1.15: Expectation values of k and \(\omega\)

The expectation value of k is obtained by integrating the wavefunction over all k. This must be performed in k-space

\[ \langle k\rangle = \frac{\frac{1}{2\pi}\int^{+\infty}_{-\infty}A(k)^{*}kA(k)dk}{\frac{1}{2\pi}\int^{+\infty}_{-\infty}A(k)^{*}A(k)dk} = \frac{\langle A|k|A\rangle}{\langle A|A\rangle} \nonumber \]

From the Inverse Fourier transform in k-space

\[ \psi(x)=\frac{1}{2\pi}\int^{+\infty}_{-\infty}A(k)e^{ikx}dx \nonumber \]

note that

\[ -i\frac{d}{dx}\psi(x)=\frac{1}{2\pi}\int^{+\infty}_{-\infty}kA(k)e^{ikx}dx \nonumber \]

Thus, we have the following Fourier transform pair:

\[ -i \frac{d}{dx}\psi(x) \Leftrightarrow kA(k) \nonumber \]

It follows that\(^{†}\)

\[ \langle k\rangle = \frac{\langle A\rangle|k|\langle A\rangle}{\langle A\rangle|\langle A\rangle} = \frac{\langle \psi | -i\frac{d}{dx}|\psi\rangle}{\langle \psi|\psi\rangle} \nonumber \]

Similarly, from the Inverse Fourier transform in the frequency domain

\[ \psi(t)=\frac{1}{2\pi}\int^{+\infty}_{-\infty} A(\omega)e^{-i\omega t} d\omega \nonumber \]

We can derive the Fourier transform pair:

\[ i \frac{d}{dt} \psi(t) \Leftrightarrow \omega A(\omega) \nonumber \]

It follows that

\[ \langle \omega \rangle = \frac{\langle A\rangle|\omega |\langle A\rangle}{\langle A\rangle|\langle A\rangle} = \frac{\langle \psi | i\frac{d}{dt}|\psi\rangle}{\langle \psi|\psi\rangle} \nonumber \]

We define two operators

\[ \hat{k} = -i \frac{d}{dx} \nonumber \]

and

\[ \hat{\omega} = i \frac{d}{dt} \nonumber \]

Operators only act on functions to the right. To signify this difference we mark them with a caret.

We could also define the (somewhat trivial) position operator

\[ \hat{x} = x \nonumber \]