Fundamentals¶

Why Quantum Chemistry¶

This is a good question, which doesn’t get enough attention. For most of us, we have to learn quantum mechanics because it’s in the course curriculum, or because “classical mechanics fails at the microscopic level” or some BS like that. Chapter 1 of the Feynman Lectures, Vol.3 explains this amazingly, and if you’re still not satisfied and want a more hardcore explanation, check out the afterword of Griffiths’ Introduction to Quantum Mechanics.

In short, Quantum mechanics primarily arises to explain the behaviour of things that cannot be measured without disturbing their state. Like the electron or photon in a double slit experiment, trying to measure their position would cause a transfer of energy, however infinitesimal. This would disturb them from the state that they were in, which would defeat the purpose of the measurement.

How to go Quantum¶

Explaining the behaviour of things that cannot be measured sounds easier than it looks. If we have a box and a ball, and we need to describe the position of the ball with reference to the box, for simplicity’s sake we have two options:

The ball is in the box
The ball is outside the box i.e. it is not in the box

Since this is quantum chemistry, we cannot measure the location of the ball. We then use probabilities to describe the location of the ball: if \(p_1^2\) is the probability of the ball being in the box and \(p_2^2\) is the probability of the ball being outside the box, then the position of the ball is defined by something we call a wavefunction. The wavefunction in this case would be

\[\psi = p_1\ket{1} + p_2\ket{2}\]

Here we have described the simplest quantum system: A 2 state quantum system. Don’t be surprised by the \(\ket{1}\) and \(\ket{2}\) notations: they merely represent the different states the system can take, and we’ll go into more detail on them in a second. If they make you uncomfortable (they made me :P), you can think of them as the unit vectors \(\hat i\) and \(\hat j\). The wavefunction would then define a vector which was inclined at \(\arctan \frac{p_2}{p_1}\) to the x-axis.

Deriving Probability from the wavefunction¶

You must have noticed that rather than putting the probability directly in the wavefunction, we took the square root of probability. This is called the Probability Amplitude, and is by definition. According to the born interpretation of the wavefunction \(|\psi|^2\) gives the probability of an event occuring. In this case, the probability of state 1 is \(p_1^2\) and that of state 2 is \(p_2^2\). A few things to note:

The probability amplitude can also be complex. In this case, the probability is the absolute value of the wavefunction, given by \(\psi^*\psi\), \(\psi^*\) is the complex conjugate of \(\psi\).
The total probability of the wavefunction should be 1. In this case, that implies \(p_1^2 + p_2^2 = 1\). In general, for a given wavefunction psi, it’s square integral \(\int \psi^* \psi d\tau\) should be 1 over it’s entire domain.

Observables, Operators and their Notations¶

In quantum mechanics, an Observable is a dynamic variable of a system that can be experimentally measured, for example position, momentum and kinetic energy. An observable quantity is generally enclosed in a ‘ket’, which is the bracket-like thing we used to denote states of the ball earlier. However, I won’t be using kets and further formalism in these notes, except maybe in the Mathematics part.

Every observable quantity has an operator associated with it. Thus, the position of a particle \(\ket{x}\) has an operator \(\hat{x}\) associated with it. The operator defines an operation to be performed on an observable. For example, the operator \(\hat x\) is defined as

\[\hat x = x\]

which is a simple multiplication operation. The momentum operator, however, is defined as follows:

\[\hat p = -i \hbar \frac{\partial}{\partial x}\]

Note that in multiple dimensions, \(\frac{\partial}{\partial x}\) becomes \(\nabla\).

One point of confusion is the notation \(\hat p^2\): this does not mean that we are squaring the operator, but that we are applying the operator twice in succession. If we take an example,

\[\begin{split}\begin{align} \hat p^2 U &= \left(-i\hbar \frac{\partial }{\partial x}\right) \left( -i\hbar \frac{\partial U}{\partial x}\right) \\ \hat p^2 U &= -\hbar^2 \frac{\partial^2 U}{\partial x^2} \end{align}\end{split}\]

An interesting thing to note is that using these two fundamental operators (momentum and position), we can define all of the operators related to classical mechanics as a function of \(\hat p\) and \(\hat x\). As an example, kinetic energy is

\[\hat K = \frac{\hat p^2}{2m} = \frac{-\hbar^2}{2m}\nabla^2\]

Operators have a few important properties, which are discussed in the next section

Properties of Operators¶

Eigenvalues and Eigenfunctions¶

If for an operator \(\hat X\), the following equation holds:

\[\hat X f = K f\]

then \(f\) is called an eigenfunction of \(\hat X\) and \(K\) is called an eigenvalue of \(\hat X\). The terminology derives from linear algebra (eigenvectors and eigenvalues), and we’ll explore the mathematics in the next section.

Hermiticity¶

All the quantum mechanical operators corresponding to observables are called hermitian: a hermitian operator \(\hat X\) is one for which the following is true:

\[\int \psi_j^*\ \hat X\ \psi_i\ d\tau = \left\{ \int \psi_i^*\ \hat X\ \psi_j\ d\tau \right\}^*\]

This will be covered in some more detail in the math section, but for now, there are two important properties of hermitian operators:

The eigenvalues of hermitian operators are real
The eigenfunctions of hermitian operators are orthogonal

The second point will be explained in greater detail in the Mathematics section.