Author: Haroon Khalil
-
SUPERPOSITION AND SCHRODINGER’S CAT
Let’s revisit the original particle-in-a-box problem (Figure 109). We had assumed that the system had a specific energy that did not vary with time. This allowed us to use a simple form of Schrödinger’s equation that gave us the wavefunctions of the system that depend only on the position x within the box, and the system’s quantum…
-
QUANTUM TUNNELING TIME
An interesting question to ask is: what is the time that tunneling particles spend inside the barrier? The experimental setup to demonstrate photon tunneling can be modified, as shown in Figure 117, to measure the total tunneling time of the photons, that is tv + th. This strange, two-part timing is due to the nature of frustrated total internal…
-
QUANTUM TUNNELING
chSo far, our analysis of the particle-in-a-box problem has assumed that the barriers around the box are infinite: which allowed us to force the particle to always be within the box. We were thus able to make Ψ = 0 for x ≤ 0 and x ≥ L. What would happen if the walls didn’t have infinite potential? Well, the…
-
REAL-WORLD PARTICLE IN A BOX
There are very few real-world systems that behave like the simplified mathematical model we used above for a particle in a box. However, recent developments in the nascent field of nanotechnology have made it possible to produce nanostructures of only 2–10 nm in size that exhibit behavior close to the simple particle-in-a-box model we discussed. These tiny structures are…
-
Introduction
We learned that |Ψ|2—the square of a system’s wavefunction—is itself a function that gives us the probability of finding a particle at a certain time and position. However, in the simply assigned the fancy symbol |Ψ|2 to the probability of finding a particle at a certain place and at a certain time. For example, in the…
-
BYE, BYE CLOCKWORK UNIVERSE
Isaac Newton realized that the laws of motion he had discovered, including the Law of Universal Gravitation, could explain, at least in principle, the behavior of all physical objects in the universe known at the time. These physical laws were so powerful they allowed scientists of the time to calculate with accuracy the movements of…
-
FOURIER ANALYSIS
When we separated light into its constituent lines, or binned scintillation photons according to their energy, we were performing spectral analysis. Since light and gamma rays most commonly act as waves, what we have been doing is deconstructing a complex electromagnetic wave into many simple sine waves. The same method can be applied to the…
-
TIME–ENERGY UNCERTAINTY
Bohr realized that if there is an uncertainty relationship between momentum p and position x: which comes from quantizing momentum as a function of wavelength: A similar uncertainty principle must then exist between energy E and time t. This is because: where frequency is the inverse of a time interval f = 1/t, so: Indeed, an identical relationship exists for the product of…
-
EXPERIMENTAL DEMONSTRATION OF THE UNCERTAINTY PRINCIPLE
A demonstration of the Uncertainty Principle is simple to conduct using a laser pointer and an adjustable slit consisting of two razor blades. The idea is that the photons emitted by the laser travel without diverging. As shown in Figure 104, the photons exiting the laser can be assumed to have momentum only along the x axis (so py =…
-
THE UNCERTAINTY PRINCIPLE
In 1925, German physicist Werner Heisenberg was an assistant to Niels Bohr at the Institute of Theoretical Physics at the University of Copenhagen. Heisenberg’s research related to the development of mathematical operations to calculate the expected results of experiments on hydrogen atoms based only on observables—that is, using only quantities that could be experimentally measured,…