A foundational course for understanding quantum mechanics.
This course introduces the basics of quantum mechanics, the wave function, Schrodinger's equation, Hilbert spaces, perturbation theory etc.
As you progress through this course, you will find links to related topics in other courses (such as calculus and partial differential equations). This interconnected structure is designed to help you quickly revisit prerequisite ideas without breaking your workflow.
The wave function plays a prominant role in quantum mechanics, it is the object that allows us to calculate (generally highly accurate) probabilities for important observed quantities in the quantum mechanical realm. In one dimensional space (we will call that the \(x\) dimension), the wave function is greatly simplified as it is a function of 1 dimensional space and time, and typically written as $$\Psi(x,t),$$ (we could have just as easily chosen \(f(x,t)\), but the notation with \(\Psi(x,t)\) has become quite entrenched in quantum mechanics). Generally, as can be seen from the notation, the wavefunction can change in space and time, and how it changes is governed by the Schrodinger equation which is given by, $$i\hslash \frac{\partial \Psi}{\partial t} = - \frac{\hslash^2}{2m}\frac{\partial^2 \Psi}{\partial x^2}+V\Psi,$$ where \(i = \sqrt{-1}\), \(m\) is the mass, \(\hslash = \frac{h}{2\pi}\) and \(h\) is Plank's constant, finally \(V\) is the potential energy function. If we write this in operator form, we get $$\hat{E}\Psi = \hat{H} \Psi,$$ where \(\hat{E} = i\hslash \frac{\partial}{\partial t}\) is the Energy operator and \(\hat{H} = - \frac{\hslash^2}{2m}\frac{\partial^2}{\partial x^2}+V\) is the Hamiltonian operator, note that the first part of the Hamiltonian (the part not including \(V\)) is the kinetic energy term and of course \(V\) is the potential energy term as already noted.
So essentially the partial differential equation (PDE) (check out the pde course), guides how the wavefunction changes in time and space. So what the heck is this wave function? Well, unfortunately this in some sense remains one of the mysteries in quantum mechanics, at least philosophically speaking. While we can't say exactly what it is, we can say what it does, and it remains an extremely important part of making quantum mechanics an accurate theory. As I mentioned above, the wave function allows us to calculate probabilities. In fact, $$\int \limits_{a}^b |\Psi(x,t)|^2 dx,$$ gives us the probability of finding the particle of interest (say an electron for example) between \(x = a\) and \(x = b\) in one spatial dimension. Notice however that we have to square the "length" of the wave function (you might wonder why we need the absolute value sign here when we are squaring, well that is because \(\Psi\) is a complex valued function). So the wave function itself does not measure the probabilities, rather the area under it's square (hence the need for the integral above) does. This is referred to as the statistical interpretation, and it has been extremely effective at giving us accurate probabilities of observing events. It is so effective in fact, that for a long period of time, most people working on quantum mechanics didn't bother asking those philosophical questions about the wave function and what it means and the "indeterminacy" that arises from the statistical interpretation. That period of time was the "shut up and calculate" era of quantum mechanics. Thankfully there is now a very robust discussion about these philosophical questions, but no explanation has emerged as the one true explanation. We will discuss these details as the course moves along.
Since we are discussing the statistical interpretation, we should note that it is natural to expect that the particle in question must exist somewhere in space, and this means that, $$\int \limits_{-\infty}^\infty |\Psi(x,t)|^2 = 1.$$ To restate, if we look everywhere in space, between negative and positive infinity, there is a probability of \(100%\) of finding the particle. Now, it may be useful to review your understanding of linear operators, (linear homogeneous PDEs), to recall that if you multiply an solution (the solution to Schrodinger's equation is the wave function \(\Psi(x,t)\)) by an arbitrary constant (in this case it can be a complex constant, not just real) \(A\), then this is also a solution. That is \(A\Psi(x,t)\) is also a solution, and the trick here, for this solution to be physically meaningful, (assuming things such as energy conservation, conservation of probability etc.) is to choose that constant \(A\), so that, $$\int \limits_{-\infty}^\infty |A\Psi(x,t)|^2 = 1.$$ Choice of \(A\) that satisfies this is called normalizing the wave function. Some wave functions are simply not physical in the sense that they cannot be normalized, including those that square integrate to \(\pm \infty\) and those that integrate to 0. These solutions are called non-normalizable, and we will then work in spaces (namely the Hilbert space) where square-integrable solutions exist.
You should note that the integral above covers all of space but there is no mention of time. So, how can we be sure that solutions remain square integrable through all of time? (Note that the implicit assumption here is that we are operating in a closed system, in an open system there are no guarantees that something remains square integrable for all time!)
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More to come!