Introduction: When Eigenvalues Are Continuous
So far, we've worked with discrete bases: \(\sum_n |n\rangle\langle n| = I\). But position \(x\) can take any real value—it's continuous. For continuous spectra, sums become integrals and the formalism extends beautifully.
The Position Eigenstates
The position operator \(\hat{x}\) has eigenstates \(|x\rangle\) for every real \(x\):
\[\hat{x}|x\rangle = x|x\rangle\]The completeness relation becomes an integral:
\[\int_{-\infty}^{\infty} |x\rangle\langle x| \, dx = \hat{I}\]Expanding States in Continuous Bases
Any state can be expanded:
\[|\psi\rangle = \int_{-\infty}^{\infty} |x\rangle\langle x|\psi\rangle \, dx = \int_{-\infty}^{\infty} \psi(x)|x\rangle \, dx\]where \(\psi(x) = \langle x|\psi\rangle\) is the wavefunction!
Worked Examples
Example 1: The Wavefunction as Inner Product
The wavefunction at position \(x_0\) is:
\[\psi(x_0) = \langle x_0|\psi\rangle\]This is the probability amplitude to find the particle at \(x_0\).
Example 2: Normalization
The total probability is:
\[\langle\psi|\psi\rangle = \int_{-\infty}^{\infty} |\psi(x)|^2 \, dx = 1\]Using completeness: \(\langle\psi|\psi\rangle = \langle\psi|\hat{I}|\psi\rangle = \int \langle\psi|x\rangle\langle x|\psi\rangle \, dx = \int |\psi(x)|^2 \, dx\).
Example 3: Momentum Basis
Similarly, momentum eigenstates satisfy:
\[\hat{p}|p\rangle = p|p\rangle\] \[\int_{-\infty}^{\infty} |p\rangle\langle p| \, dp = \hat{I}\] \[\tilde{\psi}(p) = \langle p|\psi\rangle\]The momentum-space wavefunction \(\tilde{\psi}(p)\) is the Fourier transform of \(\psi(x)\).
The Quantum Connection
The transition from sums to integrals is essential for describing particles in real space. The wavefunction \(\psi(x)\) and its Fourier transform \(\tilde{\psi}(p)\) are just different coordinate representations of the same abstract state \(|\psi\rangle\). Position and momentum representations are related by:
\[\langle x|p\rangle = \frac{1}{\sqrt{2\pi\hbar}} e^{ipx/\hbar}\]