Dirac Notation#

alias: Bra-ket notation

The dirac notation is widely used in quantum mechanics similar how vectors are widely used in electromagnetism. We use this because quantum mechanics is natural a linear algebra problem.

A vector in Dirac notation is written as \(|\alpha\rangle\) which is called a bra vector defined to be a general vector in any basis. For example we say that \(|\alpha\rangle\) may be represented in a \(N\)-dimension space using the cartesian basis,

\[\begin{split} |\alpha\rangle \rightarrow \boldsymbol{a} \equiv \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_N \end{pmatrix}\end{split}\]

  • \(\boldsymbol{a}\) : a vector in the cartesian basis.

  • \(a_1, a_2, ... a_N\) : the weights of their corresponding basis.

  • Recall in cartesian basis each row correspond to the basis vectors \((\hat x, \hat y, \hat z, ...)\)

Conjugate Vector - Ket#

The ket vector is the conjugate of the bra vector such that,

\[\langle \alpha | \;\equiv\; |\alpha \rangle ^*\]

Inner Product#

alias: dot product, braket

The inner product of two vectors is the product of a bra and ket vector.

\[ \boxed{\langle \alpha | \beta \rangle \equiv a_1^*b_1 + a_2^*b_2 + ... a_N^*b_N}\]

Properties#

  • \(\langle \alpha | \beta \rangle^* = \langle \beta | \alpha \rangle\)

Hilbert Space#

Many times it is not very useful to think of inner product as of the definition above because it is not necessary that \(\alpha\) and \(\beta\) is a easy vector to comprehend. In this case how do you represent a function like \(\Psi(x,t)\) as a vector?

The answer is you don’t; we take a look at the fact that for a square-integrable function which is defined as a function who’s conjugate square has a non-infinite area thus follows,

\[ \langle f | f \rangle = \int_a^b{|f(x)|^2 \;dx} < \infty \tag{square-integrable}\]

Then, the inner product of square-integrable functions is,

\[ \langle f | g \rangle \equiv \int_a^b{f(x)^*g(x)\;dx} \]

In quantum mechanics we deal with only wavefunctions that are normalizable. You may check that all normalizable functions are also square-integrable since the area is non-infinite. We may introduce such a space where the wavefunction and all square-integrable functions live. This is called the Hilbert space or the complete inner product space.

Properties#

  • Self inner product: The self inner product is defined as,

    \[ \langle f | f \rangle \equiv \int_a^b{|f(x)|^2 \;dx} \]

    The self inner product is always non-negative and real,

    \[ \langle f | f \rangle > 0, \qquad \langle f | f \rangle \in \mathbb{R} \]

    The self inner product is only zero for the zero function \(f(x)=0\),

  • Schwarz inequality:

    \[ \left\lvert{\int_a^b{f(x)^*g(x)\;dx} }\right\rvert \le \sqrt{\int_a^b{|f(x)|^2\;dx}\int_a^b{|g(x)|^2\;dx}}\]