Author: Haroon Khalil
-
Solving the Schroedinger Equation
The funny thing about a differential equation (including the Schroedinger equation) is that its solution is, in and of itself, another equation. For example, the differential equation for the position of a macroscopic object that is moving in one dimension looks like this: . Here x(t) is the object’s position at time t, a represents how much the object is…
-
Wave Functions and the Wave Equation
Schroedinger’s wave equation is sometimes just called the “wave equation” in quantum mechanics. This can be a source of confusion, because a wave function itself will look like an equation when we write it down. But the Schroedinger wave equation is a very special kind of equation called a differential equation. We know this because…
-
The Schroedinger Equation
The missing link in our picture so far is some way to determine the dynamic behavior of particle/waves like electrons, protons, and neutrons. Recall that in order to understand dynamic behavior–that is, how things move and change with time–we need to factor in the impact of forces. Newton’s laws of motion tell us how classical particles…
-
Introduction
We get into the real nitty-gritty of modern quantum physics. We will learn about the powerful equation that quantum physicists have come up with to describe how particles behave when they interact with other particles and forces. We’ll learn what this equation signifies, how it is solved, and what those solutions mean. Then, we’ll see…
-
The End of Determinism
Once you do get to the scale of atoms and smaller, things certainly do get counterintuitive and distinctly nonclassical. We have seen many examples now. Particle position gets blurry. Particles seem to be governed by invisible wave functions. Particles interfere with themselves after passing through or around obstacles. We can’t tell which slit the photon…
-
Uncertainty in a Macroscopic World
All of this weird uncertainty doesn’t seem to bother us in everyday life. Why not? Once again, it’s all a matter of scale. Heisenberg showed that the product of the uncertainties of conjugate pairs of observables can be about as small as Planck’s constant, but no smaller. But since Planck’s constant itself is extremely small…
-
Conjugate Pairs
It turns out that Heisenberg’s uncertainty principle applies to pairs of observable things, and only to certain pairs. The technical name for these is conjugate pairs. We don’t need to go into all the conjugate pairs of observables or why they are conjugate, or even what in the world “conjugate” really means, but it will be…
-
The Heisenberg Uncertainty Principle
The position of a moving particle is not the only thing that is a little fuzzy in quantum physics. It turns out that there is a very interesting interaction between the position and the momentum of a particle. The relationship between the uncertainties in these two quantities is something that does not appear at all…
-
The Role of Probability
It is not possible to predict exactly where any individual photon (or electron) will land after traversing the double slits. But since we observe a striped pattern, we can say that the probability of seeing a photon must be higher at certain positions than at others. The probability is high where the amplitude of the…
-
Quantum Waves
We made a big deal about the use of mathematical models to describe physical phenomenon. Now we’ll take a crack at seeing not only how this can be done, but why it turns out to be so useful. And, given the emphasis we’ve been placing on matter waves, it’s probably no surprise that our starting…