Quantum mechanics (QM) is an interesting and strange topic. It is interesting because it is widely applicable in so many different areas, for example the semiconductor which dominates our lives. It is strange because no consensus has been achieved in its meaning. In here I'm not going to talk about the physical meaning of quantum mechanics, only illustrate how we can use it, the mathematical framework of QM.

Postulates of Quantum Mechanics

• Wave function

Wave function is the central quantity in quantum mechanics. It is usually denoted as \(\Psi(\vec{r})\). The physical meaning of the square of the norm of wave function was proposed by Born: "\(|\Psi(\vec{r})|^2\) represents the probability density of finding the particle in the area of \(\vec{r}\rightarrow\vec{r}+d\vec{r}\) ".

• Schrödinger equation

The famous Schrödinger equation illustrate the evolution of wave function with respect to time (\(t\)):

\begin{equation}i\hbar\frac{\partial \Psi}{\partial t}=\hat{H}\Psi\end{equation}

where \(\hat{H}\) is the Hamiltonian of the system, usually it can be written as:

\begin{equation}\hat{H} = \hat{T} + \hat{V}\end{equation}

where \(\hat{T}\) and \(\hat{V}\) represents the kinetic energy and potential energy, respectively.

The Schrödinger equation we have listed is called "time-dependent Schrödinger equation", since we have term explicitly involved time (\(t\)). There is an easier form, called "time-independent Schrödinger equation", which is commonly used in electronic structure theory, can be written as:

\begin{equation}\hat{H}\Psi=E\Psi\end{equation}

where \(E\) represents the total energy of the system. Solving this equation is usually called "Eigenvalue problem", as we will show later that the Hamiltonian (\(\hat{H}\)) is a matrix and wave function (\(\Psi\)) is a vector in Hilbert space (a complex high(infinite)-dimensional linear space). So this problem is similar to the eigenvalue problem we have encountered in linear algebra:

Assume \(A\) is a matrix, find all pairs of scalar \(\lambda_i\) and corresponding vector \(\vec{v}_i\) that satisfies:

\begin{equation}A\vec{v}_i = \lambda_i \vec{v}_i\end{equation}

• Matrix mechanics (Heisenberg's version of QM)

In this theory, any wave functions are considered as a vector in the Hilbert space. Any physical quantities (e.g. energy, angular momentum, momentum) are considered as a matrix (or often called operator).

Given a physical quantity (e.g. energy), we can solve the eigenvalue problem of its corresponding matrix. After that we get eigenvalues and eigenvectors. These eigenvectors could be orthogonalized and then normalized, so we can get an orthonormal basis vector set. This set is usually called energy representation.

We can have different kinds of representations, since they come from different physical quantities. But given a representation, all wave functions can be written as vectors (\(\Psi = \sum_{i}c_i\phi_i\)), where \(\phi_i\) are the basis vectors of a certain representation. We will discuss its physical meaning in the next section.

• Linear combination of basis function

Assume we have an orthonormal representation \(\{\phi_i\}\) and corresponding eigenvalues \(\{\epsilon_i\}\). Any wave function (\(\Psi\)) could be expanded as the linear combination of the basis vector set (this is because the representation set is not only orthonormal, it is complete, which means all vector can be represented as linear combination of basis vectors):

\begin{equation}\Psi = \sum_{i}c_i\phi_i\end{equation}

If we want to measure the physical quantity (e.g. energy in this case), then we will get random numbers which in the set of \(\{\epsilon_i\}\), and if we do the measurement large enough times, then the probability distribution will be proportional to \(|c_i|^2\). This is the physical meaning of linear combination.

The mean value of the physical quantity (\(\hat{A}\)) in state (\(\Psi\)) can be written as:

\begin{equation}\bar{A} = \frac{\int\Psi^*\hat{A}\Psi d\vec{r}}{\int\Psi^*\Psi d\vec{r}}\end{equation}

---

Dirac notation

The famous "Dirac notation" is proposed by Paul A. M. Dirac in a paper published in 1936 titled "A NEW NOTATION FOR QUANTUM MECHANICS". He proposed two symbols, namely "bra" (\(\langle|\)) and "ket" (\(|\rangle\)).

Simply speaking, "ket" (\(|\rangle\)) represent a vector in the Hilbert space:

\begin{equation}|\Psi\rangle=\begin{pmatrix}a_1\\a_2\\\vdots\\a_n\end{pmatrix}\end{equation}

Where as it's counter part "bra" (\(\langle \Psi|\)) represent a horizontal vector:

\begin{equation}\langle\Psi|=\begin{pmatrix}a_1, a_2, \cdots, a_n\end{pmatrix}\end{equation}

where \(\{a_i\}\) are the coordinates in the Hilbert space.

So everytime we see expressions envolving bra and ket, we can convert them into vectors with different shape. We can treat the physical quantities (operators) into matrices, which means the whole expression is just a product of matrices and vectors, which will be really simple to understand.

The following expression will be very easy to understand:

• \(\langle \phi|\psi \rangle\)

\begin{equation}
\begin{aligned}
&\langle \phi|\psi \rangle\ \\
&= \begin{pmatrix}\phi_1, \phi_2, \cdots, \phi_n\end{pmatrix}\times
\begin{pmatrix}\psi_1\\\psi_2\\\vdots\\\psi_n\end{pmatrix}\\
&=\sum_{i=1}^n\phi_i\psi_i
\end{aligned}
\end{equation}

which means \(\langle \phi|\psi \rangle\) is the inner product of two wave functions

• \(|\psi\rangle\langle\phi|\)

Using the same reasoning, we can find that this expression represents a matrix:

\begin{equation}
\begin{aligned}
&|\psi\rangle\langle\phi|\ \\
&= \begin{pmatrix}\psi_1\\\psi_2\\\vdots\\\psi_n\end{pmatrix}\times
\begin{pmatrix}\phi_1, \phi_2, \cdots, \phi_n\end{pmatrix}\\
&=\begin{pmatrix}
\psi_1\phi_1 & \psi_1\phi_2 & \cdots & \psi_1\phi_n\\
\psi_2\phi_1 & \psi_2\phi_2 & \cdots & \psi_2\phi_n\\
\vdots & \vdots & \ddots & \vdots\\
\psi_n\phi_1 & \psi_n\phi_2 & \cdots & \psi_n\phi_n\\
\end{pmatrix}
\end{aligned}
\end{equation}

So next time if you are not sure how to interpret the Dirac notation, just write them as vectors and you will find what the final result looks like.