Conjugate momenta motion

The phase space for three-dimensional motion of a single particle is defined in terms of three cartesian position coordinates and the three conjugate momentum coordinates. A point in this six-dimensional space defines the instantaneous position and momentum and hence the state of the particle. An elemental hypothetical volume in six-dimensional phase space dpxd Pydpzdqxdqydqz, is called an element, in units of (joule-sec)3. For a system of N such particles, the instantaneous states of all the particles, and hence the state of the system of particles, can be represented by N points in the six-dimensional space. This so-called /r-space, provides a convenient description of a particle system with weak interaction. If the particles of a system are all distinguishable it is possible to construct a 6,/V-dimensional phase space (3N position coordinates and 3N conjugate momenta). This type of phase space is called a E-space. A single point in this space defines the instantaneous state of the system of particles. For / degrees of freedom there are 2/ coordinates in /i-space and 2Nf coordinates in the T space. 

It is now necessary to separate the reaction coordinate, denoted qi, from the other motions of the system, which will be denoted q+. At given values of the reaction coordinate q and its conjugate momentum pi the kinetic energy due to motion along the reaction coordinate is p l lp. and the contribution of the reaction coordinate to the potential energy can be written V( i). The quantity jUi is the reduced mass for motion along the reaction coordinate. Hence the energy can be divided... 

At this point, let us recall that the equations of motion for a particle i characterized by the generalized coordinate and conjugate momentum tVj are derived from a Lagrangian described in Eq. [48] as... 

So x does not appear, that is, it is an ignorable co-ordinate, and its conjugate momentum, X, is a constant of the motion, in fact corresponding to the constancy of the total angular momentum. 

If we denote the coordinates corresponding to the translational and rotational motions of the molecule by Rj and its conjugate momentum by pg. 

Another useful form of the equations of motion, in addition to newtonian and lagrangian forms, are Hamilton s equations of motion. To see these, we first define the so-called conjugate momentum pi as... 

The generalized coordinate q and its conjugate momentum p vary periodically, and the line integral is evaluated over one cycle. (A similar quantization condition on angular momentum led to the famous Bohr postulate L = nh for the hydrogen atom in the old quantum theory [16].) For vibrational motion subject to a potential U K) in a diatomic, the classical energy is E = p llp -f- U R). The integral in (4.80) then translates into... 

A little algegra wiU show that Qj obeys a classical wave equation homologous to Eqs. (3.13) and (B3.1.7), and that Qj and Pj have the formal properties of a time-dependent position Qj) and its conjugate momentum Pj) in Hamiltonian s classical equations of motion ... 

Hamilton s method is similar to that of Lagrange in that it provides equations of motion that have the same form in any coordinate system. It uses conjugate momenta instead of time derivatives of coordinates as state variables. The conjugate momentum to the coordinate qi is defined by... 

One distance, r, is needed to define the size of the bond and a conjugate momentum, pr, can define the vibrational motion of the molecule. 

Notice that these equations explicitly include derivatives and the complex conjugate f of the wavefunction. The expression for the momentum even includes i = f—W Complex numbers are not just a mathematical convenience in quantum mechanics they are central to the treatment. Equation 6.6 illustrates this point directly. Any measurement of the momentum (for example, by measuring velocity and mass) will of course always give a real number. But if the wavefunction is purely real, the integral on the right-hand side of Equation 6.6 is a real number, so the momentum is a real number multiplied by ih. The only way that product can be real is if the integral vanishes. Thus any real wavefunction corresponds to motion with no net momentum. Any particle with net momentum must have a complex wavefunction. 

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