Concept Deep Dive: Quantum Mechanics

Well, maybe Deep Dive isn't exactly a good way to put this. This one is going to be more of a narrative overview of how the Quantum Mechanic solutions to problem develop. The rigorous proofs of this are partially based in linear algebra and I will only sketch those out here, otherwise this piece will turn out too long.

We left off at Hamiltonian mechanics and how they technically transition into the Quantum physics. There are more parallels between classical, especially the Hamiltonian formulation, physics and Quantum Mechanics that are probably best remembered, but which I will try to rigorously derive here. Before we do that however, it should be mentioned why we even feel the need to introduce a complicated, and admittedly a little broken form of mechanics into the standard syllabus of physics.

When We Sweat the Small Stuff

Simply speaking, classical mechanics is kinda wrong. It's only not wrong, because of stochastic reasons, but if we remove the luxury of looking at object moving through space at a macro-scale, we will find some issues with the way we've looked at things. Case in point: The Double-Slit experiment. There are ways to have a medium emit a single photon (we'll get back to why) and let's assume there is a screen with two very thin slits very close to one another in the path of the photon. Covering one of the slits makes it pretty clear which of the slits the photon passes through. The photon, when projected on a screen moves in a way so that one could draw a straight line from the slit and its projection. On the other hand, if we leave the screen the way it is, suddenly we get seemingly random incidences on the projection. One can use a laser to get a striped pattern, which will remind the physicist of an interference pattern of waves. It then makes sense to at least partially model quantum objects (for a photon is a quantum object) as waves, rather than particles. They do however still show qualities of having momentum, which requires mass, i.e. particle characteristics. One can conclude that quantum objects are waves, unless they interact with matter. Then we think of them as particles. This is the wave-particle duality. A large part of quantum mechanics is concerned with being able to describe quantum objects in the first place. That means identifying the wave function.

Measure for Measure

First we want to know how to even do that in theory. Remember, measurement is always interaction with matter, so it will always destroy the wave-characteristics of the object. That raises the question: What even can we know about a quantum object? Naively, we can always make a measurement, but then we run the risk of destroying previous information. Well, it turns out there's a way to circumvent that problem - kinda. Some quantities can be known simultaneously, others can only be known with a certain uncertainty. How we do that is, using so called states and operators.

We can imagine a state as a vector containing all the information about an object that we could possibly want to know, and an Operator (or observable) a matrix that will pull the information about some specific quantity out of a state. The product

then is considered a measurement of the quantity A of the state alpha, with the result a. To those familiar with linear Algebra, this will be familiar as the Eigenvalue equation, and that's because that's what it is. a here is an Eigenvalue of A, so successful measurements can only ever yield Eigenvalues of their operators. There is one result though, that the Eigenvalue would find distressing, and that is zero. That would be exactly the case for a measurement failing, i.e. the operator finding no valid measurement. This is maybe best explained using filters for certain states.

Any system has a certain probability to return some eigenvalue on any measurement, until it has been measured. Then this eigenvalue is set. Assume now we filter all particles with the eigenvalue a and then attempt to filter again for some different eigenvalue b. Obviously, then we would have no particles left. This is what a "failed" measurement would look like. By the way of filters we might also attempt to construct something (exclusively) theoretical, namely the pure state, a state in which all quantities have been determined by filters. The notion of course assumes filtering without measurement, but if that were possible, stringing one filter for all measureable quantities behind one another would return such a state.

Some Linear Algebra

We would like our results to be defined well enough to be useful, but not retain as much exploitable attributes of both states and operators as possible. For this it's convenient to find a mathematical framework to conduct our computations in. In classical mechanics that is the operator field of the real numbers (don't worry about the terminology, or check out my algebra series), in quantum mechanics, that will be the Hilbert space. The reason for this is more or less empirical, or for the purpose of brevity "because it is". State are then Hilbert-vectors and each Hilbert-vector has an inverse with respect to the bi-linear form of the outer product in Hilbert space. We associate this inverse vector with the same state, but keep in mind its construction, i.e. that it's a row-vector and conjugated. We can informally introduce the bra(c)ket or bra-ket notation at this point, where the conjugated vector is referred to as a "bra-vector", and the normal state vector as a "ket-vector". A whole bra-ket of any vectors, then gives a complex number.

