Guide The Harmonic Oscillator in Modern Physics

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Simple Harmonic Oscillator

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Simone Camosso. Date: March 8, Complete Matching. The emitted wavelengths were early on associated with a set of discrete energy levels E n described by:. The maximum photon energy emitted from a hydrogen atom equals This energy is also called one Rydberg or one atomic unit. The electron transitions and the resulting photon energies are further illustrated by Figure 1.

However, at the time there was no explanation why the possible energy values were not continuous. No classical theory based on Newtonian mechanics could provide such spectrum. Further more, there was no theory, which could explain these specific values. Niels Bohr provided a part of the puzzle. He assumed that electrons move along a circular trajectory around the proton like the earth around the sun, as shown in Figure 1.

He also assumed that electrons behave within the hydrogen atom as a wave rather than a particle.

The generalized harmonic potential theorem in the presence of a time-varying magnetic field

Therefore, the orbit-like electron trajectories around the proton are limited to those with a length, which equals an integer number of wavelengths so that. The Bohr model also assumes that the momentum of the particle is linked to the de Broglie wavelength equation 1. The model further assumes a circular trajectory and that the centrifugal force equals the electrostatic force, or:.

Note that all the possible energy values are negative. Electrons with positive energy are not bound to the proton and behave as free electrons. The Bohr model does provide the correct electron energies. However, it leaves many unanswered questions and, more importantly, it does not provide a general method to solve other problems of this type.

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The wave equation of electrons presented in the next section does provide a way to solve any quantum mechanical problem. Starting from a classical description of the total energy, E , which equals the sum of the kinetic energy, K. To incorporate the de Broglie wavelength of the particle we now introduce the operator, , which provides the square of the momentum, p , when applied to a plane wave:.

Without claiming that this is an actual proof, we now simply replace the momentum squared, p 2 , in equation 1. Prior to that, we discuss the physical interpretation of the wavefunction. However, the physical meaning of the wavefunction does not naturally follow.

Lecture 7 - Modern Physics: Statistical Mechanics

This probability density function integrated over a specific volume provides the probability that the particle described by the wavefunction is within that volume. This normalization enables to calculate the magnitude of the wavefunction using:. This probability density function can then be used to find all properties of the particle by using the quantum operators.

To find the expected value of a function f x , p for the particle described by the wavefunction, one calculates:. Where F x is the quantum operator associated with the function of interest. A list of quantum operators corresponding to a selection of common classical variables is provided in Table 1.

The one-dimensional infinite quantum well represents one of the simplest quantum mechanical structures. We use it here to illustrate some specific properties of quantum mechanical systems. The potential and the first five possible energy levels an electron can occupy are shown in Figure 1. As a result one solves the following equation within the well.


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Where the coefficients A and B must be determined by applying the boundary conditions. Since the potential is infinite on both sides of the well, the probability of finding an electron outside the well and at the well boundary equals zero. Therefore the wave function must be zero on both sides of the infinite quantum well or:. These boundary conditions imply that the coefficient B must be zero and the argument of the sine function must equal a multiple of pi at the edge of the quantum well or:.

Where the subscript n was added to the energy, E ,to indicate the energy corresponding to a specific value of, n. The resulting values of the energy, E n , are then equal to:. The corresponding normalized wave functions, Y n x , then equal:. Note that the lowest possible energy, namely E 1 , is not zero although the potential is zero within the well. The energy difference between adjacent energy levels increases as the energy increases. Both quantum numbers, n and s , are the only two quantum numbers needed to describe this system. The wavefunctions corresponding to each energy level are shown in Figure 1.

Each wavefunction has been shifted by the corresponding energy and scaled with an arbitrary magnitude as is commonly done. These probability density functions are shown in Figure 1. The electron is most likely to be one quarter of the well width away from either edge. The lowest energy in the quantum well equals:. The hydrogen atom represents the simplest possible atom since it consists of only one proton and one electron. The potential, V r equation 1. The potential V x , y , z is the electrostatic potential, which describes the attractive force between the positively charged proton and the negatively charged electron.

Since this potential depends on the distance between the two charged particles one typically assumes that the proton is placed at the origin of the coordinate system and the position of the electron is indicated in polar coordinates by its distance r from the origin, the polar angle q and the azimuthal angle f.

Harmonic Oscillator/Modern Phy

A more refined analysis includes the fact that the proton moves as the electron circles around it, despite its much larger mass. The stationary point in the hydrogen atom is the center of mass of the two particles. This refinement can be included by replacing the electron mass, m , with the reduced mass, m r , which includes both the electron and proton mass:.

In addition, one assumes that the wavefunction, Y r , q , f , can be written as a product of a radial, angular and azimuthal angular wavefunction, R r , Q q and F f. This assumption allows the separation of variables, i. Where the constants A and B are to be determined in addition to the energy E. The solution to these differential equations is beyond the scope of this text. Readers are referred to the bibliography for an in depth treatment. We will now examine and discuss the solution. The electron energies in the hydrogen atom as obtained from equation 1.