the most fundamental equations in quantum mechanics. Energy level
diagrams were introduced in Section 6.3. Allowed energies illustrated by
energy level diagrams satisfy the Schrödinger equation. At least for simple
atoms and ground state energies, energy level diagrams can be derived by
solving Schrödinger equation.
11.8 Problems
11.1. In the examples below, identify whether f is a function or a functional.
• A parabola is described by f(x) =x 2 .
• Given two forms of energy anda path y(t), f is the Lagrangian
of the system L ( )
t, y, dy
dt .
• Given the magnitude of the velocity | −→ v (t)| of an object, f represents
the distance that the object travels from time 0 to time
3600 seconds.
• Given the position (x, y, z) in space, f(x, y, z) represents the
distance from that point to the origin.
11.2. A system has the Lagrangian L ( ) (
t, y, dy
dt = dy
) 3
dt + e 3y . Findan
equation for the path y(t) that minimizes the action ´ t 2
t 1
L ( )
t, y, dy
dt dt.
(The result is nonlinear, so don't try to solve it.)
11.3. A system has Lagrangian L ( ) (
t, y, dy
dt =
1 dy
) 2
2 dt +
1 · 2 y−2 . Findthe
corresponding equation of motion. (The result is nonlinear, so don't
try to solve it.)