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.

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!