Oscillators, 3-dimensional harmonic

Now consider a system of N one-dimensional harmonic oscillators with the Hamiltonian... 

Fig. 1. Optimization of the Onsager-Machlup action for the two dimensional harmonic oscillator. The potential energy is U(x,y) = 25i/ ), the mass is 1...

The model consists of a two dimensional harmonic oscillator with mass 1 and force constants of 1 and 25. In Fig. 1 we show trajectories of the two oscillators computed with two time steps. When the time step is sufficiently small compared to the period of the fast oscillator an essentially exact result is obtained. If the time step is large then only the slow vibration persists, and is quite accurate. The filtering effect is consistent (of course) with our analytical analysis. Similar effects were demonstrated for more complex systems [7]. 

Figure 4-3 Potential Energy as a Function of Compression or Stretching of a One-Dimensional Harmonic Oscillator.

The quantum mechanical hamiltonian for a one-dimensional harmonic oscillator is given by... 

Einstein9 was the first to propose a theory for describing the heat capacity curve. He assumed that the atoms in the crystal were three-dimensional harmonic oscillators. That is, the motion of the atom at the lattice site could be resolved into harmonic oscillations, with the atom vibrating with a frequency in each of the three perpendicular directions. If this is so, then the energy in each direction is given by the harmonic oscillator term in Table 10.4... 

Exercise 3.4. Evaluate the free-energy difference between two one-dimensional harmonic oscillators with potentials U1 = (X- l)2 and U2 = 0.3(X —... 

The Schrodinger equation for this three-dimensional harmonic oscillator is... 

The differential equation for X(x) is exactly of the form given by (4.13) for a one-dimensional harmonic oscillator. Thus, the eigenvalues Ex are given by equation (4.30)... 

The energy levels for the three-dimensional harmonic oscillator are, then, given by the sum (equation (4.53))... 

Derive the result that the degeneracy of the energy level E for an isotropic three-dimensional harmonic oscillator is (n + l)(n + 2)/2. 

As shows is Section 5.14, in a conservative system the force can be represented by a potential function. The force is then gives by / = -dV(jr)/dx, where V(x) = for this one-dimensional harmonic oscillator. 

Tacoma Narrows bridge % tangent 16 Taylor s series 32-34 tests of series convergence 35-36 thermodynamics applications 56-57, 81 first law 38-39 Jacobian notation 160-161 systems of constant composition 38 three-dimensional harmonic oscillator 125-128... 

The classical harmonic oscillator in one dimension was illustrated in Seetfon 5.2.2 by the simple pendulum. Hooke s law was employed in the fSfin / = —kx where / is the force acting on the mass and k is the force constant The force can also be expressed as the negative gradient of a scalar potential function, V(jc) = for the problem in one dimension [Eq. (4-88)]. Similarly, the three-dimensional harmonic oscillator in Cartesian coordinates can be represented by the potential function... 

Fig. 3 Bneigy levels of the three-dimensional harmonic oscillator. The degree of degeneracy of each level is shown in parenthesis.

The choice of the momenta, rather than of the forces, is illustrated by the simple example of a two-dimensional harmonic oscillator described by the parametric equations... 

In Eq. (3.1), //<, is a three- or four- dimensional harmonic oscillator Hamiltonian... 

Some simple models for V(r) are shown in Fig. 2.1. Two crude approximations, the infinite square well (ISW) and the 3-dimensional harmonic oscillator (3DHO), have the advantage of leading to analytical solutions of the Schrodinger equation which lead to the following energy levels ... 

Fig. 2.1. Approximate potentials for the nuclear shell model. The solid curve represents the 3-dimensional harmonic oscillator potential, the dashed curve the infinite square well and the dot-dashed curve a more nearly realistic Woods-Saxon potential, V(r) = — V0/[l + exp (r — R)/a ] (Woods Saxon 1954). Adapted from Cowley (1995).

This is the classical Hamiltonian of the three-dimensional harmonic oscillator. By letting p — 0, one has... 

The residual energy (designated of a harmonic oscillator in the ground state. The Heisenberg Uncertainty Principle does not permit any state of completely defined position and momentum. A one-dimensional harmonic oscillator has energy levels corresponding to ... 

Our discussion of vibrations has all been within the context of the harmonic approximation. When using this approach, each vibrational mode can be thought of as being defined by a harmonic oscillator. The potential energy of a one-dimensional harmonic oscillator is... 

Problem 8-17. To show that the general conclusions suggested in the previous example are false, consider the case of the one-dimensional harmonic oscillator, for which H = T + V = T + kx. The energy levels of the one-dimensional harmonic oscillator with frequency v are ... 

Figure 3.1 Phase-space trajectory (center) for a one-dimensional harmonic oscillator. As described in the text, at time zero the system is represented by the rightmost diagram (q = b, p = 0). The system evolves clockwise until it returns to the original point, with the period depending on the mass of the ball and the force constant of the spring...

Eqn (9-3.3) is the same as the well known one-dimensional harmonic oscillator equation and has as its solutions... 

Now consider a one-dimensional harmonic oscillator of charge q. We must evaluate q(m x n). The harmonic-oscillator wave functions are given by (1.133). Using (1.133) and the Hermite-polynomial identity (1.138) with z = a 2x, we have... 

For example, consider the one-dimensional harmonic-oscillator states. These states are either even or odd, depending on whether the quantum... 

---