Schrodinger Equation
Background Information
PDE Class Spring 2012
1.Ψ is the wavefunction. The square of the wavefunction for a particle describes the probability of finding the particle at a particular location in space. m is the particle's mass. V is the potential energy. ħ is the reduced Planck constant.
E = (h/i) (d/dt) is the operator that describes the total energy of the particle.
V(x) is the potential energy of the particle
Kinectic energy is equal to momentum of the particle squared divided by two times the mass of the particle. Substituting p with ħ/i d/dx gets - ħ ^2/2m d^2/dx^2.
These operators, applied to the wavefunction, produces the Schrodinger equation which gives the quantized energies of the system.
The two sides are equal because of energy conservation. The total energy of the particle is equal to the sum of its potential and kinetic energy,
2. For the particle-in-a-box problem, we assume that the potential within the box is 0 and the potential outside the box is infinite. This extrapolates to the assumption that the solutions to the Schrodinger equation will be zero at the sides of the box. Additionally, we assume the solutions will be continuous, and will have at least a second derivative. Without the assumptions listed, the Schrodinger Equation is unsolvable for a specific solution.
3. As described by quantum mechanics, the wave/particle in the box cannot have zero translational energy because all wave/particles can be described as having a wavelength, and since E = hv, all wave/particles which have wavelengths must have energy. Waves cannot exist without translational movement, and therefore translational energy. In classical mechanical models, the particle (NOT A WAVE) can have zero energy, and therefore sit still in a way that is neither allowed (for WAVE/PARTICLES) by quantum mechanics nor observed by wave/particles in Nature. Zero-point energy is the lowest possible energy in the quantum physics system.
4. The ultraviolet catastrophe was an example of classical physics fail - a scientific breakthrough around the 1900s that helped bring in quantum mechanics.
Back in 1900, physicists Lord Rayleigh and James Jean were exploring black bodies, which are idealized physical objects that absorb and emit all light. The Rayleigh-Jeans law describes radiation from a black body in classical physics. The problem arose when physicists tried to calculate how much power came out of black body radiation. Classical physics predicted that a black body above absolute zero would emit an amount of radiation approaching infinity, most of it in the ultraviolet range. Data showed that there is actually a maximum radiancy beyond which radiancy falls and approaches zero. The Rayleigh-Jeans law is closer to data with longer wavelengths, but nevertheless there was classical physics fail, and by 1900 scientists were questioning basic concepts of thermodynamics and electromagnetics that were the basis of the prediction.
German physicist Max Planck came up with a solution to the catastrophe. The physicists had been using theories like classical harmonic oscillators and the equipartition theorem. Planck thought that the previous formula predicted too much high-frequency radiancy. He wanted a way to limit the high-frequency oscillations in the atoms, thereby reducing the corresponding radiancy of high-frequency waves. Planck put forth the idea of quantized emission. Atoms emit and absorb energy in discrete packets proportional to radiation frequency. For example, the photon is a discrete packet of electromagnetic radiation. The result is that emission goes to zero at high frequencies instead of infinity. His formula is now known as Planck's law of black body radiation, and it allows scientists to estimate the temperature of objects by examining their radiation. It is the start of quantum physics. Einstein later went further with it.
5. Initially, Newton thought that light was a stream of small particles and that it would travel faster in a refracting medium like water. Huygens, however, thought that light was wave-like and would slow in a refracting medium. Although both models explained reflection and refraction, Fizeau and Foucault measured the speed of light as slower in water than in air, thus supporting Huygens model of light as a wave over Newton’s. Shortly thereafter, Thomas Young created the Double-Slit Experiment which showed the destructive interference of light when shined through an arrangement of two parallel slits. The interference pattern is indicative of light acting as a wave. Then, Hertz discovered the photoelectric effect, which describes the emission of electrons from a metal illuminated with ultraviolet light. Light’s ability to dislodge electrons from metal indicates that light has particle-like momentum. These experiments together illustrate the wave-particle duality of light’s behavior.
6. The above experiments are not specific to photons because they can be repeated using any wave/particle. Water, sound, and electrons can all be used to achieve the same conclusions.