Since we have now set up a system with quantities as elements of a vector, we might as well open this up to a discussion of the usual linear algebra. These quantities could be picked out to assemble a basis for the space, the so-called CON-basis, which can be used to express the Operators as a matrix in Hilbert space. Here, as with matrices in general, it makes sense to take care to identify the kind of matrix one is dealing with. Unitary and Hermetian matrices are particularly interesting for Quantum Mechanics. If one is fine using an operator that's a little bit ugly though, one could always construct one from two vectors multiplied in a way that they evaluate to a matrix (ket-bra, so to speak). That would be the so-called dyadic product. As this form is usually a bit unruly, we won't work with it much, but the possibility should remain present in the reader's mind.

We've touched on the possibility of measurement with (significant) uncertainty before, and this is maybe a good time to state when this becomes relevant. The bra(c)ket has that c in the middle of it. If that is a scalar, that can be moved outside the product, but it were to be an operator, one would form the expectation value of that operator with those states. The expectation value basically already tells you that it's not going to be an exact measurement coming in, but it will turn out to have several interesting and powerful forms that will justify it becoming somewhat of a standard-Ansatz.

Physics tends to operate under a couple of assumptions that will become relevant once we want to understand the motion of states in configuration space: We know some quantities to be continuous and square-integrable. This is especially convenient, because the space of square integrable functions is a Hilbert space (I'll omit the proof). This means that there could be an alternative way of writing bra-ket products in the space of square-integrable functions. This will look like this

Some Probability Theory

Since we are working with probabilistic objects (we will later refer to the wave function as a probabilistic function, or probability wave), it will pay off to establish where we can exploit the rules of probability theory in this construct as well. Returning to the filters: Application of a filter to a state is basically removing a certain subsection of the ensemble from the system. It would then make sense that the sum of all filters should evaluate to one. Writing this in terms of states with respect to some operator, it must always be true that

Having a 1-element for convenient multiplication is always nice, and it's also interesting to note that it's a sum of dyadic products. Who would have thought. Especially however,

where w is a function for probability and P is a filter for its argument.

Uncertainty & Return of an Old Friend

Where there are probabilities and uncertainties, the mean square deviation is not far away. Specifically, the square of the mean square deviation of a quantity is important for determining its uncertainty. In the terms of bra-ket notation it reads

The product of two such square mean deviations can be estimated using the Schwarz-inequality, but due to non-trivial substitutions, I'll omit the proof here, suffice it to say that at some point we will get something that looks like this

The content of the bra-ket looks awfully familiar though... that's right, that's the commutator I've foreshadowed last time. For some coordinate and associated momentum then, we can pretend that this evaluates to greater than or equal to half Planck's constant. The imaginary unit is lost by taking the absolute of the expression. This specific relation is known as the Heisenberg-Uncertainty Relation.

A Few Interesting Operators

We of course already know coordinate and momentum operators, which I like to also group in with the Hamiltonian operator, which evaluates the Energy of the system. This is because the Hamiltonian operator will usually consist of terms of coordinate and momentum operators, with the occasional constant. This makes them intuitive to keep together.

Operators that can also be intuitively understood are for example the time evolution operator U and the displacement operator T, each displaces the system along one of these axes.

From this definition of U one could formulate an infinitesimal time-translation, the second term of which is equal to the Hamiltonian with some factor. This makes the Hamiltonian the generator for the time evolution operator. In this way, the time-evolution operator can also be written as a matrix exponential.

This however implies that the observables are time-dependent, which isn't intuitive at the first instance, I think. Usually we would think of the state to be time-dependent, right? We think of an object's coordinate and momentum for example to have a time-dependence, and those would be chiefly attributes of the object. However, since this picture is never complete in quantum physics without a state and an operator, one could construct a formulation in which the operators are time-dependent, but not the state, the other way around, or both have a time-dependence. At least one of them needs a time dependence though, so there's not really a trivial case for this supposition.

The picture in which only the state is time-dependent, is called the Schroedinger Picture, the one in which only the operator is time-dependent is the Heisenberg Picture, and the one in which both are time-dependent is the Dirac Picture.

These formulations are equivalent and bridged by the time-evolution operator. To remove the time-dependence from either state or operator, we fix their time-coordinate to some point in time t. Of course there's logic to choosing which is convenient. The Total time-derivative of an operator in Schroedinger-picture is always zero, for example. That can turn out a little odd for computation.

The Dirac picture however needs a little more effort put into formulation. The Hamiltonian in the Dirac picture is decomposed into a time-independent part of the operator and a time-dependent perturbation term. Through Hamilton-Jacobi, we know that this is always possible. We tend to use the time-independent Hamiltonian for the time-evolution operator of choice, the one of the free system. Technically, the full time evolution operator in the Dirac picture looks as follows.

Where the Operator with the S in the index is that of the Schroedinger picture. This will be the construction of all Dirac-picture operators.