## Lecture: Compare & Contrast Kets & Wavefunctions

In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.
• This ingredient is used in the following sequences
What students learn
• To translate entities and calculations from Dirac bra-ket notation to wavefunction notation.

## Quantum States

The wavefunction is related to writing the state in the position basis (or the position “representation”) by inserting a completeness relation for the position basis.:

\begin{align*} \left|{\psi}\right\rangle &= (1) \left|{\psi}\right\rangle \\ &=\left(\int_{\tiny\mbox{all space}} \left|{x}\right\rangle \left\langle {x}\right|dx\right)\left|{\psi}\right\rangle \\ &= \int_{\tiny\mbox{all space}} \left|{x}\right\rangle \left\langle {x}\middle|{\psi}\right\rangle dx \\ &= \int_{\tiny\mbox{all space}} \underbrace{\left\langle {x}\middle|{\psi}\right\rangle }_{\mbox{wavefunction}}\left|{x}\right\rangle dx \\ &= \int_{\tiny\mbox{all space}} \underbrace{\psi(x)}_{\mbox{wavefunction}} \left|{x}\right\rangle dx \\ \end{align*}

Bra-ket Notation: $\left|{\psi}\right\rangle \doteq \left\langle {x}\middle|{\psi}\right\rangle$

Wavefunction Notation: $\left|{\psi}\right\rangle \doteq \psi(x)$

The dot equals sign indicates that the right hand side is an expression of the state in a specific but unwritten basis.

### The Hermitian Adjoint of a Quantum State:

\begin{align*} \left\langle {\psi}\right| &= (\left|{\psi}\right\rangle )^*\\ &\doteq (\left\langle {x}\middle|{\psi}\right\rangle )^*\\ &\doteq \left\langle {\psi}\middle|{x}\right\rangle \\ \left\langle {\psi}\right| &\doteq \psi^*(x) \end{align*}

Bra-ket Notation: $\left\langle {\psi}\right| \doteq \left\langle {\psi}\middle|{x}\right\rangle$

Wavefunction Notation: $\left\langle {\psi}\right| \doteq \psi^*(x)$

## Calculating a Probability Amplitude

For example, for an energy measurement: Bra-ket Notation: \begin{align*} c_{n} = \left\langle {E_n}\middle|{\psi}\right\rangle \end{align*}

Wavefunction Notation:

Go to wavefunction notation by inserting a completeness relation for the position basis.

\begin{align*} c_{n} &= \left\langle {E_n}\right|1\left|{\psi}\right\rangle \\ &= \left\langle {E_n}\right| \left( \int_{\tiny\mbox{all space}} \left|{x}\right\rangle \left\langle {x}\right| dx \right)\left|{\psi}\right\rangle \\ &= \int_{\tiny\mbox{all space}} \left\langle {E_n}\middle|{x}\right\rangle \left\langle {x}\middle|{\psi}\right\rangle dx\\ &= \int_{\tiny\mbox{all space}} E_n^*(x) \, \psi(x)dx\\ \end{align*}

## Calculating a Probability

Bra-ket Notation: \begin{align*} \mathcal{P}(E_n) &= | c_n |^2 \\ &= |\left\langle {E_n}\middle|{\psi}\right\rangle |^2 \\ \end{align*}

Wavefunction Notation:

Using the result for the probability amplitude in wavefunction notation:

\begin{align*} \mathcal{P}(E_n) &= | c_n |^2 \\ &= \left| \int_{\tiny\mbox{all space}} E_n^*(x) \, \psi(x) dx \right|^2\\ \end{align*} Notice the norm square is outside of the integral!

## Normalizing a State

Bra-ket Notation: \begin{align*} 1 &= \left\langle {\psi}\middle|{\psi}\right\rangle \\ \end{align*}

Wavefunction Notation:

Go to wavefunction notation by inserting a completeness relation for the position basis.

\begin{align*} 1 &= \left\langle {\psi}\right|1\left|{\psi}\right\rangle \\ &= \left\langle {\psi}\right|\left( \int_{\tiny\mbox{all space}} \left|{x}\right\rangle \left\langle {x}\right| dx \right)\left|{\psi}\right\rangle \\ &= \int_{\tiny\mbox{all space}} \left\langle {\psi}\middle|{x}\right\rangle \left\langle {x}\middle|{\psi}\right\rangle dx \\ &= \int_{\tiny\mbox{all space}} \psi^*(x)\, \psi(x) dx \\ &= \int_{\tiny\mbox{all space}} |\psi(x)|^2 dx \\ \end{align*} Notice norm squared inside the integral!

## Finding the Probability for a Range of Values

Bra-ket Notation: \begin{align*} \mathcal{P}(a \le n \le b) &= \sum_{n=a}^b| c_n |^2 \\ \end{align*}

Wavefunction Notation:

Reinterpret $|\psi(x)|^2$ as a probability density: \begin{align*} \mathcal{P}(a \le x \le b) &= \int_{a}^b |\psi(x)|^2 dx \\ \end{align*} Notice norm squared inside the integral!

Keywords
Bra-Ket Notations Wavefunction Notation Completeness Relations Probability Probability Density
Learning Outcomes