Question

What phase-space function is a constant of motion??? Give examples and the equations.

Answer 1

Phase space coordinates means position,velocity or position,momentum.

The phase space is the space of all possible states of a system; the state of a mechanical system is defined by the constituent positions p and momenta q. p and q together determine the future behavior of that system. In other words if you know p and q at time t you will be able to calculate the p and q at time t+1 using the theorems of classical mechanics - Hamilton's equations.
To describe the motion of a single particle you will need 6 variables, 3 positions and 3 momenta. You can imagine a 6 dimensional space; three positions and three momenta. Each point in this 6 dimensional space is a possible description of the particles' possible states, of course constraint by the laws of classical mechanics.
If you have N particles to describe the system, you have a 6N-dimensional phase space.
Let's make a simple example. The Pendulum. The Pendulum consists of a single particle mass that swings in a plane. The pendulum is thus fully described by one position and one momentum. Its momentum is zero at the top and maximum at bottom. The position perhaps is denoted by angle and varies between plus/minus a. If you draw states p and a in a Cartesian plane coordinate system you will get an ellipsoid (or if chose adequate coordinates a circle) that fully describes all possible states of the pendulum.
In quantum mechanics the term phase re-appeared: it refers to the complex phase of the complex numbers that wave functions take values in.
In quantum mechanics, the coordinates p and q of phase space normally become operators in a Hilbert space.

A quantum mechanical state does not necessarily have a well-defined position or a well-defined momentum (and never can have both according to Heisenberg's uncertainty principle). The notion of phase space and of a Hamiltonian H, can be viewed as a crucial link between what otherwise looks like two very different theories. A state is now not a point in phase space, but is instead a complex valued wave function. The Hamiltonian H becomes an operator and describes the observable quantity.