Quantum Mechanics - Prof. Eric R. Bittner - University of Houston

So, we can take the quantity in the parenthesis to be a constant.
�
E = ˙x ∂L
�
− L = const. (1.31)
∂ ˙x
is an integral of the motion. This is the energy of the system. Since L can be written in form
L = T − V where T is a quadratic function of the velocities, and using Euler’s theorem on
homogeneous functions:
This gives,
˙x ∂L
∂ ˙x
= ˙x∂T
∂ ˙x
E = T + V
= 2T.
which says that the energy of the system can be written as the sum of two different terms: the
kinetic energy or energy of motion and the potential energy or the energy of location.
One can also prove that linear momentum is conserved when space is homogeneous. That is,
when we can translate our system some arbitrary amount ɛ and our dynamical quantities must
remain unchanged. We will prove this in the problem sets.
1.4 Hamiltonian Dynamics
Hamiltonian dynamics is a further generalization of classical dynamics and provides a crucial link
with quantum mechanics. Hamilton’s function, H, is written in terms of the particle’s position
and momentum, H = H(p, q). It is related to the Lagrangian via
Taking the derivative of H w.r.t. x
H = ˙xp − L(x, ˙x)
∂H
∂x
= −∂L
∂x
= − ˙p
Differentiation with respect to p gives
∂H
= ˙q.
∂p
These last two equations give the conservation conditions in the Hamiltonian formalism. If H
is independent of the position of the particle, then the generalized momentum, p is constant in
time. If the potential energy is independent of time, the Hamiltonian gives the total energy of
the system,
H = T + V.
1.4.1 Interaction between a charged particle and an electromagnetic
field.
We consider here a free particle with mass m and charge e in an electromagnetic field. The
Hamiltonian is
H = px ˙x + py ˙y + pz ˙z − L (1.32)
= ˙x ∂L
∂ ˙x
+ ˙y ∂L
∂ ˙y
24
+ ˙z ∂L
∂ ˙z
− L. (1.33)