Formulation: Bras, Kets, and Operators

Adapted from J.J Sakurai, Modern Quantum Mechanics

We will introduce some basic braket notation in this note and present an interpretation of a wavefunction from a braket perspective.

Kets

Let’s begin with a Vector space, which is a mathematical concept of a set of elements that satisfies certain mathematical rules. In QM, we consider a complex vector space whose dimensions are given by the nature of the physical system of interest. A physical state is represented by a state vector in a complex vector space. We call such a vector a “ket,” denoted by: $\ket{\alpha}$.
Kets are special because we postulate them to contain complete information about the physical state.

Some properties of kets are: $$ \ket{\alpha} + \ket{\beta} = \ket{\gamma}$$ $$c\ket{\alpha} = \ket{\alpha}c $$ where c is a scalar constant. The first line tells us that the addition of two kets result in a ket, and the second line tells us that the order of scalar multiplication does not matter. When $c=0$, we get an empty ket.
Another postulate is that when we multiply a ket by a scalar, the physical state itself is not changed. Think of a cartesian vector pointing at some direction, and multiplying by a scalar simply changes the length of the vector, but not the direction; i.e. the physical state of a ket can be thought of as the direction of a cartesian vector.

Bras

For each ket, there is a corresponding bra (mathematically, they are dual space of each other). A bra is denoted as $\bra{\alpha}$. Roughly speaking, the bra space (where bras exist) is a mirror image of the ket space (where kets exist). $$ \ket{\alpha} \leftrightarrow \bra{\alpha}$$ $$\ket{\alpha} + \ket{\beta} = \bra{\alpha} + \bra{\beta} $$ The bra for $c\ket{\alpha}$, however, is $$c\ket{\alpha} \leftrightarrow \bra{\alpha}c^{*}$$ where $c^{*}$ is the complex conjugate of $c$.
Furthermore, we define the inner product of a bra and a ket as: $$\braket{\beta|\alpha} = (\bra{\beta})\cdot (\ket{\alpha})$$ and the resulting product, in general, is a complex number.
Similar to typical inner products you might have seen: $$\braket{\beta|\alpha} = \braket{\alpha|\beta}^{*}$$ and $$\braket{\alpha|\alpha} \geq 0$$ Two kets $\ket{\alpha}$ and $\bra{\beta}$ are said to be orthogonal if $$\braket{\alpha|\beta} = 0$$

Operators

To complete the discussion of brakets, we will introduce operators. They mathematically act on a ket to give another ket. In QM, an operator is associated with a physical observable (e.g. position, momentum, angular momentum …). Operators can be added as follows $$X + Y = Y + X \\
X + (Y+Z) = (X+Y) + Z $$ And they are also linear, i.e. $$X(c\ket{\alpha} + d\ket{\beta}) = cX\ket{\alpha} + dX\ket{\beta}$$ They act on bras from the right hand side: $$\bra{\alpha} X$$ Now, suppose we have a ket $X\ket{\alpha}$, its corresponding bra is $$X\ket{\alpha} \leftrightarrow \bra{\alpha}X^\dagger$$ where $X^\dagger$ is called the hermitian conjugate of $X$. Particularly, when $X = X^\dagger$, $X$ is said to be a hermitian operator.
By now, you might start to see some resemblence with the regular linear algebra you learned in 54. $\ket{\alpha}$ ~ column vector, $\bra{\alpha}$ ~ row vector, $X$ ~ a matrix, $X^\dagger$ ~ transpose of a matrix.

Connection with wavefunction

Another important postulate of QM is that for a hermitian operator $A$, its eigenkets form a complete orthonormal set, and we can use them as a basis to form any arbitrary ket within the vector space they span. Think of this as diagonalizing a symmetric matrix, which is analogous to a hermitian operator, and the resulting eigenvectors are used as a basis for the dimension of the matrix. Suppose the eigenkets of $A$ are given by $${\ket{a’}, \ket{a’'}, \ket{a’''}…}$$ Then for any arbitrary ket $\ket{\alpha}$, which not need to be an eigenket, can be written as some linear combination of the eigenkets: $$\ket{\alpha} = \sum_{a’} c_{a’} \ket{a’} \label{eq1}\tag{1}$$ Now, recall that eigenvectors are orthogonal for a symmetric matrix, then similarly, the $\ket{a’}$'s are orthogonal, so we multiply $\bra{a’}$ on the left for both sides, and we get: $$c_{a’} = \braket{a'|\alpha}$$ Plugging this back into $\ref{eq1}$, we get $$\begin{gather*} \ket{\alpha} = \sum_{a’} \braket{a'|\alpha}\ket{a’} \\
= \sum_{a’}\ket{a’} \braket{a'|\alpha} \end{gather*} $$ And by equality, we see that $$\sum_{a’} \ket{a’}\bra{a’} = 1$$ This is called the resolution of identity in QM, where the 1 implies the identity operator. Think of this as taking each normalized eigenvector of a symmetric matrix, taking the outer product ($\mathbf{aa^T}$) with itself, and adding the resulting matrices, which will yield you the identity matrix.
We just need one more mathematical concept to finish our wavefunction interpretation from the braket perspective. When the eigenvalues of an operator is continuous, the vector space the eigenvectors span is infinite dimension. Hence, we make such adjustment by writing sums as integrals. $$\sum_{a’} \ket{a’}\bra{a’} = 1 \rightarrow \int d\xi’ \ket{\xi’}\bra{\xi’} = 1 \\
\ket{\alpha} = \sum_{a’} \ket{a’}\braket{a'|\alpha} \rightarrow \ket{\alpha} = \int d\xi’ \ket{\xi’}\braket{\xi'|\alpha}$$ Now consider the position operator $X$, which has eigenkets $\ket{x’}$. The eigenvalues for $X$ are continous, because the position of a physical state could be anywhere on the number line (in 1D, for simplicity). Because the dimension is infinite, we will write an arbitrary ket as: $$\ket{\alpha} = \int dx’\ket{x’}\braket{x'|\alpha}$$ If we make a connection with the finite case, $\ket{\alpha} = \sum_{a’} c_{a’} \ket{a’}$, we see that $\braket{x'|a}$ is essentially the expansion coefficients $c_a'$ of an arbitrary ket in terms of a basis. In fact, when the basis is specifically the position operator’s eigenkets, we define $$\braket{x'|a} = \psi_\alpha (x’)$$
as a wavefunction $\psi_\alpha$ for state $\ket{\alpha}$ in the position representation $x'$. As you will see later, another postulate of QM says that $|c_{a’}|^2$ is an explicit quantity that tells the probability of finding state \ket{\alpha} in state \ket{a’}, and thus, $$|\psi_\alpha(x’)|^2$$ is the probability of the state $\alpha$ at position $x'$.